Can YOU out-logic everypony in this sequel to the Pony Puzzle Pile?
Come back to cradle your cerebral cores in a new series of puzzles throughout Ponyville, Canterlot, and beyond Equestria's borders!
As Canterlot Castle's new Royal Dispute Settler, there will be many odd problems to solve with Twilight Sparkle and her friends, some of which may or may not involve the Magic of Friendship, but all are fiendish in their solutions.
NOTE: I invite readers to try solving these puzzles for themselves! Try not to spoil too much in the comments, though.
Answers are uploaded as soon as a reader gets it right. Or I get bored waiting.
You really aren't sure how it happened. It all started with an ordinary citizen of Ponyville such as yourself. Then one day, through an extremely bizzare series of coincidences that didn't go unnoticed by royalty, the citizens and a cosmic force of magic contained within a map, you found yourself being called in to help rid an entire country of a national epidemic.
Yeah, it doesn't really make any more sense on paper either. But we'll try to contextualise it for you. See, the six most important ponies in Ponyville (Princess Twilight Sparkle and her entourage) get called in to wherever they need to be across the globe to resolve some sort of crisis, usually guided by a big glowing table in the throne room of her castle, and...okay, that's weird, even for us. Nonetheless, thanks to a little bit of international cooperation and your brilliant powers of deduction, the country is now rapidly recovering, and for your commendable efforts, you have been officially promoted to be Equestria's official Royal Dispute Settler, endorsed by Princess Celestia herself!
It's a pretty sweet gig, all things considered. You get to dress up in a fancy outfit, as well as get your very own badge. Sadly, it does mean that you ended up having to move house to Canterlot (although you have been promised express delivery of pecan muffins from Ponyville's Sugarcube corner, you addict), and it can get a bit tiresome listening to ungrateful ponies gripe about problems that could have been solved with a bit of communication, but at least it promises to not be dull.
What the next six puzzles will do is test whether your powers of deduction have remained just as polished as they were back in Ponyville, or whether the life of decedent luxury among Canterlot's aristocracy will leave you downed like a chocolate gateau with an unattended Princess Celestia.
Settling yourself into the new role of Royal Dispute Settler, your first item on the agenda concerns a young filly who happened to catch your advert in the paper and decided to seek your advice. You ask why she never asked her parents, but she explains that they're part of the problem.
Her father's birthday is coming up, apparently, and as he is an avid pool champion (six times winner of the Cue Shot Equestria Championships, Senior Division), she decided to send a commemorative solid gold pool cue: she had previously thought about a solid gold surfboard for the summer, but decided against it on the grounds gold doesn't float).
Anyway, her problem is this: the cue is a good solid two metres long exactly, but due to the Canterlot Post-Office's rather unreasonable health and safety policy, no package that exceeds one and a half metres in length, width or depth can be put through the regular parcel delivery service. What's more, as it's a present, she really doesn't feel like cutting the cue into smaller pieces, bending it, or otherwise damaging this frankly ludicrously expensive gift just to slip it past regulations.
Amazing what rich fillies do with their pocket money, am I right?
Anyway, your first problem (or puzzle) is to figure out how she can mail this gold ball-tapping stick through the mail without the post-pony getting on either of your cases. How?
Obviously putting the cue straight as it is won't work. The trick is to exploit the lack of ruling regarding DIAGONAL measurements.
What you should do is pack the cube diagonally. It will certainly be as long as 2 metres diagonally, but the length and width of its wrapping will be just under one and a half metres.
Theoretically speaking, a noble's wealth exists to contribute to society, practicing charity through making the citizen's world a better place. I say theoretically, because if you ever wanted to see a more selfish and pompous noble in all of Canterlot, you don't need to look much further than Princess celestia's niece, Prince Blueblood.
The concept of sharing seems to be the absolute last thing on his mind: last year, he spent a large proportion of Equestria's taxes on securing a solid gold bathtub for himself, and most recently he has earned the nickname amongst Canterlot's common citizens as the Prince of Anti-Fun, as his crusade on nightclubs and other "more seedy nonsense" needed to be done away with. He has tried several petitions to his aunt, with no success. He is also incredibly rude, condescending, has no sense of chivalry, and can basically be described in scientific terms as "an absolute jerk-face".
No, there's little that is tolerable about him, and as soon as you meet the stallion personally, you suddenly appreciate why Rarity, an old friend of yours from Ponyville, would have rather overly violent fantasies about the guy. Nonetheless, the Royal Dispute Settler's Oath prevents you from turning away anypony in need of assistance, and so you agree to give him a chance...while keeping a mental note to yourself to double your standard fee for this jerk afterwards.
Much to your relief, this ISN'T about the whole nightclub thing. As it turns out, there is to be a special charity auction to be held at Canterlot Castle tomorrow, and according to his intel, there is to be a special exclusive portrait of Star-swirl the Bearded, after a recent foray into Tutti-Fruity Sugar Punch, listed as one of the items for sale. It is a very obscure painting, largely due to how it depicts the ancient wizard during his less flattering moments (mooning a rather embarrassed Princess Luna), and Blueblood estimates it could be worth quite a pretty bit...and it would look positively fabulous in his private chambers, because of course you'd never display a portrait of Star-swirl showing his flank at Princess Luna PUBLICLY. He informs you that about there will be twenty-nine other nobles attending the auction, and including himself, that makes thirty ponies total vying for Star-swirl's buttocks in a frame.
When you question him about how much he values the painting, he replies that he would never give away how much out of fear of any eavesdroppers in the room, while giving the stink-eye to the royal Guards stationed right by the doors of your chambers, before finally admitting that he prices it anywhere between zero and one million bits. So far, nothing suggests that you need to do anything special, as per the golden rule: whoever has the gold makes the rules, and surely Prince Blueblood can easily afford it, except you soon learn that the auction is going to have a degree of randomness thrown in the mix.
The auction works by having everypony who wants to bid on the painting write down how much they're willing to spend, along with their name, and then sealing their bid away in an envelope, which will then be handed to the auctioneer. Afterwards, the envelopes are opened, and whoever bid the most money wins the painting, and must pay whatever was written on that paper.
However, as much as Prince Blueblood wants that painting, bidding any amount of cash would involve dipping into his funds necessary to exterminate all the nightclubs across Equestria, and so he asks that you suggest a way to minimise any losses he could incur while still offering decent odds of getting the painting. And so, he concludes, this is the task that has been given to you. How much should you tell Prince Blueblood to bid on the painting to minimise his losses? And don't go suggesting the highest amount just to drain his money pool just because he's a jerk-face: your oath is to help all ponies to the best of your ability, and frankly, getting sued by this guy would be a bit more trouble than it's worth. So how should he play this?
Prince Blueblood should bid 29/30s of whatever he values the painting at.
Imagine an auction that was down to just him and another bidder. First obvious fact is that he should never bid more than his valuation of the painting, otherwise if he loses, he's back where he started, and if he wins, he overpays for the painting. Nor should he pay EXACTLY how much he thinks it's worth, because no matter if he wins or loses, he breaks even profit-wise. Might as well have stayed home. So he has to bid below the valuation and hope to win.
Your expected payoff is the probability you win multiplied by the profit should you win. So in this scenario where it's down to two ponies, half of the painting's value is the happiest compromise between the two values.
Using similar logic, for this auction of thirty ponies including Blueblood, he should pay 29/30s; higher bids to compensate for the increased number of rivals, in this case thirty.
The role of Royal Dispute Settler is not just limited to giving advice to ponies: as you are now technically part of Canterlot's nobility now, on occasion you are called in to attend some sort of fancy do that really has nothing to do with whatever you're interested in. At least there's generally cake, which was worth sticking around for.
Anyway, while you were attending a birth-iversary of a mother and child (one of the nobles married his wife on her birthday, and she had his son on her birthday too, which resulted in the two of them sharing the same birthday. One wonders if he had this planned all along, which would be dedication if nothing else), three of the party guests happen to catch your eye: a grey stallion in his best green suit, who seems to be fascinated by an azure mare wearing an elegant white dress, who in turn seems enraptured by a brown stallion in a black suit and tie, at the nibbles table nursing a punch away from the others, the poor awkward guy.
Fancy Pants, another guest who got invited to the statistically improbable event, introduces the three to you: Upper Crust, Charity Case, and Quiet Time, in that order. There's not a whole lot to do at noble parties except gossip about all the OTHER nobles, and you soon learn about all the get-togethers and events Jet Set and his wife Upper Crust seem to attend (including some beef with Rarity), and how Quiet Time remains an eligible bachelor, for all too obvious reasons. Despite his well-connected self however, Fancy Pants confesses he is at a loss regarding Charity Case's own personal life: she seems to be more reclusive than even Quiet Time in regards to that.
At that moment, Fancy Pants's wife Fleur calls him over, leaving you to mull over what the blue stallion told you. Being the great puzzle enthusiast you are, you immediately start forming all sorts of conclusions in your head, basically inventing a puzzle for yourself. Three ponies staring at each other in a line: how could you resist?
Surprisingly, about 72% of people who get hit with this puzzle for the first time pick C: CANNOT BE DETERMINED. Because we don't know Charity Case's marital status, and it can't be determined, people assume that it infers that the solution can't be determined.
In fact, Charity Case's marital status doesn't matter. If she is married, then a married pony (Charity Case) is looking at an unmarried one (Quiet Time). If she isn't married, a married pony (Jet Set) is looking at an unmarried pony (Charity Case).
Hopefully those of you who got caught out on this will take better care not to make lazy assumptions.
The exact nature of your job as Royal Dispute Settler means you can't really make any appointments for anypony: the best you can do is tell them to turn up on a specific day and then wait in a queue. This is largely because puzzles such as THIS one come along and devour most of your morning, if not the entire day.
It's a very crisp morning in Canterlot, and your client for the moment is Secret Agent Sweetie Drops (or Bon Bon to her friends), a budding voice actress turned government worker (the exact details are far too embarrassing and/or sordid to mention within this fanfic, or in the presence of royalty), and in a rather embarrassing moment of failing to rise to the occasion, finds herself unable to crack a code. She fervently insists that it's because her maths totally stinks and that whoever sent the code stinks too, but you're not here to question the life decisions. Also, nopony else in the agency could figure it out either anyway, so there, nyeeeh.
After swearing an Oath of Confidentiality, Fidelity, Blood Feud and Office to Bon Bon that you will not leak whatever is revealed within these chambers, you examine the code. Bon Bon refuses to give the exact context as to what it all means, but she does mention that the code represents a numerical sum:
Rather unfortunately, several of the letters were smudged during transportation to headquarters, despite being packed in the most technologically advanced wrapping available this side of Ponyville - a brown envelope - and are now completely unreadable: in fact, what you're holding is a reconstruction of the equation, with the dashes representing the smudged parts. All she can tell you is that every digit, all the way from 0 to 9, are all used in the equation at least once.
Naturally, since it would be hopelessly embarrassing for you to not know the answer either, and because you're not about to enter a game of numerical-code chicken with this pony, you decide to have a bash at it. What does each letter mean, can you fill in the blanks, and can you spare Bon Bon and her secret agency the SHAME, the top secret, unaccountable SHAME?
The sum is comprised of six letters, as you can see if you squint: E, X, M, R, K and H. Since we know that the sum uses all ten digits, that means all of the smudged parts of the sum must represent a different digit each.
It's going to take a long and steady stream of logic, but it'll make sense in the end. For example, we can immediately deduce that the first dash, representing the ten millions slot, must be 1, because the sum of two digits with carryover can’t be greater than 19.
We also know that E must be even, because adding two of the same number will always produce an even number (specifically, two Ks). We also know that E + E must be pretty big, because it creates carryover on the left side of the equation. So E is either 6 or 8, which means that K must be 3 or 4.
Let's imagine the scenario where K = 4 and E = 8. this works for the ones digit of the answer, but when we hit the millions, we run into problems: 8 + 8 = 16, which has a 6 in the ones place (the K value in the sum), and we know K isn't 6. Therefore, K is 3 and and E is 6: you can check everything and it's still numerically legal.
Now let's evaluate a bit more. In the tens slot, subbing K and E, we see 6 + 3 = 9. No carryover here. The sum in the hundreds slot is 6 + 6. That's a total of 12, so there is carryover of 1 for the thousands slot, and X must therefore represent 2: the last digit of 12.
So we can re-write the sum, with the info we have so far, as follows:
(1) (1)
6 2 M R 6 6 3
+ 6 H 3 R 6 3 3
__________________________
1 3 - H - 2 9 6
So far, we know:
K =3
E = 6
X = 2
The ten millions slot number is 1.
The tens slot number is 9.
Now let's move on to the millions. Since we know X =2, H must either be 8 or 9, because we need carryover to make the 1 in our ten millions slot. But we know H can't be 9, because we've already used 9 in our tens slot, and with only 5 variables left to fill and 5 digits left to assign, it leaves no room for another 9. So H must therefore be 8.
Now, we have:
(1) (1)
6 2 M R 6 6 3
+ 6 8 3 R 6 3 3
__________________________
1 3 0 8 - 2 9 6
Just a little further! As we can see, M + 3 = 8, or 1 + M + 3 = 8. So M is either 4 or 5. But using similar logic outlined when solving for K and E, M cannot be 5, therefore it must be 4. That leaves us with enough info to determine that H = 8, and R = 7. We just need to fill in the rest of the blanks, and we have our complete sum!
I Live In A Magnificent World Of Grey And Darker Greys
During the very slow moments in which there's no job to do, Princess Luna has on occasion rather helpfully offered to appease your boredom by throwing random puzzles at you to occupy yourself until the next cretin/client comes knocking at the chamber doors. As it happens, on this occasion, Luna swings buy your chambers to inform you of a client, but rather than go through the usual process, the client has asked to see that you live up to the hype before letting you take on a case.
Intrigued, you follow Princess Luna to your office (well, throne room, but you need to relate it to something ordinary just to get through the day), where you see three ponies, sitting in a circle, on what looks like very comfortable chairs: your own personal chair doesn't look that comfy, which inspires some jealousy. Princess Luna then tells you that one of the ponies is your client, and your task is to identify that pony.
Luna introduces them as Azure Sky, Big Blue and Cirrus Spin: a team of pegasus ponies from the Wonderbolt Academy, and thrice winners of the not-at-all coveted Most Spectacular Crash Award. Flight deficiencies aside, you have a job to do. Luna then sets about explaining the puzzle.
She promptly blindfolds all three ponies, before slapping a silk scarf around your poor eyes, and then, from a box next to them (a box made from cardboard, the classic material for a reason) she pulls out three hats, which she arranges on their heads. She then turns to you, and explains.
Each pony is wearing a hat on their head, which is either white or black. The box started with two white hats and three black hats. You have been blindfolded, so you have no idea what colour hat each pony is wearing. She will then remove the blindfolds from the three ponies, and then ask, one at a time, if they know the colour of their hat. Your task is to listen to each pony's answer, and then deduce the hat colour of your client.
However, just to make things a little trickier, or perhaps easier, it's all a matter of perspective, she then informs you of a rather peculiar detail: one of the ponies is in fact completely color blind (which may explain why they won that award three times)! That colour blind pony is your client, and you need to announce both the name of your client and the colour of their hat, before any of the three do. They're also no slouch when it comes to logic either, so you gotta be on the top of your game for this. Being close friends though, Luna clarifies that all the ponies involved in this little game know who the colour blind pony is, which you're allowed to use in your answer.
Ready? Then Princess Luna will take off the blindfolds, and ask each one in a circle, starting with Azure Sky...
Big Blue is colour blind, and he's wearing a black hat.
To understand the logic, first consider what the chain of events would be if nopony was colour blind. If Azure Sky could see two white hats, she would immediately know that all the whites were taken and that she therefore must be wearing a black hat. If she says no, it means she's either seeing all blacks, or one white and one black.
The next pony down the line, Big Blue, knows this. If he sees that Cirrus Spin is wearing a white hat, he'd know that Azure must be seeing one white and one black, and so therefore he must be wearing black. His lack of a yes answer can only imply that Azure is seeing two blacks and eliminates the possibility of Cirrus wearing white.
Finally, Cirrus Spin realises that because neither know the colour of their own hat, he (and both of the other ponies) conclude Cirrus is wearing black.
Now let's add the colour blind pony, and see where that takes us.
Cirrus cannot be the colour blind one, because using the logic train described above, he would be able to deduce his hat's colour through logic alone, and whether he was colour blind or not wouldn't even matter.
Similarly, Azure cannot be the color blind one either. Since all the ponies involved know who the colour blind pony is, both Big Blue and Cirrus Spin would know Azure saying she doesn't know the colour of her hat would yield no useful information; no duh, she's COLOUR BLIND. Therefore, the train of logic to work out the hat's colour would start with Big Blue, and using the logic train above, Azure would realise she was wearing a black hat: she wouldn't have said that she didn't know her hat's colour a second time.
Therefore, the colour blind one must be Big Blue, and he must be wearing a black hat.
Congratulations, you managed to successfully identify and out the pony with the sight deficiency. I hope you're very proud of yourself. Seriously, you're lucky you didn't have to do that in public.
Anyway, Big Blue, satisfied that you have both the logic and two perfectly functional pairs of eyes, has a job for you: the sort that mostly requires manual labour, more to the pity.
He had recently been chosen by the Wonderbolts to help build a commemorative tower back at the Academy, showcasing the colours of all current members of the Wonderbolts on every floor. Being colour blind, naturally Big Blue can do little to assemble the tower himself, and so it falls to you to build it in his stead, at least by proxy. He does, however, ask that you be discreet about it: he has his own reasons for hiding his colour blindness, which you reckon would make for some hilarious misunderstandings, if the writer of the life that was his story was any good at comedy. You suspect they probably aren't.
Anyway, he then goes on to inform you that he's working with a builder who, after a very recent foray into necromancy, time-travel and a dash of magic, came to be cursed with a rather unusual deficiency: she cannot actually say any words other than numbers, and until she can visit some sort of cursebreaker, she's stuck with this curse for the remainder of the project, since Celestia forbid, she insists on seeing the project through to the end.
However, Big Blue realises that there is a workaround to this curse: all you have to do is ask how many of the floors are in the right colour after assembling the tower, and use her answer as your guide. So with the problem explained in one go, he then offers to fly you to the building site, and to hide you out of sight in a small and cramped section of the grounds to guide Blue's hooves.
You soon realise that you may have to cancel any lunch plans you have for the day, as in the event of the worst case scenario, you might be there for a good long time: Big Blue and his crew first have to build the tower, then ask the builder if they got the order correct, and if the answer is no, they'll have to disassemble it and build it again from scratch, and that could take hours. But what, exactly, is the worst case scenario here?
In the worst case, you’ll need five consultations from the builder.
It's easy for the more maths orientated to think the answer is twenty-four times, arguing that the total number of possible tower arrangements — 4×3×2×1 — is also the number of consultations required. But that isn’t true at all. Every single answer you receive from the builder tells you how many tower blocks are in the right order, and this information can be used to minimise the number of consultations by learning about the structure as you go.
Here's one such example for your perusal:
Think of the blocks as lettered and start by stacking them ABCD.
The builder says “zero.” Swap A and B, and ask again.
The builder says “zero.” From this, we know neither A nor B can be in the first or second position. Swap B and A for C and D: the tower now is structured as CDBA. Ask again.
The builder says says “two.” Swap C and D's positions, then ask about DCBA.
The builder says “zero.” Swap C and D for B and A: Ask about CDAB. This is now the correct position, as you can deduce from logic, but the tower cannot be considered built until the builder confirms that this is the order, so you ask a fifth and final time.
The builder says “four,” and you’re done!
If you're still not convinced, here is a handy diagram showing every possible route:
Rarity, AKA Miss Rarity AKA Sis AKA Marshmallow Pony, is a rising star among the Canterlot fine establishments, teacher at the Ponyville School of Friendship and Capitalist Queen Extraordinaire. Despite her surprising prestige and connections, few can ever out-charity Rarity: she'll give what she can give to anypony who needs it, even if they don't appreciate it at the time. Well, within reason. Don't ask Rarity-senpai for the tissue she just finished throwing away, you disturbing stalker.
Still, all this generosity has to be tempered with enough horse sense to retain the bits to pay for the next meal and keep the roof over her head, and it just so happens that opportunity is coming-a-knocking with a special competition being held in Canterlot: a treasure hunt sponsored by one of the descendants of the Collector, Canterlot's greatest gem collector! Whoever manages to find the legendary treasure will have access to rare gems that barely ever get seen even in Canterlot, as well as a not-so-inconsiderate sum of a million bits, just in case anyone like Spike would rather sample some gourmet gems and commits a rather grave geological faux pas.
Gleeful at the opportunity for a chance to use these rare gems in her dresses to rake in the bits for a limited time only, Rarity decides to exercise her magic talents, physical prowess, and willingness to crush the weak under her hooves, all well-exercised in previous battles of wits such as Monopoly, to seize the prize. Can she do it? Only time (and a lot of horse sense) will tell...
It all started, as your day normally does, with a rather unfortunate meeting of chance. One morning you were out shopping for a client (your particular train of logic involved shopping for purple and green horseshoes of different sizes, don't ask why), when suddenly you bump into an old acquaintance from Ponyville: Rarity, proprietor of the Carousel Boutique.
After enthusiastically and uncomfortably making very loud small talk, the white unicorn brings up the treasure hunt competition that is starting today, and in a stroke of genius, she decides to exploit a rule in the game: all competitors are allowed to take one thing that will help them in their quest to win the rare gems offered as a prize, and she decides to make YOU the thing. Congratulations, you're Rarity's property for the day. Get on your knees and bow to your new mistress.
Accepting that you're going to have to put off your shopping for the client off for the day, particularly after she bribes you with the bait of free pecan muffins via a pass received from Pinkie Pie (a slip of paper that says 'good for one favour from Pinkie Pie', because it's the lack of thought that counts), you decide that it would probably be more efficient to just play along and maybe hope for a cut of the gems in the end.
The two of you head straight to the venue where the competition is about to start, and the organiser, you believe her name is Highly Stakes, welcomes all the competitors, and then sets about giving you your first clue. She gives each of the competitors a map of what looks like a large cluster of several islands, and tells all the competitors that the treasure is located on the island where it is possible to see 2, 3, 4, 5, and 6 trees, from different points on the island, without moving from the spots, but by still swivelling the head.
Looking at the map with Rarity, you notice that all of the islands in the archipelago have exactly six trees each. To make matters a bit more complicated, according to the notes scribbled on the side of the map, the trees are of the genus Quercus Convelutious Contrivius Coincidenceus, and are said, no matter the growing conditions, to grow in exactly the same shape and of the same height as each other (it's magic, and thus removes the need to explain this phenomenon). However, it is noted that all the islands are totally flat, and have no obstructions other than trees, just to make things a little easier to the geometry-impaired. All the competitors have to do is visit the airship port corresponding to the number of the island, and it will take you to the spot where your next clue awaits you.
Pretty rock way to weed out the greedy but dim competitors from the serious players like you and Rarity. But where the heck is the treasure supposed to be?
The treasure is buried on the island with two non-parallel straight lines of trees.
Regarding the five spots on the island, we're looking for one where we can see a different number of trees each. That must mean that there must be something blocking the view for all but one of the spots. Since trees are the only things on the island, trees can only be obscured by other trees.
You would see fewer trees that there actually are if two or more trees are arranged in a straight line along your vantage point, so the island must be one where five different ponies could stand in five different spots and see different numbers of trees. That would narrow down the possibilities by quite a bit: virtually all the islands will have at least one point where all six trees can be seen, and most if not all will have a point where five trees can be seen, if two are lined up straight, one hiding behind the other. So you're looking for an island that has a lot of trees arranged in a line from a viewer's position. That's tricky: the hardest locations to find require more trees to line up with the view of the pony.
Which one, though, ticks all the boxes? Six trees arranged in one straight line is out: you CAN see one tree by standing on the far right or left of the line, two by standing right in the middle, and all six by planting yourself above the line, but nowhere on the island will let you see exactly three, four or five trees.
What if we had two lines of trees? Parallel lines of trees are right out, using similar logic above, but as long as they AREN'T parallel and their lines intersect over land, the point where they converge is where you can see two trees, and arranged properly, you can also see three, four, five or six trees, no matter how the trees are arranged or how many there are in a line - as long as they're grouped two and four, or three and three.
...And Everyone Uses It. They're Like "Oh, A Bridge, That's New"...
The trip on the airship to the designated island is a largely uneventful one, save for Rarity griping about the in-flight food, but you have to admit that for an airship privately rented for this treasure hunt, at least it's cheap, if not cheerful, because in your experience there's no such thing.
Finally, the airship touches down, and you, Rarity, and the one pony who appears to have gotten the last puzzle correct disembark. The lone pony then decides to wander away from the pair of you, hoping to find the treasure herself, or if not that, discover nature, connect with her inner self, shoot and tag a monkey, etc. She then wanders off, leaving you two to take the other direction.
After some walking, it has become apparent that you have not landed directly on the island you're supposed to be on to find the treasure. However, you see in the distance what appears to be a bridge leading to the island where your designated riches appears to be. A pillar of blue light is rising from the centre of the island from a stone slab on the sandy ground, and a series of mini islands, arranged in a perfect rectangle with smaller bridges connecting them, lead to the shore on the southern side where the light is shining, but before either of you can attempt to cross, the winds suddenly pick up and it starts to absolutely chuck it down.
With nowhere to take shelter, you promptly dig out your emergency "interdimensional pocket in a can" magic scroll you save for emergencies like this one (or in the event of capture by illegitimate figures of authority, like THAT one), and the two of you dive into the dimensional crack to escape.
Curious to be sure: there shouldn't be any weather ponies this far out to conjure such a storm. It's possible one of the treasure hunters who has experience in weather control is trying to impede your progress, but you decide to...well, cross that bridge when you get to it. (Cliches, amirite?)
You peek out of the crack to check that the storm is over, and for the most part there is now, once again, not a cloud in the sky. However, you fear that the storm may have inadvertently affected your ability to cross to the other side, and the pair of you step out.
Rarity notices that the blue light from the stone slab is shining as brightly as before, but as you feared, several of the bridges appear to have been destroyed by the storm, hampering your progress. The ones that survived don't exactly look like the most stable footings, and before you rush ahead and possibly break something that can't be fixed, you think carefully about your odds of crossing safely after the sudden and inexplicable wind and rain. Looking around, you don't appear to have anything on hoof to fix the bridge, there's no boat, and the suddenness of the storm has resulted in a splinter in Rarity's horn, temporarily sealing off her magic, so no levitating yourselves to the bank for you. So what are the chances you can cross now?
There are many ways to tackle this problem, but infuriatingly, there's a very simple explanation that barely uses any fancy maths at all.
You want to cross to the other side in this puzzle via the mini-islands and bridges. But imagine that there's a pony in a boat, in the water, who wants to pass through the bridges and islands, but can't unless enough bridges are destroyed to clear a path. Both of you are staring down the exact same problem, except in reverse: you can only pass to the other side if the pony in the boat CAN'T pass to the other side. Since both of your problems are not only mutually exclusive, but also structurally identical, it is inevitable that only one of you is going to get what you want, and since both outcomes are equally likely, both of you must have a fifty percent chance, and that's true regardless of the number of islands and connecting bridges.
Just before you can actually attempt to cross, the pony who separated from your group earlier appears to be heading your way, clutching a vividly pink umbrella in her hooves. Before either you or Rarity can actually attempt to warn the white earth pony of the perilous crossing, she steps onto one of the bridges, and through a ridiculous series of what can only be described as cartoonish chain reactions, causes the bridge to break and sink beneath the waves - followed by every other mini-island and bridge connecting them. At the same time.
While she profusely apologises to the pair of you, and Rarity begins to toss up trying to comfort the poor dear while also resisting the urge to yell at her for her blatant carelessness, you take stock of your new problem. The stone slab with the pillar of light is still intact, but now it seems to be located directly at the centre of a perfectly square moat of seawater, stuck on a very small island. By your estimation (well to be clear, it's a very exact estimate, puzzles like this engage the brain in many ways) the distance between the three of you and the slab is three metres worth of sea water. You could probably make a running start and jump to the other side, but you suspect you probably couldn't make the crossing back, and your white earth pony baggage has said she can't swim.
Fortunately, not all is lost, for at that exact moment, Rarity spots two incredibly splintered trees, most likely exposed to the elements for too long, and proposes that they be fashioned into two planks of wood, and that the three of you use them to build a bridge to the other side. Having nothing better to suggest, you watch Rarity uproot the trees with impressive force (without magic, impressive what a bit of greed can motivate a pony to do) and whittle down the ruined trees into two planks, armed with only her filly Scout pen-knife and the aid of a time-skip found only in literature works, until finally she has two identical planks of wood.
Slightly less fortunately, when the time comes to actually test the planks, you discover that they are only 2.9 metres long: less than the 3 metre distance to the slab, and no matter how much you stretch, you can't do better than that.
Now what? You have absolutely nothing to tie the planks together securely: no rope, tape, or its big older brother GAFFA tape, and none of you fancy your chances at swimming for it. So how can you reach the stone slab that will take you to the next part of the treasure hunt, without getting your hooves wet or falling in the water?
As much as you'd like to float your way across, you're almost certainly going to fall off and get wet. Or drown. They're not unrelated.
First, lay one of the planks diagonally, so that each end is touching each of the perpendicular sides of the moat. From there, rest the other plank on top. The first plank, acting as a foothold, allows the second plank to reach to the other side, leaving some room for overlap where the planks rest on each other and the bank.
Despite the very slow going, your improvised planks prove enough to get to the centremost island, and the three of you reach the light being emitted from the stone slab, and as any mysterious artifact shining with and odd light is want to do, swallows you in the radiant glow.
By the time you can actually see again, you have found yourself in the middle of a completely different island. You can see what appears to be a small green pegasus wearing a business suit and tie, and carrying a suitcase, and standing next to her, you see what appears to be a large, square-based grid, with enough room for a pony (even a very fat one) to stand in one with room to spare.
The pegasus, who apparently is acting as referee, welcomes you, along with another orange earth pony who happened to also make it across, it seems, to the final challenge of the treasure hunt: one where only the most worthy can obtain riches beyond their wildest imagination* (*Based on typical wild imaginings of previous ponies matching the demographic profile. Additional terms and restrictions may apply).
Sadly, as you're only taking part in the contest as part of Rarity's belongings, you must unfortunately sit this one out. The referee guides Rarity and the other two ponies to places on the grid, and then hands all three competitors a piece of paper each. She then explains the following rules to them:
Each pony has been given a piece of paper with a number written down. The number represents the number of squares a pony must travel horizontally or vertically to reach the square where the treasure is buried, and the number always represents the shortest distance to the treasure.
Everypony can see where each pony is standing, but they only know the number that is written on their own card. The referee will ask everypony at the same time if they know where the treasure is located, and all three must answer at the same time. After that, they must pick a square where they think the treasure is, and start digging. If it's there, congratulations, you win the gems. If not, you auto-lose the hunt and will have to be sent home. In a box. If everypony doesn't get the prize then they'll be made to stand in the corner. In a box.
So on the count of three, after she asks each of you, "Do you know where the treasure is?" the replies are thus:
Scarlet Pimple: "Nah." Pearl Wisdom: "Nope." Rarity: "No!"
Rarity sends a very discreet signal to you to indicate that by no, she does in fact mean yes. You realise that she's figured out where the treasure is. "Ah, my child, you have remembered your training" is what you're tempted to say, except Rarity isn't your child and you have not in fact ever trained her.
Having said that, you do know Rarity has made an effort ever since the Gazelle Plains incident not to disappoint you, so you're decently confident that she's logical enough to have worked out the answer. So where IS the treasure, anyway?
All the cards have different numbers on them, so we can eliminate any squares that has the same number of steps between two ponies: in other words, the six squares equidistant between Pearl Wisdom and Scarlet Pimple (A5, B5, C4, D4, E4, and F4).
We also know none of the cards have a 0 written down, otherwise one of the ponies would immediately know the treasure was buried under them, so we can eliminate the squares each pony is standing on.
We also know Scarlet Pimple doesn't have 7 written on his card, as there's only one square seven steps away from his location (F6). Likewise, Pearl cannot have 8 and Rarity cannot have 8 or 9, because if they had either number, they would have responded that they knew where the treasure is. So we can eliminate A6, B6, F1 and F6 as well.
Finally, for Rarity to know where the treasure is, she must have a number that provides the distance to only one possible square, and it just so happens that there is only ONE square seven steps away from Rarity, and that's the square A4.
The referee of the game then asks each contestant to pick a square, and then dig. If they scored the treasure, great! If not, the buck passes to the next pony until all the squares are dug up, or the treasure is found, whichever is sooner.
As Rarity is the only one who knows where the treasure is, you proudly watch as she digs up a large bronze treasure chest and hauls it out of the ground, but before she can even begin to examine the prize inside, the ref draws a big pointy and very sharp looking metal stick resembling a cutlass (which makes it in all likelihood PROBABLY a cutlass) tackles Rarity to the floor and threatens her to fork over the treasure or else be repurposed into a very beautiful sharp object repository.
It turns out the treasure is in fact a bunch of ill-gotten gains, and the referee was hired by the pony who organised the whole event to find the treasure and dig it up, since she couldn't remember the exact spot where it was buried. Before you can even begin to question the sheer implausibility or the convolutedness of it all, Rarity manages to summon up enough magic to throw her off...right next to you. Well, horseapples.
She draws out several more cutlasses (one for each hoof, and yes, including the back ones somehow) and lunges at you. However, you are able to give her a good swift kick, and knock three of the cutlasses out of her hooves, with one landing to the left, and the other two to your right.
The kick also sends her reeling back and has sprained one of her wings, thankfully grounding her. However, she's still got a mean jump, and looks poised to strike...until the cutlass she's holding suddenly snaps in two. Your assailant curses out loud that her collection of sharp pointy swords seem to be easier to snap than initially thought, with half of the swords she got from the blacksmith (well, her apprentice, hence the less than professional job) seeming to fall apart at the slightest hit of forceful impact. She then goes through her Pinkie Pie-esque hammerspace looking for a cutlass that won't fall apart in her hooves.
Not much time to ponder yourself, unfortunately. You are seemingly without a weapon and therefore have to make for either the lone cutlass to your left, or try and grab the two on the right. (You admittedly don't have a lot of experience in sword-play, but you figure the basic principle of the pointy end going in the other pony shouldn't be too hard to grasp.) However, each cutlass has a 50% chance of falling to pieces at the slightest hint of great force. You recall that as you kicked the crazy pony away, you heard a slight sound of one of the swords cracking in your right direction, so you know that one of the swords in the pair is a defective brittle one, but not which one: you can't see it from where you're standing, and there's not much time to examine it before you are set upon once again.
So to avoid having more holes in you than a cheese grater, which one should you pick: the lone cutlass, or the pair, with one guaranteed defective one?
You should dive towards the pair of cutlasses, for your best chance at survival.
Surprisingly, you have a 2/3 chance of grabbing a sword that's useable if you spring for the pair, as opposed to the 1/2 chance you'd be gambling on from the lone cutlass to your left.
It sounds straightforward: if there is roughly the same number of brittle cutlasses as useable pointy sticks, then the odds of the single cutlass being brittle is 50:50.
The same, however, cannot be said for the pair of cutlasses on your right. Many people's first mistake is assuming that there is a 50:50 chance of the other cutlass being brittle assuming you know one of them is: since each cutlass's probability is independent, there too must be a 50:50 chance of the other cutlass being brittle, when in fact that's completely wrong.
It's flawed because in this case, their probability ISN'T independent. This is where a little trick called conditional probability comes in, where knowing certain information changes the odds of each possibility. Let's assume you were going in blind, and had no information on the cutlasses other than the odds it will be a glorified metal stick:
Both are brittle: BB
Both are useable: UU
The cutlass on the left is brittle and the other is useable: BU
The cutlass on the right is brittle and the other is useable: UB
People make the second mistake here, thinking they can treat BU the same as UB, since the intent is to grab both cutlasses anyway. In fact, they should both be treated as two completely different outcomes: you DON'T know which cutlass is brittle, and so without that knowledge you must consider both possibilities separately.
However, the sound of a cracking sword tells you that AT LEAST ONE is brittle, so you can eliminate the possibility of two useable swords. That leaves us with:
Both are brittle: BB
The cutlass on the left is brittle and the other is useable: BU
The cutlass on the right is brittle and the other is useable: UB
As we can see, out of the two outcomes, two out of three have a useable sword for us, hence the probability of a useable cutlass, based on what we CURRENTLY know, is 2/3.
One last unusual observation; if you actually DID know which of the swords was brittle, then that information actually LOWERS your odds to 1/2, because then we'd have to eliminate one of the possibilities (either BU or UB), leaving us with BB or the remaining scenario. And who said there's no such thing as too much information?
Thankfully, one of the cutlasses you grab is serviceable, and you manage to put in a decent show of flailing it around to avoid getting riddled with holes, up until the moment she forgets the other three ponies (Scarlet, Pearl and Rarity) sneaking up behind her, whereupon she promptly gets knocked out with a coconut. Bad one there.
With the villain safely trussed up, you call for help with a Royal Emergency Beacon you stashed away, and on cue, two Royal guards escort you back to Canterlot Castle, where Rarity can officially receive her prize, and the crazy pirate pony can await a trial in the dungeons.
At the award ceremony, Princess Celestia recognises Rarity as the officially dubbed winner, but recognising your part in keeping Rarity safe and for being the sharp pencil for solving the puzzles with her, she proposes you split the prize...but the distribution will be decided by (what else) a battle of wits. Because it always is, nowadays.
She then sits both you and Rarity down, facing each other at a table, with a partition blocking the sight of both of you. She then reaches into the treasure chest you brought back, hands you one hundred of the precious gems inside, and explains the rules of the game.
The gems will be used to play a game of war, where you, the defender, must defend ten towers, with the gems representing soldiers. Each of the ten towers is worth a certain number of points, from 1, 2, 3...9, and 10. Both you and Rarity will then send soldiers (or gems, which make for very shiny soldiers) to a tower, and whoever has the most soldiers wins the points that tower is worth. In the event of a draw, the points are split. Whoever has the most points once all ten towers have engaged in battle wins, and receives two gem for every point earned, and any leftovers fall under the crown's ownership, as restitution for the public.
It sounds like this could go either way, but Princess Celestia then announces a twist: YOU will arrange your one hundred soldiers as you see fit, whereas Rarity can win the game (and the ENTIRE stock of gems) by using the fewest number of gem-soldiers possible: if she's forced to use all one-hundred, however, all the gems will fall under YOUR possession. Sounds harsh, but it'll motivate Rarity to try her best, at least. On the other hand, if she can successfully identify the fewest number of soldiers she needs before your distributions are revealed, she will win the whole game and thus all the gems.
Imagine you are Rarity. Taking into account you are playing a perfectly logical ex-partner, what is the fewest number of gem-soldiers you need to take home the entire prize? You could do it the old fashioned way, but that would involve taking home a smaller prize...and before Rarity's sense of generosity kicks in, it's pointed out that if she wins all of them, she can just freely give them out as she wishes afterwards, so it's in her best interest to win. So what to do?
Help Rarity defeat...well, yourself, and take home the long-coveted prize!
Rarity needs 56 soldiers minimum to win the contest.
The total number of winnable points when we add the value of the towers (1+2+3...+9+10) is 55 points. Therefore, to secure a victory, Rarity needs to win 28 points in total at least.
Since Rarity is proceeding under the assumption she has all the knowledge in this contest, we need to first discern what the opponent's most optimal strategy would be, and then devise a countermeasure. Since the opponent won't want to yield 28 or more points, they (that is to say you) will naturally make the highest scoring towers the ones that are the most difficult to conquer - in other words, to make it so that every tower has an equally poor number of soldiers required to conquer it.
Luckily, 100 is the perfect number, and so you'll most likely set up each tower with a distribution of 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19, since that works out to 2 soldiers spent per victory won. As this is the worst case scenario, Rarity can earn the 28 points EXACTLY she needs by conquering towers 1-7, by sending 2, 4, 6, 8, 10, 12 and 14 soldiers out to the first 7 towers, which in total works out to 56 soldiers.
Just like any good war, this one is decided on economics.
Part 3: Applejack's Got Apple-Crabs (And They're Huge)
Applejack, AKA AJ AKA Silly Pony, is farmer who faces down many, MANY problems. These problems include, but are certainly not limited to: financial worries, family support, crop yield, and crabs the size of apples who devour entire orchards in the blink of an eye. And possess poisonous shells. So, just like any other pest a farmer can reasonably expect to attract.
The Apple-Crab (not to be confused with the Crab-Apple, which is more annoying than fatally poisonous and highly aggressive) is a pest that recently has been plaguing the farming community of Equestria ever since some thoughtless berk tried to introduce some comically amusing animals into the wilds of Equestria, apparently for a bet. Sweet Apple Acres has so far been lucky to not be visited by such a creature, but unfortunately their luck seems to have run out, as almost all of the entire East Orchard is being chowed down by this rather geographically inaccurate crustacean.
And it couldn't have happened at a worse possible time either, because it just so happens that you were on an errand to Ponyville that day, fixing somebody's roof via sudoku and advanced trigonometry (it's a heck of a roof, don't ask further), and you are obliged, as a sort of acquaintance and official Royal Dispute Settler, to track down the elusive and dangerous creature, while also not being expected to charge full price for your services to Applejack, curse the friend discount.
Oh well. Better get that roof fixed quickly then. Applejack's livelihood is at stake, and Celestia be darned if her cider and apple pie isn't worth preserving...
Luckily for you, it would seem Applejack was considerate enough to not let you start entirely from scratch when it comes to catching the Apple-Crab, and she has recently, with Applebloom's help, constructed what she reckons is a pretty fearsome trap for the elusive beastie. It appears to be an extremely large grid of squares, a hundred by a hundred by your count, with several arrows pointing in all different directions, along with walls of several revolving doors.
So how, exactly, does this implausibly elaborate device function, you ask? Applejack proudly explains. The trap is to be baited with several apples, and when the crab attempts to nibble on one, it will instantly activate the device and force the crab to move. The arrows on each of the squares will move the crab in the same direction as that arrow, after which, once the crab has left that square, the arrow will rotate by ninety degrees: if it's pointing north, it will move the crab north and then change its arrow to face east, and then if the crab should visit that same square again, it will be forced to move east, and the square with that arrow will rotate again to face south, and so on.
Such a device is necessary as the Apple-Crab must remain in almost constant motion, lest it gain its bearings long enough to use its venomous and acidic bubbles that they naturally possess to simply melt their way out of the trap and wreak havoc on the orchards once again. The only catch, however, is that in the event that the crab hits an edge square and it points towards the wall of the trap, nothing will happen, and the crab will be able to escape. However, she remains decently confident that such a trap will be for the most part largely unescapable, and her crop shall be saved.
Is she right though, you think. Is this trap actually foolproof, and if not, how is the best way to carefully and deliberately destroy her hopes and dreams?
It's unfortunately true: the trap can be escaped from.
Let's start by assuming the opposite: that it's impossible to leave. The starting square, assuming the crab's doomed to wander the trap forever, must be visited an infinite number of times. That would then, using recursive logic, lead us to believe that all four squares surrounding it must also be landed on an infinite number of times, and the square surrounding THEM to be visited an infinite number of times, until finally we come to the edge pieces. And then we return with a contradiction:
How can you visit an edge piece an infinite number of times, travelling in all four directions as a result, and NOT leave the grid?
So, since we proceeded under the assumption that escaping is impossible, and the train of thought ended with the conclusion that you can escape, this must be false and so the trap must be possible to escape from.
With Applebloom and Applejack's efforts thoroughly undermined (way to go, by the way), both of you conclude that you'll have to think of a better trap. However, just as you sit down to begin brainstorming ideas, Ponyville's mailpony flies by Sweet Apple Acres to drop a leaflet on your heads, advertising what appears to be an all-purpose pest capturer. Any who are interested should visit Ponyville Square today, for a limited time only get half price off all fuel and self-repair kits.
Thinking that this sounds far too good to be true, but unwilling to ignore it and think of conceivably worse ideas to catch the Apple-Crab, the pair of you hurry to the square. Since the crab is nocturnal, you reckon that you should have until dusk to check if the capture mechanism is legit before heading back to the farm in time for lunch to bemoan a wasted morning.
Unfortunately, it IS too good to be true, as Applejack instantly recognises the salesponies pitching the device: a very poorly disguised Flim and Flam: they appear only to have swapped the moustache to the other brother, which has been dyed a very fetching bright red.
Before Applejack can even begin to give the brothers a good earful, they point out their cider machine actually DID work, so she shouldn't write them off just yet. You do recall that Flim and Flam first arrived in Ponyville on a strange locomotive/cider squeezer, and for the most part were able to demonstrate a working product...until things sort of escalated and the brothers were chased out of town by an unruly mob who found sticks in their drinks. To be fair, if they weren't so hasty, they probably would have succeeded.
Willing to give them a chance even if Applejack won't, you ask for the price. It's admittedly a bit much for your average pony's wages, but Applejack's desperate, and no matter who loses, you win, since you can still charge Applejack, so you decide to purchase it...after a long series of anticipated haggling. However, eager to make a quick bit, Flam offers to skip the tedious price haggling and play a game with you: should you win, you get the product for 75% off the usual price, but if they win, you pay the full amount, plus standard VAT. They say it's a game that offers a 50:50 chance of winning, so it's all fair.
Taking two playing cards out from under his hat, Flam shows that one of them is your standard Ace of Spades, the other, a misprint he got in his pack, which is a card with two backs. He places them in a small burlap sack and then shakes the bag. As he shakes, he explains the rules:
The cards are randomly mixed inside the bag. You will then draw one of the cards at random from the bag, making sure that the back of the card is facing you. You will then flip the card over: should you reveal the Ace underneath, you win a point, but if it's the double-sided card, Flam gets a point. First to ten points will win the game, and will suffer all consequences associated therein.
Applejack urges you to just play, since you two need to get back and the chance to only pay a quarter of the price is too good to pass up. Silly honest pony, you think. Still, now that you're resigned to playing, how should you exploit this game to actually win and save yourself quite a lot of cash?
First of all, definitely don't go into the game thinking that the odds of you winning is 50%. Although the odds of you drawing the Ace from the bag certainly is 50:50, consider this: what are the odds of you pulling the Ace, with its back facing you?
If you pull out the Ace, and the front is facing you, you will inevitably have to put the card back in the bag and re-shuffle, so 50% of the time you draw it, it's going back in the bag. In comparison, the double-sided card has two backs, so no matter which side faces you, it will never be put back in the bag. Therefore, the possible outcomes are:
Since we're not counting re-shuffles, we eliminate that possibility to realise that we now have two scenarios where Flam wins, compared to your measly one. So the probability of you winning a single round is one in three. Multiply that over the course of ten games, and your odds of winning only come out to about 6%.
Doesn't sound like it's possible to win without cheating, does it? Well, if it isn't possible without cheating...no duh, you cheat.
But what's the best way to do it? When playing a pony who knows the game is rigged in their favour, a lot of straight wins will certainly tip him off. But there is way to limit suspicious activity to only four rounds.
First, mark the double-sided card. Marking the front or the back won't help, and it'll probably get you caught. What you need to do is make a tiny dent on the SIDE of the card. Even tiny dents can be detected by touch (try it out on a train ticket or store loyalty card later if you find the time), and they're difficult to spot with sight alone.
When starting the game, pay close attention to the 2:1 ratio of pulling the Ace. Flam knows the game is stacked in his favour with those odds, so you need to intentionally let him win with a score of about 9-5, so he won't figure out you're cheating.
From there, when it looks like you're on your last legs, simply keep drawing the Ace until you win. On 9-6, he won't suspect anything. 9-7, he might get suspicious but chalk it up as bad luck. 9-8, and he'll know you're cheating, and may even inspect the Ace card, but he won't find anything wrong with it: the card you actually marked is still in the bag!
The faster the game is going, the harder it'll be for Flam to catch on, and with only two rounds left, he's very unlikely to figure out the trick within that time.
Grumbling, Flam allows you to hand over what is effectively chump change in exchange for the small device, and with triumph, you and Applejack make your way back to Sweet Apple Acres, clutching the small robot thing in your hooves, ignoring the cries of vengeance and the sounds of the other brother trying to calm down for the next customer from over your shoulder.
However, as is the case with any new product that needs to be assembled and configured, trying to figure out how to actually start up the device soon proves to be an absolute chore. As soon as you get back, open the packaging and attempt to read the instruction manual inside (the instruction manual is the size of a Jeremy Brenthoof novel, so Applejack has foisted the job of making heads and tails of it to you), you encounter a problem: starting it up requires inputting a password.
Luckily, it would seem that out of graciousness in defeat, Flim, as a reward for besting his brother, has input most of the password already and has left a note saying that the password requires inputting a series of characters according to a pattern in the numbers. The display screen reads 9, 10, 19, 24, 31, 40, 51, 64, 79, 90, ???. Right underneath, you see what appears to one of those weird QWERTY keyboards that you see on Equestria's most advanced typewriters, or on one of Twilight Sparkle's weird machine things in her basement that she vehemently denies the existence of...
Confessing that she has no plum clue as to what the numbers actually mean, Applejack decides to leave it to the expert and have you figure it out, cheers for that. No time to grumble though, as dusk is soon upon you and the Apple-Crab will awaken to wreak terrible vengeance, havoc and all sorts of terrible nouns upon the farm. So what is the mysterious missing piece in the sequence that will allow you to finally activate the machine and deliver sweet justice to your crustacean-shaped problem?
The pattern is going to be really confusing (especially since we seem to be introducing LETTERS in our numbers) to most until you realise the sequence of numbers isn't being represented in the standard Base 10. Rather, this sequence is in Base 16, or a hexadecimal system. Base 10 only uses 10 digits, so for Base 16, computer programmers represent the numbers 10-15 with letters, A to F.
The pattern is (N + 2)2, where N is the position in the sequence. So the first number is 32, or 9. The second number is 42, or 16, which in hexadecimal is represented as 10. 25 is written as 19 in hexadecimal, and so on until we reach the last number, which is 169, or A9 for short.
And Then We Entered A Completely Different Genre Altogether
Upon hitting the Enter key on the innocuous device, the trap suddenly springs to life, and from the little package, it suddenly transforms into what appears to be a giant mechanical crab about twenty-four feet high. Before you can argue that such a feat should be impossible for a small cube about six inches wide, it emits a high-pitched whine, and from who knows where, the Apple-Crab emerges from its hiding place and begins to square down the crab.
You recall from the instruction manual that upon sighting its target, the package will execute what the manual referred to as "Protocol B-10-Up", which causes the device to enter a combative stance, warning that it would become "a swirling vortex of metal, fleshy remains of your target pest, and violence", and could create "profoundly disturbing scenes".
Unfortunately, nopony could ever have known that Apple-Crabs have a spontaneous biological process inside their bodies that automatically trigger when encountering a threat larger than itself: it can expand its body to match the size of the threat, and very soon you have two twenty-four foot tall crabs brawling in the middle of Sweet Apple Acres.
The farm is quickly evacuated, to Big Macintosh, Applebloom and Granny Smith's confusion, with only you and Applejack choosing to remain to monitor the situation. The brawl is long and intense, but suddenly, the Apple-Crab grabs your mecha in a vice, and to the shock of everypony watching, the mecha directly assimilates the Apple-Crab directly into its own body.
This however proves to a costly mistake and soon the crab's consciousness begins to override that of the mecha, and recognising a target, it begins to devour the apples in the trees at a rapid pace, its new mechanical body protecting it from the heat of the day. The only way now to contain the threat once and for all is to destroy the mecha-Crab entity down to its very foundations.
From what you could remember from the instruction manual, the device has one-hundred exhaust ports, and you propose that blocking up some of them would cause the mecha-Crab to cease to function and destroy itself. Suddenly, the crab turns on the pair of you, but in a display of strength and sheer unadulterated stones (if a mare like Applejack had any), she leaps onto the bio-mechanical foe and tells you to look up where the exhaust ports can be found.
After a few minutes (and several bruises inflicted on the orange mare), you read that the exhaust ports are located at the base of the legs. Rather less fortunately, the mecha will automatically close such ports upon sustaining damage, but that further damage would then cause every second port to remain open, while the other half closed off, then every strike would cause every third exhaust port to toggle open or closed, depending on position.
You recall that in the fight between the Apple-Crab and what is rapidly turning out to be a waste of bits, the mecha was struck a total of exactly one hundred times. As soon as you reach that conclusion, Applejack is thrown off the Mecha-Crab, and is forced to retreat back to the barn, leaving you to be chased around the farm for a good five minutes, before she returns with a plank of wood, a log, and several boulders. To launch at the exhaust ports.
Improvising a see-saw, the pair of you load up and prepare to fire. Applejack assures you that she's got a good eye, and won't miss: just point out the legs with the open exhaust ports, and she'll let the rocks fly. So hurry up! Which legs need to be attacked to bring the crab down?
Ports 1, 4, 9, 16, 25, 36,49, 64, 81 and 100 will remain open.
The ports will only remain open if they are toggled an odd number of times, and only if the leg's number has an odd number of factors. Factors have to come in pairs, since you need to multiply two factors to make a number. Square numbers are also the only numbers where there's only one factor involved (multiplying by itself, so it counts as one).
Consider the number 16: 1 x 16, 2 x 8, 4 x 4. Its factors are therefore 1 and 16, 2 and 8, and 4, which makes a total of five factors, an odd number.
Square numbers are the only numbers with an odd number of factors and therefore only those corresponding exhaust ports will remain open after one hundred strikes from the Apple-Crab.
With the last boulder fired, the Crab-Mecha (or was it Mecha-Crab?) immediately starts to sputter and cough, before finally shrinking to about the size of a lemon by ejecting spare parts (mechanical and biological, by Celestia the mess and the smell!). Recognising that it's been cornered, it chooses to scuttle away to safety, with you and Applejack in hot pursuit.
The pair of you manage to chase the crab to a section of the farm where there are five mini-orchards in the western orchard lined up in a row. On Applejack's instructions, the rest of the Apple Family have completely surrounded the area, with the crab having seemingly no hope of escaping. However, knowing how wily the foe truly is, Applejack instructs her family not to leave their designated guarding post, lest the crab somehow escape.
Unfortunately, the sun isn't going to remain high in the sky forever, and you estimate it will be another six hours before Princess Luna raises the moon in the sky to mark the start of the night. By then, Granny Smith must turn in for the day, Big Macintosh has chores to complete, and Applebloom needs to finish up on some homework. On top of that, Applejack's night vision absolutely stinks, largely because of her shunning of carrots in favour of apples, the one and only answer to all of life's problems. Except this one.
You don't have a torch, and only you and Applejack are free to move around. However, you do have a slight advantage in your favour: judging from the crab's behaviour, you reason that as soon as you start to search the five mini-orchards, the crab will move from one adjacent mini-orchard to the other in an attempt to escape, and will never remain still for very long. You and Applejack working together however are free to search any of them in any order you like, but it will take an hour to comb that orchard from head to tail for any suspicious scuttling or acidic bubbles.
Better hurry it up though: you only have six hours to catch the crab, and you really need to be getting back to Canterlot shortly to report on the roof you were meant to be fixing today. So where should you start searching?
Check mini-orchards 2, 3, 4, 2, 3, 4, in that order.
To find the crab before dusk, you can't make assumptions. Assume the worst and prepare accordingly. Simply searching each orchard at random, or the same ones over and over won't work.
Let's imagine that the crab started in one of the even-numbered orchards - in other words, either 2 or 4. If we search orchard 2 and find the crab, great! If not, that must mean the crab started in orchard 4, and moved to either 3 or 5.
So now we check 3. If it's not there, it must have moved to 5. Now the crab is cornered - it can only move to orchard number 4 from there, so you simply search orchard 4 and you'll find the crab for sure.
All this, of course, assumes the crab was in orchard 2 or 4. but what if it was in 1, 3, or 5? If it started in any of those, in the second hour it will have moved to either orchard 2 or 4. Now we discover an incredibly important pattern: after three hours, if the crab started in an odd-numbered orchard, it will be definitely in an even-numbered orchard at the start of the fourth day.
So if after three hours you haven't found the crab following the strategy for even-numbered starts, all you have to do is repeat the pattern again, and you'll catch the crab for sure!
Having finally secured the crab, you and Applejack eventually manage to find a way to subdue it: Apple-Crabs can apparently be rendered catatonic when exposed to oranges, and although it does nothing to not fuel Applejack's belief that apples are the only answer, it does at least solve your current crisis.
Applejack has half a mind to march up to Flim and Flam and get your money back, but as it's been seven hours since you purchased the mecha-trap, you reason they must have skipped town and taken all the cash with you. Your poor bits. You will never see them again...
...or will you? As it happens, as the brothers were trying to skip town, they were intercepted by Princess Twilight Sparkle, along with a mob of angry ponies whose livelihoods were almost destroyed by these transforming robots, and made to pay restitution for all their meddling. A mecha battle in the middle of Sweet Apple Acres was hardly going to go unnoticed, after all, especially not by the Princess of Friendship. (It does make you ask why nopony tried to help, but you know pony herd mentality.)
After you shake down the pair for all the almost pennies you paid for the device, the brothers try to hastily leave...only to discover that the locomotive they arrived in Ponyville with won't start. Much to their embarrassment, it would seem that the angry mob has vandalised their precious vehicle, and will no longer start. They implore you to help, as they are currently a little too preoccupied being splattered with various fruits and vegetables and being jeered at by the townsponies to fix the machine themselves. Sadly, your oath is to help everypony, so keeping a mental note to fleece the brothers for every single last bit on them as your fee, you open it up.
You are greeted by what appear to be a jumbled up mess of four identical looking wires. when you crack open the right side of the locomotive, you see the other ends of the wires, but you have absolutely no way of telling which end of the wires are which from just a cursory glance. Flam just about manages to splutter out that you can get the vehicle to start again as long as two of the cables are joined at one end together, and then linked to form a loop. Form two complete circuits, and it should work again, allowing Flim and Flam to flee from their overstayed welcome.
As much as you would love to dawdle and allow the brothers more time to have so much abuse heaped on them, time is precious to you, and so you want to waste as few trips as possible walking from one side of Flim and Flam's locomotive to the other. Your first test is to tell you that two cables are either A and B, or B and A. So what's the quickest way to get the thing fixed so everypony can be on their way?
First, label each end of the cables on one side A, B, C and D. Then, connect any two of them (let's say A and B for simplicity) and go around to the other end.
On this end, label the cable ends on this side 1, 2, 3 and 4. Test away pouring power into each until you hit a complete pair which forms a circuit. Let's say it's 2 and 4.
From there, disconnect one of them and attach it to another unused cable - let's say 2 to 3 - and then go back to the other side. Disconnect A from B and test again for a different complete circuit. One of them will be A or B (the other end of 2) and C or D, the other end of 3. Let's say it's A and C.
Now, you have all the information you need to match up the rest. A is 2, so B is 4, C is 3 and D must be 1. This works regardless of combinations that form circuits.
Part 4: Pinkie And The Brain (World Takeover Not Included)
Pinkamena Diane Pie AKA Pinkie Pie AKA GAHH-Where-Did-You-Come-From, is the proud counter and kitchen worker of Sugarcube Corner. But then, you knew that already, her pecan muffins are simply to die for. You nearly did just that, actually. About twice. But the risk of catching something highly infectious or possibly getting mauled by bears, or even getting mauled by infected bears is worth the risk.
As it happens, unlike her two friends Rarity and Applejack, Pinkie was apparently kind enough to actually make an official visit to you, rather than make you an unfortunate victim of happenstance yet again. It turns out that after a brief visit to Canterlot on official School of Friendship business (it's NOT a vacation, she has the paperwork to prove otherwise with a shiny official seal and everything, and no, that isn't a beach towel in her suitcase), she caught wind of a certain dessert, known as the legendary Cake of Diablos.
This incredibly sinister-sounding cake is said to have been sealed away by the Royal Sisters themselves, for reasons that they will not disclose. They have fervently denied any and all knowledge of such a cake, but Pinkie apparently managed to get the recipe for such a thing lying in the Canterlot Castle Library (under C, for Cookery). You at first want to deny such an implausible coincidence, but if the Alicorn Amulet can be found in a pawn shop of all places, anything can happen.
You don't know where Pinkie is going with this, or rather, you do and you desperately hope you're wrong. Sadly, Pinkie prepares to disappoint you when she announces that she wants to enlist your help in locating the ingredients for such a cake and bake it. For science. And cake. Especially the last part. Oddly, your oath to help everypony as Royal Dispute Settler doesn't seem to extend to such dangerous and life-threatening activities that could compromise the safety of others such as baking a cake, so you decide to play along - and twist the interpretation of the oath to Princess Celestia in the event you need to explain yourself.
Having packed all your equipment needed for such an arduous journey, like crampons, skis, helmets, etc. you and pinkie set of for the Doomdale Mountains, where you're apparently meant to retrieve something called W1ND3G0 Flour (it's apparently not pronounced how you think it is, nor is flour in the traditional sense, but this isn't what you call a traditional cake). it's not an especially easy going trip, considering that the Doomsdale Mountains are under the sea: you had to employ the use of camels for much of the journey. Before any readers point out camels for an underwater trip is mad, that is precisely why Pinkie found only MAD camels: only they can even LIVE underwater.
After you have surfaced for what you estimate is the last meal necessary, you finally reach the peak (which many have mistook for an island) sticking right out of the water, and before you is a huge stone monument towering to the sky. that, apparently, is where you will find the flour for the Cake of Diablos.
You don't know how tall the pillar reaches, but fortunately, Pinkie has a magical extendable ladder that she stole/borrowed without permission from Twilight Sparkle that should make itself tall enough to reach the top. However, there is a small problem: the ladder works by stating how long you need it to be, and it will instantly form a stepladder of the stated length, down to the millimetre. However, without knowing exactly how tall the stone pillar is, there's the risk that whatever ladder you call upon will either be too short, and you won't reach the flour, or too tall and it will impede your efforts to take the flour trying to reach around it.
When you ask whether if Pinkie has any useful equipment, Pinkie confesses, as she pulls the items from out of her ears, since there's no room for them in her mane, that all she has to offer are two one metre-long rulers. It goes without saying that such a device is certainly shorter than the pillar, but thankfully, the surface area of the mountain peak seems just large enough for the idea that you have just come up with. So how, exactly, are you going to measure how tall the stone pillar is and reach the flour at the top?
This one's going to require some cunning - and some hope no pegasi have scheduled any clouds to be moved here today.
First, stick one of the rulers into the ground. After that, it's all a waiting game. using the other metre ruler, you wait it out, periodically measuring the shadow of the ruler in the ground, until its shadow is one metre long - the same as the ruler.
By that point, the shadow of the pillar will also be the same height as the pillar itself, so using your rulers, you can now measure the pillar's height by measuring the shadow's length.
Your trip to find all the necessary ingredients for this cake is taking much longer than you realize, but after a lot of trekking, swimming, spelunking and a considerably large medical bill, you and Pinkie have arrived at what she assures you is the final stop, and then she'll have everything she needs to finally start baking the cake. With you, obviously, because you're smart. And she gives you all your precious pecan muffins. And she refuses to go down alone if caught.
You appear to have arrived at a strange temple, where an extremely wizened old pony appears to be standing guard in front of a series of stepping stones across a pond. He is apparently the keeper of the final ingredient that you need: 50M8RA extract, and although Pinkie isn't entirely sure what a 50M8RA even is or what was extracted from it (although you have a sinking feeling you know) only those the keeper deems worthy can have a cup of it. Worried, the pai of you approach the keeper, who goes though the expected spiel of how 'none may pass' and you must 'answer his riddle or die', and all that. Before you can even protest, Pinkie announces you'll do it. Jeez, what happened to manners nowadays? It's dead, dude.
Anyway, you've been shoved in the spotlight, so it's time for you to make a stand. The keeper gestures to the pond, explaining that before you is a completely straight line of twenty stepping stones. Starting on the shore of the pond, which he rather helpfully lists as Number 0, and ending precisely on the last stepping stone, which he lists as Number 20, is how anypony who wants to pass to the other side and claim the prize of the 50M8RA extract should go about it.
Pinkie then asks if the puzzle is to cross, and for her dumb question she is promptly beaned on the head with a small rock the keeper had hidden in his robes. Of course that isn't the puzzle: there's a bunch of ways to get across. The puzzle, he says with a mocking grin, is to determine EXACTLY HOW MANY ways there are to cross: to start at the shore and then end on the twentieth stepping stone exactly.
Ah. Well, that's a slightly trickier task to work with, but in your mind, certainly doable. So what's the answer? Pinkie won't be happy if you have to walk home without the last ingredient after all your trekking, swimming, spelunking and medical expenses, so spare yourself the embarrassment and get this solved.
There are 10,946 different paths that you could take.
There's only one way to "reach" Stepping Stone 0: stay exactly where you are. There's also only one way to get to Stepping Stone 1: by advancing one space.
When it comes to Stepping Stone 2, you can get there two ways: one stone at a time, or advance two spaces at once. As for Stepping Stone 3, you could advance once three times, advance one then two spaces, or two then one space.
More generally, the total ways you can reach a stepping stone is equal to the combined total ways you can reach the two stepping stones previously. This is because you can advance one or two stones at a time, so one of the two stones behind it must have been your starting point. So starting on the shore, we can see the number of ways to reach the stepping stones in sequence is 1, 1, 2, 3, 5, 8, 13...
These numbers are known as Fibonacci numbers (for the nerdier readers among you). And the twentieth Fibonacci number is our solution - 10,946.
Never Sit In A Restaurant Five Minutes Before it Closes
Dragging the last of the ingredients back to Sugarcube Corner the very next morning, Pinkie chats with Mr and Mrs Cake about her leave of absence, and gets back to business as usual, but not before asking you to come back at closing time so you can work on the cake together.
After a seemingly uneventful day around Ponyville, which only ended with one arrest on your part, you decide to show up a little early to see how Pinkie is doing. As it happens, she seems just as eager to start, but before she can start daydreaming how such a harmless-sounding cake such as the forbidden and mysterious Cake of Diablos would even taste, three ponies walk into Sugarcube Corner, asking for some of Pinkie's specially frosted sponge cakes.
With Mr and Mrs Cake having gone out ten minutes before closing, Pinkie realises that she's going to have to work overtime just to get all three cakes ready for the customers, until suddenly she spots you. Oh dear, the gleam in her eye tells you everything you need to know, and Quelle surprise, you've suddenly found yourself as an unpaid volunteer worker for the next three customers.
Thankfully, one of your jobs back in Canterlot has left you not unfamiliar on how to frost a cake, and you reckon that you should be able to keep pace with Pinkie at her best: one minute to prep the frosting for each sponge-cake, and another three to frost it over: a four minute job. Unfortunately, Pinkie isn't fond of overtime (or at least she isn't today), and with only eight minutes before Sugarcube Corner closes, she does sort of want the job done as fast as possible. Working together with Pinkie, can the pair of you frost three cakes with enough time left over to close up at precisely 8pm?
Having finally served/chased away the last customer, Pinkie wraps up cleaning and then drags your reluctant carcass down into her 'mad scientist retreat' right inside a huge cavity she dug right underneath Sugarcube Corner. before you ask, yes, she did check to see if it was safe from a construction standpoint before she hollowed out an enormous room underneath a building. Only a few moles sustained any injuries from the sudden fall.
With all the facilities set up ready to make the cake of Diablos, she sets up the exotic ingredients, brings out the recipe, and basically starts ordering you around the kitchen. However, it is in the middle of handling the 51R3N vanilla extract that she realises a small problem: according to the recipe, the butter must be added to the cake mixture while 'infusing the spirit of a true learned air. Mix until soft and fluffy'. Unfortunately, as there is no way for her to take knowledge and bottle it up (Twilight Sparkle might know, but she's not here as it is and it would be awkward to explain why you're breaking the law underneath a small confectionery shop) it falls to you to add the mixture. Apparently.
She therefore takes all 12 pints of vanilla extract, pours it all inside a measuring jug that goes all the way up to 12, and then taking out two smaller measuring jugs, she decides to set you a puzzle, hoping that your feelings can be transferred into the cake. Feelings are not an exact science you admit, but you suppose it isn't called the Magic of Friendship for nothing. Friendship can make rainbow-coloured laser beams. Anything is possible at this point.
Rolling your eyes at the puzzle in front of you, you immediately realise Pinkie gave you a classic. You had hoped for something harder, but the low props available and the immediate need to handle the N1GHT3AR3 300N butter 'with extreme care' has limited options. Nonetheless, you have your puzzle, now get cracking.
You have 12 pints of vanilla extract in a jug, enough to fill it all the way to the top. You also have a 8-pint jug and a 5-pint jug, and you need to separate the extract so that you have 6 pints of extract in two jugs each, by pouring the extract in the jugs one at a time. To make things harder, Pinkie has erased all markings on the jugs, so all you have to go on is the knowledge that you must pour until the jug is totally full to the brim.
Yawn. But nonetheless, it would be embarrassing to fail here, so get cracking, friend. Seperate the extract into two lots of 6-pints, and then pour in the batter.
The fastest method requires seven moves. SisselSanvich details the method below:
"At the start, the 12-liter jug is full and the others are empty. I'll notate this as 12/0/0, to make it easier to track. Pour from 12 to fill 8. (4/8/0) Pour from 8 to fill 5. (4/3/5) Pour all of 5 into 12. (9/3/0) Pour all of 8 into 5. (9/0/3) Pour from 12 to fill 8. (1/8/3) Pour from 8 to fill 5. (1/6/5) Pour all of 5 into 12. (6/6/0)"
In Soviet Russia (Or Pinkie's Laboratory Kitchen At Any Rate)...
You pour the divided vanilla extract into the batter, hoping that your feelings (or perhaps lack of it) will make it into the Cake of Diablos. Giving it a good mix, Pinkie pours the mixture in the cake tin, and sets the temperature at 20% Above Reasonable degrees, for 20 and pi minutes exactly.
It isn't until Pinkie tries and removes the cake from the oven that you encounter a slight setback. Namely, the cake has burst out of the oven in a blind rage and is now attempting to devour both you and your pink miscreant.
Grabbing the cliff-notes from the kitchen counter as you try and flee the baked monstrosity, you discover at the back page that the Cake of Diablos, despite its perfectly harmless sounding name, is in fact a secret project from a long-past war, to design a food for the enemy comprised of the most EVIL ingredients in existence.
Now that it mentions it, that explains why such ingredients like W1ND3G0 Flour seems suspiciously similar in spelling to Windigo Flour, but in all fairness, if you weren't smart enough to figure that out, you would have stopped long before this ever got out of hoof.
Also, what sort of cake mix comes with the instructions 'beat thoroughly with silver-spiked club', anyway?
You can't die here: Pinkie won't pay you for your job otherwise, so there's no recourse to follow other than to slay the rascally cake-monster once and for all. As it happens, lying around Pinkie's kitchen is a laser beam death ray: you don't ask where or why it's there, because client confidentiality and everything. It's mounted directly in the centre of the ceiling, and you plan to slice the monster into two to put the feral beastie down - and cook the pieces well done, like toast. Taking stock of your surroundings, you note that the kitchen is, bizarrely, in the shape of a pentagon, and you reckon the laser is maneuverable enough to allow you to slice through the kitchen anywhere, as long as it's right down the middle of the pentagon.
Warning Pinkie that it's about to get hot in here, you advise that she first move all precious belongings that the cake hasn't already absorbed into its gooey body out of the kitchen before you fire up and launch the laser. Problem is, you don't actually have time to tell Pinkie which parts of the kitchen are at most risk of being fried to a crisp, so it's up to Pinkie to figure that bit out for herself.
Pinkie better move her hooves. Where are the places most likely to be hit by the death ray?
In a room shaped like a pentagon, the area bisectors (in other words, the points where the shape is cut into equal halves) intersect there, so the centre is the most likely place to get zapped. The direct edges of the kitchen are the safest place to be, with the direct middle of each side being the safest, assuming no knowledge of where the laser will strike.
Now, when you have a shape with four sides, like a rectangle, the bisectors are directly in the centre, making that the most dangerous point. A pentagon, however, is a different story. There are places where the bisectors intersect, but they don't intersect just at one point. Examining the possible places where you can start firing the laser, you'll discover that mapping the dangerous spots actually generates a shape: a sort of curvy star. Stay away from the curvy star. At all costs.
The exact maths involved is way too complicated to list here, but here's a diagram to demonstrate:
Unfortunately, and perhaps rather inevitably, your little stint and subsequent unleashing of baked evil hasn't gone unnoticed, and Twilight Sparkle has since discovered your collaboration with Pinkie and reported it to the proper authorities - namely, your employer Princess Celestia. Despite all pleas, wit, and reason, you and Pinkie are to be sentenced by her personally. Thankfully, some of the batter escaped during the chaos and did much to help Ponyville with its local gingerbread spider outbreak, so your punishment has been reduced somewhat, but Princess Celestia has decided that the most appropriate punishment for you is not only a 5% pay cut (which is already pretty bad as it is) but also to help out the remaining group of Twilight's friends Fluttershy, Rainbow Dash and Twilight Sparkle in whatever capacity you can. You cannot envision anything more ironic and cruel, to work for such agents of chaos as these ponies.
First things first though, there's the matter of the cake itself. The recipe will be sealed away in a much safer location (and they don't come much safer than the moon), since it was in your defense a delicious cake, and too good to destroy. For your part in actually baking it and yet paradoxically nearly unleashing pure evil, you only get a slice if your common sense can prove it hasn't abandoned you entirely.
Which, naturally, has led you to be locked inside a small room with twenty-nine other ponies. Staff around Canterlot Castle, apparently, you realise.
Princess Celestia then pitches the problem to you. All of these ponies work around the castle in different jobs, such as cleaner, chef, royal back-rubber, tame cat-burglar etc. but for all their skills, they keep misplacing all the tools of their trade. This has caused no end of frustration, and so she's decided to deal with this sensibly and calmly: by locking you in a small soundproof, windowless room with the threat of job termination hanging over their heads. As you do.
Fortunately, she is not without a sense of fair play, and has managed to find all thirty items herself, but she's sealed them away randomly in thirty different boxes. They all have a picture of one of their tools, but the picture won't always correspond with what's inside.
Princess Celestia will then let one of you out at a time. That pony can then look through fifteen boxes each for their own tool, but they may not remove what's inside, mark the box in any way, pick up the boxes to check for weight, or communicate any clues to the other ponies: you're only allowed to open the box, look inside, then close it again. If all the ponies can find all their tools, then Princess Celestia will keep them on: otherwise, a castle this huge could definitely never have too many interns - she could do with a new secretary to the brigadier.
This royally sucks, most of the ponies conclude - by picking fifteen of the thirty boxes randomly, they only have a 50:50 chance of finding their precious item, and multiply those odds among thirty ponies, and for all thirty of you? An astronomically low 1/1073741824 chance.
As impressive as the fact was that you managed all that in your head, they don't seem any closer to being able to find their things. It now falls to you to ensure that you and the other ponies get to keep their jobs and that you survive to carry out the rest of your punishment: more life-and-death situations in a small country town, helping out three simple ponies. The horror. So what's the plan, chuckles?
"Open the box that has the picture of your tool on it. If it's inside, stop. Otherwise, look at the object, then open the box with the corresponding picture of it. Repeat until you find your item or you've searched fifteen boxes, whichever is sooner."
By searching the boxes this way it forms a closed loop: starting with the box with the picture of their item, and ending with the box actually containing it. Had they continued on, that would just take them back to the start.
By searching in this loop, they restrict their search to only the loop that contains their instrument, and the odds of the loop being fifteen boxes or less tends out at about 30%, and although that's not perfect, it's way better than just picking fifteen random boxes and going from there.
Part 5: The Animal Caretaker And The Logician Went In Two By Two
Fluttershy, AKA Sorry, Speak Up AKA Aaaaaaw, could very easily be mistaken for an earth pony with two wings strapped to the side, given her pretty ground-based lifestyle - one that she grasped with both front-hooves, having found companionship with several equally ground-heavy lives, her animal critters.
She's not the type to ever want to take sides, but admittedly if she were forced to choose between something like a penguin, or a cow, almost nopony wouldn't choose the penguin. they're cute. if a penguin waddled into your kitchen, you would be surprised, yet delighted. A cow could never generate the same reaction.
That, however, has seemingly nothing to do with why you've been sent into the Everfree Forest for your imposed punishment. There is an admittedly large margin for unwanted bedlam and chaos (actually, said potential would circumnavigate the planet about five times over, owing to her friendship with Discord, an embodiment of chaos, but you've been promised that he's away on vacation), but for somepony who's as reluctant to get involved in any fuss as you, you figure that the first week of your punishment should be mostly peaceful. That is, until Fluttershy announces that she needs to do her annual checkup on one of her most difficult of patients: Graham.
Who Graham even is, Fluttershy is reluctant to say, but apparently he lives in the large cave at the other end of the forest, and you need to work with her to coax him out and take his lumps like a good boy. Sounds positively joyous - you don't think. Well, medical procedures wait for nopony, so let's get started...
The first part of getting Graham ready for his checkup is to first procure the right tools for the job. With that in mind, Fluttershy takes you into her cottage to explain how she's going to go about it.
Graham, as Fluttershy explains, reacts strongly to a rather specific smell, and unfortunately, the medicine she needs with that exact aroma is starting to run rather low, forcing her to special-order some more. However, such a thing is not so easy to pull off, for there are several obstacles in the way.
Her supplier explained via their supply of transporting dragon-fire that they need to get the medicine to her, but ever since the terrible Spam-thrax attacks of...well, a week ago, the top-secret organisation known as the Ponies In Black (not to be confused for the organisation made up of ponies with black coats, the Ponies OF Black), and the store that makes the medicine can't really send any package out, out of fear of having to explain why they need to send such strong stuff out to an animal most ponies aren't sure is legal to keep, which would be bad. Very bad.
Unfortunately, email hasn't been invented in Equestria, and both sides lack a Spike to send packages that are not letters via dragon-fire, so to beat the most rudimentary delivery service of them all - the post office - the suppliers have a cunning plan. They intend to send the package to Fluttershy, secured by a Padlock of Chaos, a creation of Discord's that, should it be tampered with, releases an incredibly potent toxin that turns all who tamper with it into rubber chickens. Along with all their friends. And pets. And Prince Blueblood. Such a padlock is also designed to unleash the gas if the package itself is tampered with too. So needless to say, it's secure as heck.
However, by the same token, how is Fluttershy even supposed to open the package? The supplier cannot send her the key, otherwise it falls into the hooves of the PIB. And the PIB's reach extends pretty darn far, so Fluttershy has to assume that should anypony concoct the most fiendish plan, the PIB will know about it and react accordingly as soon as it is explained.
With the only means of communication being their own personal supply of dragon fire exclusively for letters, how can Fluttershy receive her shipment of medicine without the medicine itself falling in the hooves of the PIB? They're so good, they can intercept dragon-fire mail, so assume whatever plan you concoct will be picked up by the organisation, so make it good.
First, the supplier fits the package with a padlock and sends it to Fluttershy.
When Fluttershy gets it, she now does something counter-intuitive: she secures the package with ANOTHER padlock of her own. This sounds crazy, but that's where the genius step 3 comes in.
Fluttershy then sends it back, where the supplier can now remove their own padlock, then send it back once again. After that, all Fluttershy needs to do is remove her own padlock, and she can open it up and get Graham's medicine!
You might wonder how Fluttershy would even know to do this, however. Well, simply put, the supplier can tell her whatever he likes, because even if the PIB knows about the plan, without knowing the combinations to the padlocks, they can't do a thing about it. Bad one PIB.
Having finally procured a series of medical stones, several powders, and a bottle of what smells suspiciously like barbeque sauce, the next step comes into play. Fluttershy explains to you that recently, Graham has come down with a bit of joint pain in his arms, and so a special brew needs to be concocted to restore vitality. Unfortunately, it's beyond Fluttershy's level of expertise to brew, and so, as is the case with anything involving potions, help from the zebra Zecora is required.
You drag all the ingredients to Zecora's hut, where she sets up a cauldron for the brew. The good news is that it's possible for her to make. The downside is that the brew requires a very specific ingredient to brew properly, and although you have a lot of it, not everything is the right quality. Specifically, the medical stones (they look a lot like kidney stones, but that's neither here nor there).
The recipe calls for stones that are twenty grams or more, and while the supplier has generously assured Fluttershy that every medical stone provided does weigh a whole number of grams, all of them look exactly the same: the only difference is in the weight. Fortunately, although Zecora seems to forgo the new-fangled things like the electronic scales that Canterlot chefs (and Twilight Sparkle whens she hits a 'mad scientist' phase) seem so very fond of, she does have at her disposal a set of old-fashioned balance scales, along with enough weights, weighing 1-40 grams, to go round.
As much as it would be easy to simply set a twenty-gram weight on one side and just use whatever stones are equal or heavier, Fluttershy seems insistent that every medical stone is weighed individually and recorded, so that the subsequent data can be used for next year. As much as the thought of doing this again next year makes you shudder, it's all part of your self-inflicted punishment, so you have to go along. So, what's the most efficient way to weigh all of these medical stones?
You need four weights: a one gram weight, a three gram weight, a nine gram weight, and a twenty-seven gram weight.
How it works: you measure a 1g stone with the 1g weight. To weigh out 2g, you place the 3g weight on the opposite end of the scales, then the 1g weight on the SAME side of the scales. 3g is easy to measure, so for 4g you combine the 1g and 3g weights on the opposite end, for 5g you place the 9g weight on the opposite end, then put the 1g and 4g weights on the same end, and so on down the chain.
To make this into an easy to swallow formula, the stone's weight must be equal to the total weight on the opposite side of the scales (let's call this B) minus the weights on the same side of the scales as the stone (C):
A = B - C
More Complications (As If You Weren't Going to Get Any)
Graham's special anti-arthritis meds have now been successfully brewed, and Fluttershy must now move on to a special set of pills that aid in his digestion (he does suffer from rumblings in the night so). However, as is becoming rather disappointingly normal for you, there is, of course, a problem.
The same supplier of Fluttershy's medication has sent along ten bottles of the indigestion pills to her, each one containing the exact same number of pills - attention to detail is everything - but unfortunately, it seems that a rather zealous bunch of environmentalists and slightly mad corporate lawyers broke into their shed and contaminated one of the batches to agitate the symptoms of indigestions right before they mailed them to Flutters. The only way to tell a contaminated pill from another, the note that came with the pill says, is that the intact pills weigh ten grams, while the contaminated ones weigh eleven.
Before you posit that you simply weigh the bottles, as the extra milligrams would surely quickly add up, Fluttershy rather embarrassedly tells you that Discord seemed to take a fancy to the bottle, and made its weight completely indistinguishable from the others - while also just feeding them to unsuspecting Ponyville residents and then blocking all the public toilets. As, you know, the Spirit of Chaos is want to do.
Unfortunately, Zecora's old fashioned pan scales don't go that precise, and so the two of you are forced to use the only other set of electronic scales in Ponyville - at Twilight Sparkle's personal secret lab under the castle, that she denies exists to the public.
Always willing to help a friend out, the princess is of course giving permission to use the scales, but sensing a bit of an opportunity to test you further, she decides to spring a puzzle on you. She takes the ten bottles of meds Fluttershy brought along, measures out ten equal batches, and sets you the challenge. Due to the fact these scales are brand new and she doesn't want your dirty little hooves to damage it too badly, on top of the fact she still hasn't quite forgiven you for the whole cake of evil and death and stuff, she will only allow you one weighing to identify the contaminated batch of pills she has set out for you.
Better solve this one, or else she'll hang this over your head forever. Probably accompanied with some highly immature taunts and raspberries. So, how can you identify the dodgy set of indigestion pills with only one weighing?
Take one pill from the first batch, two pills from the second batch, three from the third, and so on until you've taken ten pills from the tenth batch. Weigh.
The total weight of the pills, assuming they are all uncontaminated, is 450 mg (1+2+3...+10). The key here is to see how much heavier the actual pile of pills is compared to the expected value.
For example, if your total weight is 455 mg, that's because five of the pills are contaminated and thus heavier. It therefore must be down to the five dodgy pills from your fifth batch.
Having finally managed to get Graham's medicine all ready for his annual checkup, the pair of you set off to the edge of the Everfree Forest to look for Graham. Eventually, after a brisk walk involving small talk and fending off several hungry manticores, you arrive at your destination. And, in a moment that would make you want to face-hoof at the sheer improbability of it all, you find that conditions are perfect to hit you with yet another puzzle.
Graham, as Fluttershy explains to you, has taken residence in a series of caves, inside a mountain. However, by a frankly startling coincidence, the mountain appears to be shaped as a layered pyramid of sorts. Nopony knows exactly how the mountain came to be that way, but current thinking favours extremely bored pegasi who wanted to see if they could just use rain, wind, and the march of time to create something so neat it couldn't have happened by chance. Or aliens. probably aliens.
Anyway, this pyramid of caves is arranged so that the inside of the cave on the top layer is one room, the layer underneath containing two rooms, the third containing three rooms, and so on and so forth. Graham's place of residence on the mountain is on the floor that satisfies the following conditions: all the rooms in the caves have exactly three doors, that lead to another room, all except the one where Graham is holed up in, which connects to only one. Fluttershy assures you that you won't need to climb any slopes, ledges or stairs to get to where Graham is, as being a pegasi, she can fly you up herself (probably), and will need to be present for his checkup anyway. Also, there are no passages connecting rooms.
You're going to have to do it as discreetly as possible though. Graham has more sensitive hearing than you'd think, and if you stomp around the caves, he'll surely try to escape his doctor's appointment and become a bit violent. Therefore, you'll only have time to search one of the layers of the pyramid cave before he detects your presence. This does absolutely nothing to reassure you that Graham won't be something harmless like a bat or something, but you feel that a bit of logic should compensate nicely.
You ask why Fluttershy doesn't just visit Graham on her own, but she explains that usually Angel Bunny goes with her, but this year he said that he had enough years taken off his rabbit life accompanying her to these visits, and no amount of cherries on top his lunch will ever make up for it.
Well, darn. Fluttershy, unfortunately, didn't draw up a map for you to follow, but nonetheless, you resign yourself to the unpleasant sounding job. So where the heck is Graham (whatever he is)?
Several commenters brute-forced their way through, which to be honest is probably the best way. We know Graham's room is on a floor where at least one room links to three others, including Graham's, making a total of four rooms, so that automatically eliminates the first three floors from the top with only three or less rooms. This also eliminates the fourth from the top once you've drawn out a plan: the non-Graham rooms,once connected, leaves us with no connections left.
The fifth floor doesn't work either, because once you draw out the rooms using a bit of graph theory, you see that all non-Graham occupying rooms use up their three connections linking with each other: you'd need FOUR to accommodate them:
But lo and behold, this array does work:
The Fifth Puzzle (So the Part Where It Goes To Tartarus, Basically)
Having located Graham's hideout, Fluttershy then takes the alarming step if smothering your body in what smells rather oddly like barbeque sauce, and then tying a rope around you, suspended on a wooden rod. She then lifts you into the air and dangles you at the mouth of the cave.
Suddenly, a roar sounds out, and you discover what it is that Fluttershy is so insistent on not going to visit alone.
Graham, as it turns out, is a tyrannosaurus rex.
Well. That explains so much and yet raises so many more questions.
But before you can finish screaming as Fluttershy attempts to slowly coax Graham out of the cave, suddenly you slip out of the ropes and land rather ungracefully on the floor. Unfortunately, your ropes weren't as secure as you'd hoped, and you soon find yourself being chased by a rather hungry and annoyingly fast dinosaur. Both of you quickly leave Fluttershy in the dust.
The chase continues until you reach a lake. By a frankly startling coincidence, you find a small rowing boat by the shore, and not wanting to look a gift pony in the mouth, you jump in and start rowing like the clappers, leaving Graham frustrated at the shore, unable to swim. However, as ill fortune would have it, as you row straight to the middle of the lake, you hear a rumbling sound, and the ceiling is starting to slowly but surely bear down on you. As if your day couldn't get any worse. Shouldn't have thought that, because at that exact moment, you spot what are unmistakably shark fins peppering the surface of the lake. And no, they don't belong to unusually large kippers.
Trapped by the lowering ceiling above, and the sharks in the lake below, death by shark-nibble seems imminent.
However, you've studied such traps under the tutelage of Princess Celestia and Twilight Sparkle, and so you know that such traps are typically handled by a series of crystals in a runic shape. You look around, and sure enough, you see a series of azure crystals lining the sides of the lake. You have, as per usual, been trapped in a puzzle most fiendish. To retract the trap and raise the ceiling again, the crystals must make contact with water, and so you have to find a way to raise the water level somehow!
Unfortunately, with the way the trap seems to be arranged, all of the crystals must register contact with water at the same time, so you cannot rock the boat to make waves to trick one of the crystals. Unfortunately, you have absolutely nothing else on your boat you CAN use, save your carnivore-attracting saucy body and the oars you rowed with to get here! So using the seeming nothing that you have, how can you raise the water level and avoid death?
Dip one of the ends of an oar with the sauce covering your body, then use it to lure the largest shark in the lake onto the boat.
Sharks have no swim bladders and are denser than water - they sink if they stop moving. As they're denser than the surrounding water, they only displace water equal to their volume while in the lake. But by luring one onto your boat, the shark's weight is now pressing down, and assuming this doesn't sink your boat, the shark's weight will be supported by the boat, and it will now displace water equal to the shark's weight - more than it would were it to remain in the water.
With more displaced water, you can now reach all the crystals and stop the ceiling bearing down on you. Congratulations - for now.
Back to Ponyville, No Longer Smelling Like Barbeque
Just as the ceiling retracts back to its original stance, you catch Fluttershy, having finally caught up tp you and Graham, before scolding the enormous dinosaur for skipping his appointment. To your rather unsurprised disappointment, she doesn't blame Graham for trying to eat you. Her fault, really. Asked for a really delicious sauce on this occasion.
Once Graham has taken his pills/been vaccinated/cured of his irritable bowels, you and Fluttershy make the long trek back through Everfree Forest (making doubly sure to not walk downwind of any hungry manticores). She thanks you for all your hard work, but really, all you want to do at this stage is just head back to the hotel you're staying at, sleep this off, then finally go see what Rainbow Dash wants the next day. Shudder...
But Flutters isn't willing to let you go just yet: she actually has the keys to your room, and Princess Celestia has informed Fluttershy that you first need to solve a rather annoying puzzle before she hands it over, thus confirming Celestia as probably the most heartless and vicious ruler of Equestria, forever cemented in your mind.
Once you arrive back at her cottage, she explains the rules to you. Basically, you and Fluttershy are going to play tic-tac-toe. You scoff at the prospect of playing such a simple game, but Flutters does mention that the game comes with a few clauses.
For starters, you will be blindfolded, and won't be able to see the board or your opponent's moves. You will play by announcing which square you will put your nought, and then Fluttershy will place her cross somewhere, but only she will know. In the event you try to place a nought on an occupied square, you will be told so and must call another square until you reach one that isn't occupied. For Fluttershy to hand over the key, you just need to avoid losing ten times in a row. Fail once, and you sleep outside Fluttershy's cottage in the cold tonight. She'd love to invite you inside, but even she, the ex-holder of the Element of Kindness, has to be cruel to be kind, as Celestia told her before you arrived in Ponyville for your exile.
She will, however, allow you to go first in every game. Just as a sporting token of kindness on her part.
The thought of sleeping out where you could be eaten by anything (even the plants under your hooves) is a rather unsettling one, so better get those grey cells working, and fast. So how can you earn the key back from Fluttershy?
There's actually more than one strategy you could employ for this. Three, in actual fact, depending on where you start on the board.
It's easier to find a strategy assuming Fluttershy will not be playing optimally, because otherwise constantly second-guessing could lead you to a loss. Now, obviously, as you can't see Fluttershy's moves, you're going to have to lock her into blocking your plays to make her moves more predictable.
Let's label the squares from 1-9, as follows:
1 2 3 4 5 6 7 8 9
The first strategy can be summarised as "start with 5, then move down the left-most branch on the diagram below. If you're blocked, move to the next branch along and so on". Coloured circles below represent draws, all others represent wins:
Here's a second strategy where you start with the corner square 1, following the same rules as the first diagram:
Finally, here's another strategy where you start with square 2, and then follow the steps going right, moving down if you're blocked:
Rainbow Dash, AKA Dash AKA Skittles, is the latest Wonderbolt in training and self-proclaimed fastest flyer in Equestria. Faster than a speeding train (though to be fair you don't get to be a Wonderbolt when you're that slow), stronger than Applejack's cider meant for the grown mares, and able to casually break the visible light spectrum at Mach 10, there's no doubting her skills.
Impressive as it sounds however, none of that happens to have anything to do with why she's enlisted your help. For you see, she seeks to get justice for the wrongs that have been inflicted on her. Oh she'll rue the day, Rainbow swears. Oh how she will rue.
Basically, she wants you to defeat Black Bird, some jerk around Ponyville, in what appears to be some sort of prank war/feud. A prank feud, if you will. She is not, technically, at war with Ponyville as a whole.
It all started when they bumped into each other in a designated low-flying zone, and both claim the other was flying the wrong way. Eventually, it got to a point where they just waited for the other to move (Rainbow was carrying a bag of chips and so decided that Black Bird would starve first) until finally Black Bird just punched her way through.
And so it falls to you to deliver sweet, sweet justice by applying your wonderful, logic-deciphering noggin...to break this pony until she can "stands no more", as Rainbow puts it.
This cannot be legal...but Princess Celestia didn't close that loophole, and so deciding to blame her if things go south, you embark on this dangerous undertaking...
In order to avoid sullying the good name of the Wonderbolts by being caught by the authorities, Rainbow Dash's first plan is to anonymously make a large donation of about 1000 tons of Prism Popper candies directly onto Black Bird's house. No, she doesn't mean TO her house. She knows exactly what she said. And by her, she means YOU will anonymously make this rather generous in metric amount by hugely ungenerous in spirit contribution. Oh joy. You're lucky Equestria lacks the means to track deliveries online here.
However, a prank worth doing is worth doing well, and so Rainbow Dash wants to actually send as close to 1000 tons of Prism Poppers to her nemesis as possible. Only problem is, she doesn't actually know how much one individual Prism Popper actually weighs, and while cracking open a pack to check would be simplicity in itself, Rainbow also wants to pack the candies into bags and glue them shut, denying her the sweet treats within, as insult to injury. She happens to have a large pile of packets for you to weigh with Twilight's sensitive electronic scales she stole...borrowed without permission with the intent to return it, but owing to the rather disturbing lack of quality control at the factory (and the fact it's moonlighting as a smuggling organisation on the side), there are some kinks.
Namely, none of the packages Rainbow has contains the same amount of Prism Poppers inside. Aside from that, each packet and all of the delicious contents are identical in every way. Unlikely, but you've been in some pretty unlikely scenarios yourself at this point.
So without being able to actually open a packet to check, how can you determine the weight of a single Prism Popper? For the sake of irony compared to the last time you were allowed a go on the scales, you get unlimited weighings this time.
Record the mass of every single pack. The smallest difference in mass between any two readings must be equal to the smallest difference in amount of Prism Poppers - in other words, the mass of a single sweet.
However, this only works if the number of sweet packets is large enough for you to be sure that you'll have packs with consecutive whole number totals.
As a result of dumping delicious candy on Black Bird's house, Black Bird responds with what many would widely consider to be a rather disproportionate revenge - she's mailed what appears to be a small ticking box to Rainbow's house in Cloudsdale, with the instruction to glue it directly to her Wonderbolt uniform. Sure the whole candy thing can't be traced back to Rainbow, but Black Bird knows she's behind it. She does.
I mean, all you did was crush her house under a huge pile of citrus sweets. A bomb's a bit much. Probably. Justice is relative, after all.
Unfortunately, despite the best efforts of the local authorities (the bomb disposal dogs were foiled as everypony remembered dogs are green and red colourblind) and so it looks like you're going to have to do it yourself. Using only your rudimentary skills in bomb disposal from the lessons at Fancy Pant's last dinner party (don't ask, the tale is too disgusting for small ears), you set about the task.
Fortunately, Black Bird doesn't appear to be completely without a sense of fair play, so attached to the bomb is a note, saying that you can only disarm the bomb by following chess and an intimate knowledge of queens. Chess is a boring game for eggheads like Twilight, so it should suit you perfectly, Rainbow reasons. Gee thanks, Rainbow.
After studying the bomb for a bit, you realise that in order to shut it off, you need to hit the squares on the grid, which is shaped and coloured exactly like a chess board, to place eight queens on the board. However, absolutely none of the 'queens' are to put anywhere on the 'board' that would allow each other to be taken, if one were to follow standard chess.
Despite the fact that this stinks of a trap, you set about disarming the bomb. Unfortunately, you spent so much time studying it that you only have about ten seconds to enter the solution and hit the switch to disarm. Move those hooves, rainbow's place of residence depends on you!
You could probably solve this through trial and error, but you can get a good basis for the solution decently quickly with logic. For each queen to not attack each other, you'll have to put them in their own row, both horizontally and vertically. We also know that the queens will be sharing four black and four white spaces between them, because any queens sharing the same colour can be attacked, if arranged improperly. To that end, here are two solutions I found:
.
Golden_Scroll found a third solution, described following the standard chessboard pattern of A1 being the bottom left corner and H8 being the top right: "A6, B4, C7, D1, E8, F2, G5, H3."
Unfortunately, in a rather irritating case of genre-savviness, the bomb was, in fact, designed to go as soon as it was successfully disarmed, and now Rainbow's home is entirely flooded in strawberry jam. That hungers for pony flesh. As jam does.
While the cleaners are trying to hoover it all up, Rainbow plots her revenge. Unfortunately, most of the budget for the prank war went into the purchase of the Prism Poppers, and so you're pretty much on a shoestring. Fortunately, Rainbow does at least have one idea, considered by many to be the most heinous scheme ever conceived: spam her letterbox. As it happens, Wonderbolts such as herself receive a lot of fanmail, and she in turn has access to a lot of stamps, letters and other items to reply to any at any time back at the Wonderbolt Academy. She therefore decides to mail Black Bird a series of seemingly useless letters, filled with only seven digit numbers, direct to her mailbox, in different fonts from a typewriter, so she doesn't realise they are from the same pony, and hopefully either it will annoy her, receiving so much junk all at once, or result in her ignoring an important tax bill out of sheer frustration and get arrested for fraud, or something.
Gathering together all the supplies she got from the Academy for free, she decides to keep you occupied with a puzzle, to keep you from going stir crazy writing out idiotic letters. She's writing down different strings of seven numbers, basically to see if Black Bird tries to find meaning in these numbers that doesn't actually exist, but what Rainbow wants to know is: if she had some sort of machine that randomly assigns seven figure numbers to each of you, what are the chances both of you would wind up with an exact scramble of the other - that is to say, each number appears with the exact same frequency, but in a different order from the other? Basically, if 5 appears three times in Rainbow's number, it will appear three times in yours as well.
For our purposes, a number can have digits in different combinations. All seven digits could be different. Or maybe they're all the same. Or maybe they have two digits that appear three times and a third only once. There are fifteen such combinations available.
For each type, we need to work out how many of the ten million numbers you have fall under this configuration, which gives us the probability Rainbow Dash has it. For example, there are 1,058,400 seven-digit phone numbers in which one digit is repeated three times, another appears twice, and two other digits appear once each. We could call that particular configuration 3A 2B C D — digit A appears three times, B twice, C and D once.
Then, we need to figure out how many exact scrambles there are. We can calculate this through the sum 7!/(3!2!1!1!)−1, for a total of 419 exact scrambles. (We minus 1 because we're not counting the ORIGINAL number.)
Finally, we can multiply the number of exact scrambles associated with each configuration by the number of your numbers with that configuration, summed up over the 15 possible configurations, and divided by 9.999999 x 1014 — that’s the number of possible numbers YOU have multiplied by the number of possible numbers RAINBOW has (one shy of 10,000,000 because one of your numbers is already taken).
It takes a while, but multiplying the possibilities for your numbers for each configuration (from 7A down to 1A 1B 1C 1D 1E 1F 1G) and multiplying them by the possible combinations for Rainbow's number, and then dividing by (10,000,000×9,999,999) gets us 0.0001676, or about 0.017%.
The whole spam-letter thing was indeed a clever prank. Rather unfortunately, the follow-up response on Black Bird's part proves to be somewhat irrefutable - she knocks you out with sleeping gas, marenaps the pair of you, and now you and Rainbow Dash have found yourselves locked in a small dungeon somewhere, with what appears to be a small, but very long table in front of the pair of you. The accommodations seem pretty roomy, but someone ought to plug that infernal leak. And there's too many spiders. Gingerbread spiders. Well, at least you won't starve. You hope.
This is why barbarians are so hard to argue with - there's no clever answer to death. Or worse.
After determining to your dissatisfaction that there are no windows and the only securely barred door cannot be opened, you spot a note right on the table's left end.
Greetings, you blue troglodyte and you ordinary looking royal flank-kisser, it reads. I have arranged for you to play a little game of mine, and I'm pretty confident that Rainbow Crash isn't lucky or smart enough to win, prepare to be stuck forever while you struggle for my amusement.
Here are the rules: on the table right in front of you are a hundred coins, in Equestrian bits. The values of each coin is different from the others - some are worth one bit, others two, and so on. The pair of you will take turns, picking off coins from each end (you can freely choose which end every turn, of course), until the last one is gone. Whoever has the most bits at the end of the game wins.
But I'm not that nice. The pair of you only get to go free if Rainbow wins the game, and I am trusting you to play at your best, Mister Royal Dispute Settler On Probation, because I'm watching, and if I suspect you went easy on her, you can enjoy rotting here forever. With the hungry GIANT gingerbread spiders in the cell next door.
And don't worry. Should Rainbow lose, the coins will randomise themselves and give you another chance, but as if Rainbow ever will win. I'll even let her go first for the first attempt.
Have fun. And don't tell her the solution, mister. I'm always watching and listening.
Yours insincerely,
Black Bird
You look up. Sure enough, there appears to be a two-way crystal ball, observing your every move. Well, horse-manure. Rainbow has to figure this puzzle out herself, and to make matters worse, you're obliged to make it as difficult for her as possible. But all the same, you and Rainbow have solved puzzles together in the past. No choice but to trust Rainbow is smarter than she appears to be and learned a few tricks. So how can Rainbow win the game and free the pair of you, so that you can continue to plot revenge in this prank-war gone too far?
The trick is to first think about the number of coins, rather than the amount of money on the table - at first, anyway. This game involves one hundred coins - an even number, in other words, and that's something that Rainbow, who's going first, can exploit.
First, label all coins from 1-100. Once Rainbow has taken the first coin at one end, all her opponent would be left with to choose from are either even-numbered coins on both ends (if she picks 1, the opponent's left with 2-100) or odd-numbered coins if she picks 100, with a 1-99 split. For example, if Rainbow wants all the odd ones, she'll pick coin 1. Whichever coin they take leaves an odd coin on one end for her to take, and then this pattern just repeats until there are no coins left. Therefore, in order to assure herself of collecting the most money, she needs to add the total value of the odd-numbered coins against the even-numbered coins, and then take ONLY those, depending on which ones have more value.
However, the strategy only works if the total value of the odd coins and the total value of the even coins are different. Even if they are the same though, the system will just reset and you can just repeat the game until Rainbow eventually wins.
To Black Bird's shock, you and Rainbow Dash kick open the door to your cell having won the game, flown up from the dungeon to the room of the ruins she's in, and you invoking Royal Protocol to arrest Black Bird for marenapping a national celebrity. And flying in the other direction of a designated flying zone. Despite all protests that you started it by mailing a shipment of candy onto her house, Rainbow blows it off - you actually used Black Bird's bank details for the purchase, but nopony else needs to know.
Frustrated at her revenge having not stuck, Black Bird decides it's time to get serious - she pulls out what seems to be a souped-up Mark V Party Mini-Cannon from her mane and decides to just bubblegum you and Rainbow to the floor. She fires and manages to bind you to the wall, but Rainbow is just swift enough to dodge in time, grab three more Party Cannons of her own, and try to fight back.
However, just as the two of them are about to engage in what must surely be the least harmful duel of the century, you hear a knock on the door, and Black Bird goes to answer. At the door is an extremely crotchety old grey stallion, who's just come round to complain about the noise. However, you can see that he's decided to deal with his extra nosy neighbours personally and with extreme prejudice - for he too is carrying a Party Cannon - Mark VII, he must be tech-savvy for an old fogey.
Suddenly Rainbow, Black Bird and the grumpy old neighbour find themselves in a truel. As per prank war tradition (it exists, for Celestia knows what reason) when two or more individuals with Party Cannons gather in one place with disagreements hanging in the air, the matter must be settled with a turn-based shoot-out. Each of you will take turns using a Party Cannon to blast one another, going in a circle - Rainbow, then Black Bird, then the old guy. That means Rainbow is going to have to choose one of the three party cannons she gathered up before, pick one of them, and start playing. Black Bird, with a hint of pride, mutters out loud she hits seven out of ten times. But the old grey pony who ain't what he used to be can hit a pony nine times out of ten.
Black Bird and Old Fogey, as you nickname him, allow Rainbow first shot, as the one with the worst aim. After some sulking over the blunt insult, you just tell Rainbow to get on with it. So, what is she going to do?
Out of the Party Cannons Rainbow grabbed, one is the Mark II which jams about 40% of the time, one's a more modern Mark IV with a jam rate of 20% and then there's the Party Rocket Launcher 2000, which will without fail generate a party for the whole family to not enjoy.
It IS worth noting however that each Party Cannon ever manufactured does come with a certain clause: after one round, if everypony is still gum-free, all the cannons are primed to explode and coat everything in gum. And cake batter. And balloons. As per standard procedure for exact situations like these, you understand.
So what should Rainbow Dash do to ensure that she has the best odds of making it out the duel without getting gummed?
Pick the Party Cannon with 60% accuracy, then deliberately miss your first shot.
First instinct would be to pick the best Party Rocket Launcher for a 100% shot at eliminating one of the competitors, but you'll soon find this strategy wanting. By picking the best weapon, you have made yourself an obvious target for Black Bird, who has a 70% chance of taking Rainbow out: the one shot would naturally be used on the opponent with the best shot: Old Fogey.
So what if Rainbow used the Party Cannon with an 80% success rate? As long as the odds of making the shot aren't the highest, you know Rainbow won't be a target until Old Fogey's out the running, but then Black Bird, again, would have a 70% chance of knocking Rainbow out, assuming she managed to actually knock out Old Fogey. If Rainbow takes out Black Bird, Old Fogey will take out Rainbow afterwards, nine times in ten.
However, what if Rainbow misses her opponent with that same 80% success-rate Party Cannon, INTENTIONALLY or otherwise? Well, then that would mean Black Bird would attack Old Fogey, the better shot, then he'd incapacitate Black Bird, leaving Rainbow free to shoot at Black Bird, and four out of five times she'd succeed...if it wasn't for the fact that Black Bird can catch on to this, deliberately fluff her shot as well, then, as your 80% success rate is higher than Black Bird's 70%, Old Fogey's going to try and take Rainbow out first - remember, he has to try and take someone out now, or else all of them lose.
The answer is therefore clear: take the weakest party Cannon, then miss on purpose. That way, Rainbow is almost guaranteed to survive after one round, then when it's her turn again. If she misses, Black Bird will have to take out the better shot now (Old Fogey) or else Old Fogey will attack back. If she succeeds, Rainbow has a 60% chance to take out Black Bird. If she misses and Old Fogey takes out Black Bird, Rainbow can take out Old Fogey with a 60% success rate, and that's the best she can do.
There IS a 3% everypony will miss and everypony gets gummed, but when all is said and done that's the most optimal strategy.
Rainbow, as the sole survivor of the duel, and thus the one that hasn't been covered in gum, cake batter and gingerbread, is forced to chew you free from the wall (and Old Fogey too I guess), and drag Black Bird to Canterlot for marenapping a national hero. She gets to now spend quality time in a dark, damp cold dungeon for years, and so she can now consider herself even for the first offense - assault - with candy destroying her house and an arrest.
And no, she doesn't really understand the phrase 'disproportionate retribution'. Too many big words. She'll ask Twilight what it means later.
However, she's spent a bit too much time away from practice with the Wonderbolts as a result of the feud, and Spitfire, her boss, has put Rainbow through her paces to make up for lost time by making her do several flying laps from the Academy to Canterlot Castle and back again, and she's not allowed a break - or to leave the Academy for paid holidays - until she can beat the Academy record.
Thankful to you for delivering sweet, sweet justice/vengeance to Black Bird, she asks you for one teeny-tiny little piece of advice. She's been thinking about trying to beat the record through strategic use of wind. As a pegasus, she can of course manipulate the weather, and was thinking of setting up storm clouds to help push her along and increase her speed. The downside is that these storm clouds will only blow at a constant speed, and only in one direction, and she wants you to figure out how best to set the wind up so that she can beat the record early.
How very devious-sounding. You wonder if Rainbow ever really needed your help the whole time - aside from commiting fraud purchasing Prism Poppers. You swear this job will be the death of you...but you digress. If Rainbow really wants to beat the record, how should the wind be blowing to best support her?
For the fastest time, there needs to be no wind at all.
Let's imagine a case where the wind blows totally in Rainbow Dash's direction for the trip outward, and against her for the return. Instinctively, the first thought is that it shouldn't matter, because in terms of speed, what the wind giveth in one direction, it taketh away in the other. If the wind speed is W, then the speed there is Rainbow's speed + W, and the speed back is RD's speed - W.
But it's TIME that's important here, not speed. Rainbow will naturally, spend a longer time flying slower than she would flying faster.
It's easy enough to prove without any tricky maths. Let's imagine that the wind speed was equal to that of Rainbow's speed. She could make the outward journey in half the time, but the return trip? Her return speed is equal to that of the wind's speed: she wouldn't actually be able to even get off the ground, because they cancel each other out!
Okay, that's a bit extreme, but you see the point. You may gain an hour by raising speed by a fixed amount, but you may lose a lifetime by decreasing speed by the same fixed amount.
What about crosswinds though? Well, they depend on how much of the wind is going against Rainbow, whether it's buffering her at a 90 degree angle or if there's enough wind blowing against her. But even then, in order to not get blown off course, Rainbow would have to exert some energy flying directly into a perpendicular wind, so naturally, some of the velocity would be spent fighting the wind, rather than travelling from A to B. So even then it will still take Rainbow longer.
That must be why Spitfire chose this demerit: there's no way to cheat that won't slow Rainbow down in the long run. It's all muscle-flying.
Part 7: Twilight Sparkle's Last Grand Rescue Adventure
Princess Twilight Sparkle, AKA The Purple Smart AKA Bookhorse, is the Princess of Friendship - and also the one keeping the closest eye of your rather unlucky reputation to fall into the most criminal of schemes. Despite your punishment being to help her in whatever capacity you can, she is insistent that she requires no help and has thus booted you back to Canterlot Castle, back to your ordinary like as Royal Dispute Settler - although you don't get to eat in the Royal Canteen with the rest of the staff. Just to remind you.
Months pass, seemingly without incident, except for that one time, but they rebuilt that pony's house and all the chickens are expected to make a full recovery, until suddenly Twilight Sparkle bursts into the throne-room.
Before you can protest that if Twilight must burst, please do it in a convenient place, but you quickly come round when she explains to you that a mysterious black fog suddenly swept over Ponyville, and has spirited away Pinkie, Rainbow, Applejack, and all the rest of her friends! To make matters worse, time has completely frozen in Ponyville, and no spell she's tried can get anypony moving again, and so in a desperate bid for help, she's come to you. YOU. By Celestia it must be bad.
She says that she'll consider everything to be even if you can help her rescue her friends and help to solve the mystery of the fog that marenapped them.
The five unfortunate victims Rarity, Applejack, Pinkie Pie, Fluttershy and Rainbow Dash are imprisoned on a crystal mountain that has devoured the town, and she believes that rescuing them can somehow shed some more light on exactly what the heck happened. This, it would seem, is the greatest test of your character, and the sum of all your time in Ponyville with them. And so, packing all the essentials to climb a mountain, like crampons, goggles, a flask of coffee and several Hint Coins you found down the back of the Royal Sofa, the pair of you set off into the jaws of danger...
Climbing the first part of the mountain takes several hours, but soon, you and Twilight come across what appears to be a cavern, with some sort of pink glow coming from within - as well as a seemingly lackadaisical humming. Identifying the voice as Pinkie's, both of you rush inside.
The scene before you is Pinkie, tied to a rope, about to be dangled into what appears to be a pit of lava. She comments that it's a bit slow...and also a bit warm in here, until the rope turns her around and she spots the pair of you. Sh'ed wave, but she's a bit tied up at the moment.
All of Twilight's attempts to try and magic Pinkie over to your side seem to fail, until you realise that the mountain itself is actively suppressing magic on every front: it would seem that wits are pretty much the only way forward in this quest. As is often the case with you, you unlucky berk, you're going to have to try and rescue her before she goes in the molten deep fryer.
Fortunately (and suspiciously rather conveniently) the rope is attached to a crane arm, which can be operated through some sort of device. A quick look at the screen in front of you shows three sequences of numbers, which you hypothesise will bring Pinkie over to your side and allow you to untie the rope, freeing her from captivity, as long as you enter the correct numbers for each sequence. The first sequence will raise the arm, the second will swing the crane over to you, and the third will lower the arm within your reach. Twilight offers to just fly up once you've brought her out the lava, but it's kind of a jerk move to interrupt you in the middle of your mojo.
Ignoring practicality for a moment, what exactly ARE the numbers at the end of each sequence meant to be?Better screw that thinking cap on, because you have a feeling these will probably be the least fun puzzles you've ever worked on...
Pretty simple to reach: all you have to do is multiply the digits to get the next number in the sequence. 7 x 7 = 49, 4 x 9 = 36, 3 x 6 = 18, 1 x 8 is therefore 8.
In the second sequence, the answer is 20.
For this sequence, you square both digits of the previous number and then add them together to make the next. So 42 =16, 12 + 62 = 37, 32 + 72 = 58, and so on down the chain until we reach 42 + 22 = 20.
Interestingly, this is where we'd loop back to the answer 4, making this whole sequence a loop.
In the third sequence, the answer is ten googol.
There doesn't appear to be any mathematical rhyme or reason here, but when you actually spell out the numbers...you realise each number in the sequence uses one more letter than the last number. Ten (3), Nine (4), Sixty (5), etc. until we reach a number that needs nine letters. But there's lots of numbers with nine letters, so which one is it?
Looking back at the previous numbers, you can see that each number in the sequence is the LARGEST known example of numbers with that many letters. So the largest number with nine letters must be ninety six...right?
Nope. There's a number that can be represented as 10100, in other words, a number followed by a hundred zeroes, called a 'googol'. Add an extra zero and we get ten googol, which has nine letters, and that's the best answer you can give.
Rescuing Applejack (With Her Least Favourite School Subject)
Having untied Pinkie, the three of you decide to venture deeper into the cave. It's very winding and rather dull, save for one moment where you travelled back in time and arrived back five minutes before you left, but the thought of rescuing Twilight's friends is the only thing on all three of your minds.
Or at least you think so, you're not quite sure from the way Pinkie seems to be babbling on about how crispy she would have been if you hadn't found her, on a scale of ashes to perfectly cooked tempura. Ashes is, like, a 1 out of 10 here.
As soon as you come out of the cave, you are greeted by the sight of Applejack, suspended unconscious inside some sort of magical cage. An electric magical cage. In the shape of barbed wire. So needless to say she's probably not getting out without help. Not even with Pinkie's electric wire trimmers. The phrase 'electric wire trimmers' has many meanings, and the tool does encompass all of them, but alas, they're just not sharp enough, or electric enough. It looks like some other way must be explored.
By the floor of the cage, however, you see some scratches and indents in the floor that look like some sort of mathematical equation. It truly makes you wants to just put a hoof to the forehead in disbelief, as it's yet another puzzle. Seriously, who keeps giving you such a sporting chance here?
On the floor is the equation abcd x 4 = dcba. Well, there's no denying it's a simple one, at least. However, trying to actually scratch out the answer repels your touch, and to make matters worse, you have frozen in place, along with Twilight. The puzzles glows in the colours of Pinkie's Cutie Mark, indicating it is her who has to solve the puzzle and allow all three of you to rescue Applejack and proceed in your quest. No giving the answers to her: your jaw have frozen in place. That's probably good news for those who can't stand the sound of you (and why would they) but there's no denying it's a bit debilitating now.
So, acting as the little voice in Pinkie's head, where her Maths Degree knowledge ought to be, can you help her free Applejack from the cage?
We can basically re-write this entire sum as (1000a + 100b + 10c + d) x 4 = (1000d + 100c + 10b + a). Now, let's simplify the entire thing down.
Let's solve a first of all. The left side of the equation is even, so the right hand side must also be even: a multiple of 4. We can also deduce a is even since 1000d + 100c + 10b is also even. That puts a limit on how big a can be, since 4000a must be lower than 9999, because otherwise we'd have a 5 digit number. And the only even number that fits this condition is 2, so a = 2.
Now let's solve d. The left side of the equation is at least 8000 since we now know a is 2, so d must either be 8 or 9. But d cannot be 9, because 4 x 9 =36, and the number on the left side of the equation ends not with 6, but a: 2, in other words. So d must be 9.
Plug both in the formula and it can be revised as: (2000 + 100b + 10c +8) x 4 = (8000 + 100c+ 10b + 2), simplified further as 13b + 1 = c.
Remember, b and c are single digits. c's biggest value is 9, so the biggest value of 2c is 18, so b can only be one, and therefore c is 7.
Forward. Four Words? Foreword? Four Wards! ...To Rescue Rarity
As the number is entered, the cage is dispelled, and Applejack rather un-majestically drops to the floor, the resulting thud snapping her awake. Swearing to give whoever is responsible an appointment with Bucky McGilliguddy and Kicks McGee, she nonetheless dusts herself off, and the four of you continue on up the mountain.
The going is slow, no thanks to the fact your stamina compared to Applejack's is terrible and unlike Twilight you can't fly, but you manage. I mean, if you didn't have any smarts then you're useless, frankly.
Suddenly, you hear a scream that unmistakably belongs to Rarity, and the four of you run towards the source. There, you are greeted by the sight of the formerly white unicorn now completely covered in some sort of black ooze. There doesn't appear to be any sort of fence stopping her from just leaving, but the fact that there doesn't seem to be any part of her that isn't caked in mud has sent the beauty and neat-obsessed pony into a rather hysteric screaming.
Apparently incapable of doing anything else, it looks like one of you is going to have to fetch her and give her a good wash. However, the only source of water nearby is a jug of water on a shelf right above Rarity's head, that can't be reached by hoof. And yes, after checking again, magic is entirely suppressed here. The only way you're going to get the jug down is to lower the shelf, which seems to be attached to a rail.
Unfortunately, to lower the rail, it needs to go down the fifty stops on the way before it reaches her. To make matters worse, it's rigged to (you guessed it) a puzzle. And judging by Applejack's glowing Cutie Mark next to it, Applejack, the...less educated mare of the current group, has to be the one to do it.
Using only four 4s, the control panel reads, can you make every number from 0 to 50? You may use the standard operators +, -, x, ÷ and (). In addition, you may concatenate digits, use decimal places, exponentiate, and use the factorial symbol !.
Twilight then has to take twenty minutes to explain to Applejack the fancy-mathematics muddying the issue before she can start. Fortunately, the puzzle seems to be solved by trial and error. The bad news is that Rarity appears to have more air in her lungs than any pony could have dreamed, and the longer it takes Applejack to solve this, the more at risk of going deaf all of you become. So please, help Applejack and save her poor eardrums.
It takes about another ten minutes to finally calm Rarity down enough that you can proceed on with your quest, but you manage to wash the mud off before your ears are wrecked beyond repair. There are only two more ponies to rescue, and with it, hopefully the mystery of the strange mountain and the unusual array of puzzles along the way may be solved.
Your next hostage turns out to be none other than Rainbow Dash, who the group encounters nearer the top of the mountain, inside what looks like a long forgotten temple. She's trapped inside an hourglass running the hour, and just to throw in a few more irony points, she seems to be coming close to drowning under Prism Poppers. And they're coconut flavoured: the absolute worst kind next to coffee flavoured. Things truly cannot be any more cruel.
Fortunately, Rarity is more than willing to step up to bat to try and free Rainbow Dash (after Applejack conclusively proves unable to break the glass with several well-timed bucks) and makes her way to a small sign. The instructions are refreshingly direct - almost disappointingly so:
Estimate 264.
Sadly, out of all the equipment you and Twilight packed for the expedition, a calculator isn't one of them. Twilight is tempted to break out her parchment and quill to write down the calculations, but is bluntly informed by Applejack that Rainbow probably won't last that long, and it says to GUESS the number. Even pinkie admits that she'd probably make a mistake somewhere if she tried it the hard way. In other words, the right logic thread will get you close enough to the answer to pass.
The real question is if Rarity can figure that out too. So how can it be done: estimating the number to a reasonable degree of accuracy?
Following logic, one can estimate the answer to be 18.4 quintillion.
The tenth term, or 210, is 1024. That's roughly 1000, a nice round number. So if 1000 is roughly 210, then a thousand times itself six times is equal to 210 multiplied by itself six times, to make 260.
10006 is one quintillion. So 260 is approximately equal to a quintillion.
The remaining 24 equals sixteen, so 264 is 260 x 24, equalling 16 quintillion.
That's not bad, but we can definitely use a bit more logic to make this more precise.
We approximated 210 as 1000. But 1024 is 2.4 percent bigger than 1000. So every time we multiplied by a thousand, we should have also added on the remaining 2.4%. Multiplying a thousand by itself six times means we should have added on six lots of 2.4%, to make about 15% total. So the answer is 16 quintillion plus an extra 15%.
Luckily, it isn't difficult to work out 15% of 16 quintillion. Ten percent of it is 1.6 quintillion, so five percent of that is 0.8 quintillion, to make a total of 2.4 quintillion, which we add on to make 18.4 quintillion.
BTW, the exact total is 18,446,744,073,709,551,616.
It takes a lot of candy munching, but eventually you and Rainbow Dash manage to eat her way to safety and freedom. She is then given a crash course on what to expect when the time comes to rescue Fluttershy, which causes no end of horrifying descriptions and images from each member of the team as you journey straight to the top of the mountain. Which is even more impressive, given it's hard to tell through the oxygen masks what you're trying to say.
However, as soon as you actually find Fluttershy, she is in fact seemingly completely unharmed, with absolutely no obstacle or nefarious trap to be found. upon being questioned, she reveals a mysterious spooky voice told her that her friends would come to collect her and was told to wait where she was.
That just raises the question of why she didn't just go back down the mountain, and she answers that she 'didn't want to disappoint whoever that was'. All six of her rescuers including you manage a perfect 10/10 in synchronised face-hoofing.
However, Fluttershy does at least manage to shed a bit more light on what the deal is with the mountain and the forces at work. The spooky voice's owner is apparently something trying to study the logic of this world, and so mare-napped the ponies who were closest to probably the most accomplished puzzle-solver in Equestria - you, in order to observe such brilliant logic in action. Oh cosmos, how they flatter you. With that said, Fluttershy also mentions that it wasn't above observing how that logic could also be taught to others, and so Rainbow Dash has been instructed to try solving the following puzzle that Fluttershy was rather graciously given written on a piece of paper:
How many zeroes are there at the end of 100! ?
At first, Rainbow is quick to just point out that they should really just grab Fluttershy and leave, but is quickly told that all who turn back without solving a puzzle will be quickly given a rather unpleasant bath in flesh-melting acid from the geysers below the mountain. Along with all their unfortunate allies. Please don't make them do that.
Having been coerced by whoever the heck it is, Rainbow settles down to think, after being told that they can go on ahead if they manage to just do this teeny-tiny favour for them. What a drag. Still, can Rainbow solve it and allow the group to meet the guy in charge?
What this question is basically asking is "How many times is 100! able to be divided by 10?". Each zero represents one time, so any number with 00 at the end can be divided by 10 two times.
So let's go through each one. 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 Are divisible by 10, so that makes a total of 11 zeroes at the end. But wait. What about the numbers that can produce numbers with zeros at the end that don't end in 0? Such as, for example, 4 x 5 = 20?
Well, there is a way to work out how many such combinations there are. Let's break down 8 x 5 as an example. This can be expressed as (2 x 2 x 2) x 5, which can be rearranged as (2 x 2) x (2 x 5) = (4 x 10).
Likewise, something like 4 x 15 can be expressed as (2 x 2) x (3 x 5) = (6 x 10). Basically, when two numbers that don't end in zero multiply to make a number that does, it must involve a 2 and a 5 in the multiplicative breakdown. So the puzzle can be rephrased as something simpler: "5 divides into 100! how many times?" We can lay out the numbers as such counting upwards:
5, 10, 15, 20, 25...90, 95, 100.
All these terms divide by 5 once, except for 25, 50, 75 and 100, which you can divide by 5 twice, for a total of twenty-four times. So that's our answer.
Having finally managed to gather all of Twilight's friends in one piece, all that's left is to confront the mastermind responsible directly at the top of the mountain. The seven of you open up the golden throne-room doors at the peak, with a good swift kick from both Applejack and Rainbow Dash, to discover...
...Starlight Glimmer, Trixie and Spike, in the middle of what appears to be a My Tiny Human Marathon on a television screen.
Flabbergasted, Twilight demands to know what Starlight did this time. Starlight objects, until Pinkie points out bad things happen when Starlight attempts to cast magic without supervision...which Trixie unhelpfully agrees with.
Sounds like worthwhile gossip. You file a mental note to talk about this with Pinkie later.
As it turns out, Starlight was attempting to break the laws of reality when a game of Truth or Dare got out of hoof, and they ended up unleashing a long-trapped spirit that seeks out great sources of logic. Figuring they weren't going to escape any time soon, and the mountain was surprisingly comfortable, they decided to wait until one or more of you rescued them.
Sadly, this does little to calm Twilight down, and she puts Starlight on probation at the School of Friendship, as well as a grounding for Spike for encouraging the whole thing.
Suddenly, a rumbling sound rings out, and Rainbow is the first to identify this as a sign the doom-fortress of the villain will soon collapse, and now's a good time to skedaddle. Grabbing on to your hapless victims, you dash down the mountain, dodging the falling crystal boulders, until the construct sinks into the ground, seemingly for good.
Rarity laments that she couldn't collect any of the gems for herself, Applejack decides to check on how much damage that mountain did back to Sweet Apple Acres, Pinkie decides to check out Sugarcube Corner, Rainbow Dash starts gathering the fallen Prism Poppers that should be enough to choke an elephant, Fluttershy needs some time at the spa to avoid quietly freaking out, and you need to report back to Canterlot to report on how everything went to the Princesses.
It isn't until everypony left that Twilight realises that the owner of the mysterious voice that visited Fluttershy was nowhere to be found while they were exploring the mountain...
Unfortunately, Twilight's bad feeling soon proves to be entirely correct, because as soon as you return to Canterlot Castle, you find the entire staff in a state of panic. As soon as the mountain in Ponyville disappeared, Princess Celestia, Princess Luna, and Princess Cadance, who happened to be visiting, suddenly became afflicted with a strange and mysterious condition. They've started babbling in some sort of strange and alien language, and absolutely nothing that any of the best experts can do can seemingly break the curse.
You borrow the tools and quickly send a letter to Twilight, who quickly teleports herself to the throne-room, having regained the ability to use her magic. As it turns out, after everypony left, Twilight discovered a strange text right in the centre of the rubble where the mountain used to be. The bad news is that Starlight Glimmer's...unfortunate accident unleashed three evil spirits who seem to have chosen Celestia, Luna and Cadance as their hosts to render the population of Canterlot into a catatonic state, as they feed on their brainwave energy and the magic of friendship all over Equestria. The good news is that these spirits were sealed away by Star-Swirl the Bearded long ago, and with it, instructions in the text on how to send the spirits back to from whence they came, thanks to a rather handy spell known as the 'Hero's Sporting Chance For The Sake Of Drama' Spell, which he perfected while vacationing in Olde Trottingham. This prevents the spirits from trying to flee should anypony make an attempt to banish them, so at least all hope isn't lost.
His text describes the problem as such: one of the spirits tells nothing but the truth, one spirit tells nothing but lies, and the other answers questions posed to it randomly. A quick asterisk at the bottom of the page states that the random spirit decides whether to answer 'yes' or 'no' through some sort of coin toss in its mind, so not even it knows how it will answer: it's truly random in every aspect. To banish them, their identities must be ascertained in two or fewer questions. However, there is a rather major problem impeding your progress.
The spirits dwell in another world, and speak in a language that bears no resemblance to any language on this planet today, and not even a pony as brilliant as he has managed to decipher even one recognisable word. In other words, you have absolutely no way to actually understand any of their answers, regardless of whether they are the truth, or a lie. Well, horseapples.
One last thing he also notes is that the spirits are omniscient and don't take kindly to those too smart for their liking, and should the questioner pose a question that is paradoxical and cannot be answered, they spirits will automatically 'rage-quit' and just render their host body totally unconscious, preventing you from asking them any more questions, for being a smart-flank. The exception is the spirit of randomness, who cares nothing for truth, lies, or even if the answer makes sense.
Having gathered all three possessed princesses in one place, it is now up to you to prevent Equestria from turning into a nation of apathetic losers and save Equestria from certain doom. Your country needs you, citizen, so cranks those brain cells like never before!
And don't go thinking that just a general summary of the answer is enough! With only one, complicated answer, we couldn't make it that simple!
Since we have no way to understand any of the princesses' answers, we cannot refer to them in questions. Knowing only one word (“yes” or “no”) would be enough. However, what little information we DO have gives us a decent place to start.
Before asking questions, we have only one piece of information—the princesses speak following a language. Logically, “yes” and “no” must sound completely different in whatever language this is. We can exploit this by structuring our questions in a certain way. First, we use embedded questions, and second, we need to establish some method of telling the answers apart as we ask. In this case, let's sort the answers of yes and no in whatever follows the alphabetical order of whatever language these princesses currently speak in.
First, in order to eliminate any ambiguity with our questions' answers, we need to use the first question to positively identify one princess we know definitely isn't Random. From there, we ask that princess the second question, depending on the outcome of the first.
Start by asking the princess on the left: "If I were to ask this question to the middle princess instead of you, asking about you instead of her, is there a chance she would answer with the word meaning 'yes' if and only if the word meaning 'yes' comes alphabetically before the word meaning 'no' in your language?"
This results in one of 4 outcomes:
If the left princess is Random, she will give some random answer.
If the middle princess is Random, then there is indeed a 50% chance she will answer yes, so if the left princess is True she will truthfully reply with the word meaning yes (if it does come first alphabetically), or no (if it comes second alphabetically). Either way, you can be sure that she'll answer with whatever comes first alphabetically in their language.
Using similar logic, if the left princess is False and the middle Princess is Random, then the answer will be second when sorted alphabetically.
If the right princess is random, the paradoxical question will cause the left princess to drop unconscious.
If the left princess drops unconscious:
We now know the middle princess isn't random. We therefore simply ask the middle princess "Will you answer this question with the word that means no in your language?"
If the middle princess is true, then the paradoxical question will make HER drop unconscious too. We can now sort the order as False, True, Random.
Or if she's false, she'll answer with whatever answer she wants, and it will be a lie no matter what. Therefore the order is True, False, Random.
If the left princess doesn't drop unconscious:
This is trickier, as there are three different possibilities as to why the left princess answered. However, we do at least know the RIGHT princess isn't random, no matter what answer we got. So we now ask the princess on the right the following (rather long-winded) question:
"Is it the case that either:
1. the left princess is Random and the statement '*answer left princess gave* means yes if and only if you will respond to this question with the same answer OR you are True' 2. the middle princess is Random AND the word meaning yes comes before the word meaning no when sorted alphabetically in your language?"
Now, one of four things happens:
If the left princess is random and the right princess is true, then the question can basically be simplified as "Does the answer to the first question mean yes in your language?" So the right princess would truthfully reply with the same answer as the first question: yes. So we now know the order to be Random, False, True.
If the left princess is random and the right princess is false, then the question can be simplified as "Does the previous question's answer mean 'yes' if and only if you will give the same answer to this question?" i.e "Will you answer this question with the word that means yes?" A false princess can't answer this question, so she drops unconscious. That means the order must be Random, True, False.
If the left princess is true and the right princess is false, then we default to case 2 for this question, basically simplifying it as "does the word meaning yes come before the word meaning no when sorted alphabetically in your language?". As left and right are True and False: polar opposites, then you're going to get the opposite answer of whatever answer you got to the first question. But the left princess already effectively answered this question before, which allows you to have both answers in hand, as well as where they come alphabetically in their language. Score! So if left is true and right is false, then the answer you get is second alphabetically, so the order must be True, Random, False.
If the left princess is false and the right princess is true, then the answer is whatever's first alphabetically, so the order is False, Random, True.
.