"Ah, well!" she replied. "Your Uncle John fell also into the trap. I am
no good at catches and puzzles. I suppose I haven't the right sort of
brain. Perhaps some one will explain this to me. Only last week I
remarked to my hairdresser that it had been said that there are more
persons in the world than any one of them has hairs on his head. He
replied, 'Then it follows, madam, that two persons, at least, must have
exactly the same number of hairs on their heads.' If this is a fact, I
confess I cannot see it."

"How do the bald-headed affect the question?" asked Uncle John.

"If there are such persons in existence," replied Mrs. Allgood, "who
haven't a solitary hair on their heads discoverable under a
magnifying-glass, we will leave them out of the question. Still, I
don't see how you are to prove that at least two persons have exactly
the same number to a hair."

"I think I can make it clear," said Mr. Filkins, who had dropped in for
the evening. "Assume the population of the world to be only one million.
Any number will do as well as another. Then your statement was to the
effect that no person has more than nine hundred and ninety-nine
thousand nine hundred and ninety-nine hairs on his head. Is that so?"

"Let me think," said Mrs. Allgood. "Yes--yes--that is correct."

"Very well, then. As there are only nine hundred and ninety-nine
thousand nine hundred and ninety-nine _different_ ways of bearing hair,
it is clear that the millionth person must repeat one of those ways. Do
you see?"