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 headmanís house number from each of his four acquaintances. They say:
1. It is the sum of squares of two distinct positive integers.
2. It does not have any repeating digits.
1. It is even.
2. It has exactly two digits.
1. It is a prime number.
2. It is one less than a triangular number.
1. It has exactly one digit.
2. It is the product of consecutive prime numbers.
Determine the headmanís house number from these statements.
I suppose the obvious (to anyone but myself ;) ) implication is that house numbers cannot be 0, but if you allow this, then 0 works as follows, also treating it as a 1-digit number:
It does not have any repeating digits, but nor is it the sum of squares of two distinct *positive* integers -- so Abe lied, then told the truth.
It is even, but does not have two digits, so Ben told the truth, then lied.
It's not prime, but it is less than a triangular number (at least, I *think* 1 counts as triangular, someone correct me if I'm mistaken), so Cal lied, then told the truth.
It has exactly one digit, but is not the product of any two consecutive primes, so Don told the truth, then lied.
Posted by Caleb
on 2013-01-30 02:00:00