A hairy problem

Logic Level 2

In the town of Podunk, the following three conditions apply.

1) No two inhabitants have the same number of hairs on their head.

2) There is no inhabitant with exactly 4,302 hairs on their head.

3) There are more inhabitants than there are hairs on the head of any inhabitant.

Consider the following two statements:

A) The maximum possible number of inhabitants of Podunk is 4,302.

B) Precisely one inhabitant of Podunk is completely bald (has 0 hairs on their head.)

What are the truth-values of statements A and B?

B is true and A is false The truth-value of A cannot be determined The truth-value of both A and B cannot be determined The truth-value of B cannot be determined Both A and B are true Both A and B are false A is true and B is false

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Jun 8, 2017

By condition 1, we know that (since hairs come in integer values) every inhabitant has a distinct integer value of hairs on their head.

By condition 3, we know that the set of values for N inhabitants has to be {0, 1, 2,... (N-1)}. Since 0 is part of the set, this proves statement B.

By condition 2, we know that 4,302 cannot be part of the set. So since all integers up to to N-1 have to be part of the set, the maximum value of N-1 is 4,301, so the maximum value of N is 4,302. This proves statement A.

So A and B are both true.

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