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You bake a nice, delicious-looking pumpkin $\pi$, cut it into 6 slices and serve it on your table. As a fan of numbers, you label each slice with numbers from 1 to 6.

Feeling very pleased and satisfied, you go back into your kitchen to grab a glass of water. Once you get back to your table, instead of finding your pumpkin $\pi$, you find a piece of paper with a pumpkin $\pi$ drawn on it, with the numbers you'd labelled it with replaced by letters (as shown above).

Oh no - that's not good news.

You look at the back of the paper and find a message that reads:

"Har har... want to get your pumkpin $\pi$ back? Not until you tell me the numbers labelled on your $\pi$."

Bummer! Having a terrible memory, you have absolutely no idea how you labelled your $\pi$.

You continue reading the message:

"Don't worry! Just in case you forgot, I left a liiiiiiitle bit of a clue on 6 separate pieces of paper. But mind you... I'm not always honest..."

Surprisingly, you find 6 pieces of paper under your table:

"1---- is adjacent to 4; is adjacent to 3"

"2---- is adjacent to 4; is opposite 6"

"3---- is opposite 4; is adjacent to 5"

"4---- is adjacent to 6; is not adjacent to 5"

"5---- is adjacent to 3; is not adjacent to 6"

"6---- is not opposite 3; is not adjacent to 1"

Each piece of paper contains 2 pieces of information about how your numbers are labelled. You also know that on every paper, one piece of information is true while the other is false.

Can you deduce the numbers labelled on your pumpkin $\pi$?

(Note: Please don't answer YES.)

(Note 2: If 2 slices are next to each other, then they are adjacent. e.g. A and B. If 2 slices have 2 other slices between them, then they are opposite to each other. e.g. A and D.)

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From (5) we know that 5 is either adjacent to 3 and 6, or adjacent to neither of them. Assume it is adjacent to both:

3 5 6 _ _ _

From (6), since 6 is not opposite of 3, it must be adjacent to 1:

3 5 6 1 _ _

From (1), since 1 is not adjacent to 3, it must be adjacent to 4, and then the last remaining spot must be filled by 2:

3 5 6 1 4 2

But this makes both of the claims in (2) true, which is a contradiction. So our original assumption is false, and we conclude that 5 is not adjacent to 3 or 6. From (3), this means that 3 is opposite of 4, and the only way that works is if 4 is adjacent to 5:

5 4 _ _ 3 _

From (4) we know that 4 is adjacent to 6:

5 4 6 _ 3 _

From (6) we know that 6 is adjacent to 1, and the last remaining spot is filled by 2:

5 4 6 1 3 2

This satisfies all the conditions of the problem.

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  • 1
    $\begingroup$ Welcome to Puzzling - this is a great answer! I hope to see you around here a lot more! c: $\endgroup$ – Deusovi Apr 18 '16 at 17:44
  • $\begingroup$ Oh yes! Also, please answer as A = x, B = y, etc. $\endgroup$ – bleh Apr 18 '16 at 17:47
  • 3
    $\begingroup$ Given that every rotation/reflection of the answer works, I don't think A=x, B=y makes a lot of sense... $\endgroup$ – ffao Apr 18 '16 at 18:35

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