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?
When you say "shortest route" would going from F2 to E3 count as 1 move or 2?
9305882
Diagonal moves are not allowed
I don't understand what you mean by point 7: All three ponies are perfectly logical. Does this mean all three ponies can now deduce the location after they've all answered no?
I've run into a problem.
All three answered simultaneously, so the only information each one has to go by is their own individual distance. Scarlet Pimple, on B3, would know where the treasure is if it was in B3 (with a 0) or F6 (with a 7). Pearl Wisdom, on C6, would know where the treasure is if it was in C6 (with a 0) or F1 (with an 8). Rarity, on F2, would know where the treasure is if it was in F2 (with a 0) or A6 (with a 9). For each of the three, any other square would be uncertain - for example, Pearl Wisdom would not be able to discern between A1, E1, or F2 as all would have distance 7. Since all three answered no, the squares A6, B3, C6, F1, F2, and F6 can be eliminated.
But now, the remaining squares are the following distances away from Rarity:
1 away - E2, F3
2 away - D2, E1, E3, F4
3 away - C2, D1, D3, E4, F5
4 away - B2, C1, C3, D4, E5
5 away - A2, B1, C4, D5, E6
6 away - A1, A3, B4, C5, D6
7 away - A4, B5
8 away - A5, B6
And no matter what the distance is, Rarity still has at least two options remaining.
Where am I going wrong?
9306218
Would any of those options for rarity make any of the others shorter?
9306218
You came to the same conclusion I did. That's why I asked for clarification of rule 7.
See, the problem is that they're all logical ponies. Well, OK, they would know which squares would be immediately disqualified. However, the phrasing implies that Rarity and only Rarity has enough information to figure out where the treasure is buried. Also, we are not given any reason to believe the other two ponies could figure out (except maybe a loose interpretation of Rule 7, which is not a valid basis) or if they did.
If we were allowed to assume that any of the 3 ponies could figure it out, then that means at least one of the ponies must have information that allows them to solve it. Going through the information, it couldn't be Rarity or Scarlet. Rarity for the reasons SisselSandvich states and Scarlet for similar reasons.
Pearl, however, has two squares that are 7 squares away. One of them (F2) was eliminated when Rarity said no. That means Pearl would then know that the correct square is E1. Being logical, both Rarity and Scarlet would realize this and draw the same conclusion.However, since we do not know if the other 2 did or are able to, Rarity has no reason to assume that one of them has the information to definitively solve the puzzle. Therefore, Rarity has other information that we're either overlooking or the other 2 do not have access to, causing our roadblock. That should be her number, except that number would have to be from 1 to 8, none of which point out a unique spot after elimination.
If Rarity and Pearl swapped places, we would know she has the number 7, regardless of what the other 2 would have.Ignore the strikethrough parts, I overlooked a square for Pearl. Thanks Sissel!
So, yeah, you're not the only one stumped.
9306429
Actually, A1 is also 7 steps away from Pearl, so even then there's no solution.
9306590
I updated my remark from earlier. Thanks for pointing that out!
9306116
Yes, ONLY Rarity knows, because she hasn't said 'yes' out loud. If she did, everypony would know where the treasure is.
Updated with one more condition. Let's see if that helps. Sorry, wasn't clear.
Okay, that new condition makes things possible.
The squares A5, B5, C4, D4, E4, and F4 would all give Pearl and Scarlet the same number, so they can all be eliminated immediately, before the question is even asked. These are the only six squares that can be eliminated this way - any other square is closer to one of Pearl or Scarlet than to the other, and since Rarity is an odd number of squares away from each of them, the distance to Rarity's square will always be different from that to Pearl's or to Scarlet's (one distance will be odd and the other will be even). As before, the squares A6, B3, C6, F1, F2, and F6 can be eliminated through the answers to the question, as well as B6, which with the elimination of A5 is now the only remaining square 8 away from Rarity. Removing all of those squares still leaves multiple possibilities 1-6 away from Rarity, but now there is only one square that is 7 away from Rarity, and that is the answer: A4.