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Amusements in Mathematics

The Paradox Party
"Is not life itself a paradox?"
C.L. DODGSON, Pillow Problems.
"It is a wonderful age!" said Mr. Allgood, and everybody at the table turned towards him
and assumed an attitude of expectancy.
This was an ordinary Christmas dinner of the Allgood family, with a sprinkling of local
friends. Nobody would have supposed that the above remark would lead, as it did, to a
succession of curious puzzles and paradoxes, to which every member of the party
contributed something of interest. The little symposium was quite unpremeditated, so we
must not be too critical respecting a few of the posers that were forthcoming. The varied
character of the contributions is just what we would expect on such an occasion, for it
was a gathering not of expert mathematicians and logicians, but of quite ordinary folk.
"It is a wonderful age!" repeated Mr. Allgood. "A man has just designed a square house
in such a cunning manner that all the windows on the four sides have a south aspect."
"That would appeal to me," said Mrs. Allgood, "for I cannot endure a room with a north
"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
"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.