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Programming Enigma Puzzles

21 December 2014

Posted by on **From New Scientist #2425, 13th December 2003** [link]

It was a dark and stormy night when the disciples took Jesus for a birthday boat trip on Galilee. He set them a puzzle, which went as follows.

“It was a dark and stormy night when Caiaphas set the priests a puzzles, which went as follows:

“It was a dark and stormy night when Pilate set the guards a puzzle, which went as follows:

“I want you to tell me how many camels are in Caesar’s stable. I will tell you the sum of all the positive integers which divide exactly into the answer.”

“He told them and they worked out the answer. Now I, Caiaphas, want you to tell me how many camels are in the stable. I will tell you the result of multiplying the answer minus 10 by the answer.”

He told them and they worked out the answer. Now I, Jesus, want you to tell me how many camels are in the stable. I will tell you the remainder when the answer is divided by 8.”

He told them and they worked out the answer. Now everyone knows that the number of camels is at least 1 and no more than 13.

How many camels are in the stable?

[enigma1269]

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This Python program runs in 30ms.

Solution:There are 10 camels in the stable.The name is Caiaphas. See, for example, Matthew 26.

Thanks. I’ve fixed it up now.

I can’t see that there’s a unique solution. The remainder on dividing by 8 is the same for 2 and 10 (as also for 1 and 9, 3 and 11, 4 and 12, 5 and 13). All numbers below 10 have been excluded by the n(n – 10) test, so there’s nothing left!

I think it works out this way…

We first consider Pilate’s puzzle. He told the guards the sum of the divisors of

n(wherenis the number of camels), and they worked out the answer. We don’t know the number he told the guards, but we can eliminate the possibility of there being 6 or 11 camels (as the divisors of these numbers both sum to 12).Then we consider Caiaphas’ puzzle. He told the priests the value of

n× (n– 10), and they worked out the answer. But they also know that Pilates’ guards worked out the answer, soncan’t be 6 or 11. From the remaining possibilities we can eliminate 1 and 9, 2 and 8, 3 and 7 (each pair has the same value). So that only leaves 4, 5, 10, 12 and 13 as possibilities.Finally we consider Jesus’ puzzle. He told the disciples the remainder modulo 8, and they worked out the answer. They also know by now that

ncan only be 4, 5, 10, 12 or 13.So, we can eliminate 4 and 12 (

nmod 8 = 4), and 5 and 13 (nmod 8 = 5), leaving 10 as the only remaining candidate (nmod 8 = 2).So it follows that Pilate told his guards the sum of the divisors was 18, so they can deduce that

n= 10. Caiaphas told his priests thatn× (n– 10) = 0, from which they would immediately be able to deduce thatn= 10 (without even needing to know anything about Pilate’s puzzle – which is a bit disappointing). Jesus told his disciples thatnmod 8 = 2, so they do need to know about Caiaphas’ puzzle (in order to eliminaten= 2), but not about Pilate’s puzzle.We need to know about all three puzzles to deduce the answer.

Thanks for the explanation, Jim. I agree it’s a somewhat unsatisfactory puzzle (and some people might find it offensive).