“I cannot conceive how it is done,” Uncle John confessed.  “I suppose he puts bay windows on the east and west sides; but how on earth can be contrive to look south from the north side?  Does he use mirrors, or something of that kind?”

“No,” replied Mr. Allgood, “nothing of the sort.  All the windows are flush with the walls, and yet you get a southerly prospect from every one of them.  You see, there is no real difficulty in designing the house if you select the proper spot for its erection.  Now, this house is designed for a gentleman who proposes to build it exactly at the North Pole.  If you think a moment you will realize that when you stand at the North Pole it is impossible, no matter which way you may turn, to look elsewhere than due south!  There are no such directions as north, east, or west when you are exactly at the North Pole.  Everything is due south!”

“I am afraid, mother,” said her son George, after the laughter had subsided, “that, however much you might like the aspect, the situation would be a little too bracing for you.”

“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?”

“Yes; I see that—­at least I think I see it.”

“Therefore two persons at least must have the same number of hairs on their heads; and as the number of people on the earth so greatly exceeds the number of hairs on any one person’s head, there must, of course, be an immense number of these repetitions.”