A remote island consists of three types of inhabitants. The liars always lie about everything, knights who always tell the truth and, knaves who strictly alternate between lying and telling the truth.

Frank, who is an inhabitant from a nearby island was visiting four acquaintances Abe, Ben, Cal and Don, each of whom are known to Frank to be knaves. Frank wants to meet the newly elected headman at his house. Accordingly, he inquires about the headman’s house number from each of his four acquaintances. They say:

__Abe__
1. It is the sum of squares of two distinct positive integers.
2. It does not have any repeating digits.
__Ben__
1. It is even.
2. It has exactly two digits.
__Cal__
1. It is a prime number.
2. It is one less than a triangular number.
__Don__
1. It has exactly one digit.
2. It is the product of consecutive prime numbers.

Determine the headman’s house number from these statements.

I think that this problem takes more than 15 seconds to solve. The solution is not 2, because of Cal's statements. 2 is both prime and one less than a triangular number, which would make both of Cal's statements true.

Here is a better solution:

If it had only one digit, then based on Ben's statements it would need to be even, so it is 2, 4, 6 or 8. But 2 make both of Cal's statements true, and the others make both of Cal's statements false.

Therefore, D1 is false and therefore D2 is true. It is the product of two consecutive prime numbers.

Therefore, B1 and C1 are false and therefore B2 and C2 are true. It is a two digit number that is one less than a triangular number. So it is 14 or 20 or 27 or 35 or 44 or 54 or 65 or 77 or 90

The only 2 that are products of consecutive primes are 35 and 77.

Both 35 and 77 make A1 false, but only one of them makes A2 true, and that is 35.

So, final answer, the house number must be 35.

All the knaves told an initial lie, followed by a truth.

Great problem !!!

*Edited on ***November 1, 2012, 4:58 pm**