Pumpkin Pi

Logic Level 5

You baked a nice delicious looking pumpkin $\pi$ , cut it into 6 slices and served it on your table, as a fan of numbers, you labelled each slice with numbers from 1 to 6.

While still feeling very pleased and satisfied, you went back into your kitchen to grab a glass of water, but once you got back to your table, instead of finding your pumpkin $\pi$ , you found yourself a piece of paper with a pumpkin $\pi$ drawn on it with the labelled numbers replaced by alphabets (as shown above).

Oh no, that's not good news.

You looked at the back of the paper and found a message that reads:

"Harhar... 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 continued reading the message:

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

Surprisingly, you found 6 pieces of paper under your table, each one reads:

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

"2---- is adjacent to 4; is at the opposite of 6"

"3---- is at the opposite of 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 at the opposite of 3; is not adjacent to 1"

(If 2 slices are next to each other, then they are adjacent , like A and B, if 2 slices have other 2 slices between them, then they are opposite to each other, like A and D)

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

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

Give your answer by concatenating A, B, C, D, E and F, assuming A is 1, forming a 6-digit number, for example, if you think A, B, C, D, E, F are 1, 2, 4, 3, 5, 6 respectively, then your answer should be 124356. Take note that 124356 and 165342 are technically the same answers except mirrored, give out the number that is smaller , in this case, you should give 124356 as the answer instead of 165342.

The answer is 132546.

4 solutions

Felipe Perestrelo
Oct 11, 2015

I made assumptions on whether the truthful information for the first 2 statements, that is, regarding pieces 1 and 2, would be A or B, thus creating 4 hypothesis (which are the combinations of each individual assumption for both statements). I did that because I'd had to test only 4 hypothesis on a, hopefully, much more restrict universe. Turns out that my hope came to be satisfied.

Let's say that H(A,B) is the hypothesis where information A is truthful for statement 1, and information B is truthful for statement 2, so:

For H(A,A) we would have 142XX5, where XX is either 36 ou 63, and statement 5 would refute this hypothesis, since both informations would have to be either true or false.
For H(A,B) we would have 146XX2, where XX is either 35 or 53 and again statement 5 would refute it for the same reason as above.
For H(B,A) we would only get to 13XXXY, where Y is not 4, by using just the first 2 statements, and I'd have to make further assumptions that seemed complex, so I skipped it and tried the last one, which came out to be a very wise decision =P.
For H(B,B) we would start with 13XYYX, where X would be 2 and 6, since they are opposite, and easily we can fill in the gaps using other statements and check that the resulting sequel satisfies all conditions.

nice solution, but you could have discovered the answer based only upon the truthfulness of the statements about piece #1

Jacob Huebner - 5 years, 5 months ago

Let me finish your case H(B, A).

Currently you have 13XXXY where Y is not 4, looking at statement 6 we know that 6 has to be either adjacent to 1 or opposite to 3, which is either 13XXX6 or 13XX6Y.

Case 13XXX6: Now from statement 5, either 5 has to be adjacent to 3 and 6 at the same time or not adjacent to 3 and 6 at the same time, only the latter case is possible, so we now have 13X5X6, the X's would be 2 or 4. However, this contradicts H(B, A) where 2 has to be adjacent to 4. Ouch.

Case 13XX6Y: Since 2 has to be adjacent to 4, we know that XX must be 24 or 42, Y is 5. But this will make both information in statement 5 to be false at the same time. Hooray.

Also one more thing to mention, there are actually 2 answers, the other answer is 164523, this answer is obtained by simply mirroring the pie, that's why the question suggests you to give out the smaller answer since 132546 is smaller than 164523.

Kenneth Tan - 5 years, 4 months ago

Pablo Torres
Jan 6, 2016

It was awesome! Very entertained.

Tom Van Lier
Oct 11, 2015

I made a table with on every row the number and in the columns which information was true and which was false, but I got lucky, so I didn't have to fill the table in.

I started with the opposites, because they limit a lot for the rest. I tried to figure out whether they were true or false, by wondering whether they were true. So fill in 3 on the piece of cake B.

Now assume 3 is opposite of 4 (fill in 4 on E), then 3 is not adjacent to 5. This menas 5 needs to be on D or F, let's try D.

Because 5 is not adjacent to 3 in this setting, 5 cannot be adjacent to 6 (one of both expressions is true), so fill in 6 on F.

Because 6 is not opposite of 3, 6 has to be adjacent to 1 => 1 on A and 2 on C. This gives 132546.

For those wondering what would be a good approach when I didn't get so lucky => because you know if this doesn't work that 3 is adjacent to 5, you have extra information.

You can also do the same with 2 and deduce even more information and hence solve the puzzle piece by piece.

Shivam Puri
Oct 5, 2015

132546 satisfies the conditions!!

154263 also satisfies

Bilal Zaheer - 5 years, 8 months ago

No 154263 doesn't satisfy the condition, since in this case "6 is not at the opposite of 3" and "6 is not adjacent to 1" are both true, but one of these statements must be true and the other false, as stated in the problem.

Kenneth Tan - 5 years, 8 months ago

164523 satisfies the conditions

Nigel Chu - 5 years, 4 months ago

@Nigel Chu – You can get the answer 132546 by simply mirroring the pie, and 132546 is smaller than 164523.

Kenneth Tan - 5 years, 3 months ago

Pumpkin Pi

The answer is 132546.

4 solutions

0 pending reports