It started off as just another ordinary day in Ponyville: you had finally come back from your well-earned vacation and were just about to head to Sugarcube Corner for your regular box of pecan muffins, when suddenly Derpy, the local mailmare, arrives with a letter for you.
The contents are, it has to be said, startling. Not least because you have apparently been summoned directly to Canterlot fro an audience with Princess Celestia herself, but because the matter involves a Friendship Problem, and despite your absolutely ham-hoofed attempt to solve the last one you weren't even supposed to be involved in, the Cutie Map has, oddly, glowed with YOUR Cutie Mark, along with all six residents of the castle, so evidently the Map either has a sense of humour, or it trusts all seven of you not to collectively botch the whole thing.
Regardless, as soon as you have finished reading the letter, Princess Twilight Sparkle teleports right in front of your eyes, but just before you can make to flee for your life, she ropes you back with magic and promptly frog-marches you, with help from her friends, to the train to Canterlot.
It's an awkward ride to Canterlot, as Twilight is reluctant to explain exactly what happened to her friends, and keeps giving you the stink-eye for reasons known only to you and her...nonetheless, it is otherwise uneventful, save for Rainbow Dash pranking the ticket collector, and all seven of you soon find yourselves right inside the throne room of Canterlot Castle.
Princess Celestia explains to you all that there is an national crisis taking place within the Gazelle Plains, but before she elaborates any further, she notices you've apparently been invited along as well. This, perhaps, is a necessary component to see it resolved, but first she would like to test your wits.
Everypony seems a bit confused, but Celestia explains: if you've been called to summon a Friendship problem, you should first demonstrate you have what it takes. She acknowledges your smarts, but she claims that the puzzle she is about to set you will test...more than that. She then points her hoof at you, Rainbow Dash and Applejack, and announces that the three of you will take part in her little test.
She then calls one of the Royal Guards over, and whispers something in his ear. He nods and departs from the chambers. Celestia then tells the remaining ponies to wait, and leads you, Rainbow and Applejack down into the dungeons.
Reassuring you that this isn't for what happened during your vacation, she explains to the three of you the following test:
Down in the dungeon, there are three cells. Each cell has a certain number of apples inside, distributed by the royal Guardspony from earlier, with no fewer than one and no more than nine. No cell contains the same number of apples. Celestia will now guide each pony to a cell, where they can see the number of apples inside, but they will not be allowed to view the contents of another pony's cell.
Celestia will now allow each of you to ask her one question, to which the answer is either yes or no. She will answer truthfully to each one, and you must now work out how many apples there are in total in the three cells. A correct answer from one of you would see all three of you pass the test.
As you are being guided to your own cell, Rainbow raises her hoof, asking what's to stop all three of you from just looking in their cells, forcing her to clarify the apples inside by asking "Are there -blank- apples in my cell?" and just passing the test that way,to which Celestia replies, what a good idea: let's not allow you to do that. Smooth move, Rainbow.
So with that question out, she says Applejack will ask first, then Rainbow Dash, and then you. Certainly wanting not to fail, Applejack decides to lead by asking whether if the total number of apples is an odd number, to which Celestia replies yes.
Rainbow then asks whether if the total number of apples is a prime number. She sounds a little proud of herself for knowing about numbers that can't be divided exactly except by themselves, and one: probably trying to impress you. Celestia replies no.
Now it's your turn to ask. You turn to face your cell, and you can see five apples total inside.
You have everything you need, but what, exactly, did Celestia mean by the puzzle testing more than just your wits...? What should you ask Celestia to pass?
There are not that many odd numbers that are not prime
The only ones above 5 are
7
9
15
21
25
27
It can't be 7, because that would mean that the other two cells have 1 apple each.
It can't be 25 or 27, because that would mean that one of the other cells have at least 10 apples.
That would leave us with 15, 9, and 21.
From here, I am stuck.
My best guess is this
"Seeing that the only answer is 9, 15, or 21, are there 15 apples altogether?"
She says yes, the answer is 15.
She says no, we wait to see if Applejack or Rainbow Dash calls out 21.
If they don't, the answer 9.
I ask: "What exactly is the point of all this, Princess?"
9263373
I think you've got the right idea there.
Really, any question that singles out the number 15 from 9 and 21 will do (I like "Is the total number of apples a multiple of 5?" myself). If yes, we have the answer - 15. If not, sadly you are unable to determine what the total number of apples is, but that's not a problem - there are two other ponies there that can figure it out. In fact, you know they'll both be able to figure it out - since no two of you have the sane number of apples, you know that the others either have 1 and 3 (if the total is 9) or 7 and 9 (if the total is 21).
Furthermore, if the total were 9, then the most apples any cell can have is 6 (with the others having 1 and 2), and if the total were 21, then the fewest apples any cell can have is 4 (with the others being 8 and 9). So whether Applejack and Rainbow Dash have 1 and 3, or 7 and 9, they will be able to figure out which of the two remaining totals is the correct one.
5+RD+AJ=X
X is odd non-prime number
RD != AJ
RD,AJ != 5
0<RD,AJ<10
X = {9(1,3),15(1,9),15(2,8),15(3,7),15(4,6),21(7,9)}
the question asked should be : "does the number of our apples create an aritmethic sequence with difference of 2?"
if yes: 1. the pony who has either 1 or 9 apples know the answer : either 9 or 21
2. if nopony gives an immediate answer, the total is 15
if no : the total is 15
Oh, this is a good one. Here's my best guess.
The question you should ask is: “Is the total 15?”
Here’s why:
Each cell has between 1 to 9 apples in them, and no two cells have the same number. So the lowest possible total is 1 + 2 + 3 = 6 apples, and the highest possible total is 7 + 8 + 9 = 24 apples.
If we eliminate all the even numbers and the prime numbers we are left with only three possible totals: 9, 15 and 21.
Case 1: the total is 15.
Celestia will answer yes to the question ‘Is the total 15?’. Since we know she is truthful, any one of you can tell her that the total is 15.
Case 2: the total is 9
Celestia will answer no to the question ‘Is the total 15?’
If the total is nine, and you have five apples. Then Applejack and Rainbow have four between them, so one has a single apple and the other has three. The person who has a single apple knows that the total cannot be greater than 1 + 8 + 9 = 18, and the person who has three apples knows that the total cannot be greater than 3 + 8 + 9 = 20. So, the total cannot be 21, nor is it 15, so both AJ and RD can deduce the total is 9.
Case 3: the total is 21
As above, Celestia will answer no to the question ‘Is the total 15?’ And following similar reasoning, AJ and RD have 16 apples between them. Which means one must have 9 and the other 7. And if they do, they can both deduce that the total must be more than 9, which means they know that it must be 21.
So far I managed to get it down to either 9, or 15, or 21 apples total, but what should I ask Celestia to determine which ? If only I knew !
Guess, I won't accompany the ponies to a friendship problem ...
9264894
Well, that makes sense .
...!
Good luck solving the friendship problem ()!