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Amusements in Mathematics

Geometrical Problems
"God geometrizes continually."
"There is no study," said Augustus de Morgan, "which presents so simple a beginning as
that of geometry; there is none in which difficulties grow more rapidly as we proceed."
This will be found when the reader comes to consider the following puzzles, though they
are not arranged in strict order of difficulty. And the fact that they have interested and
given pleasure to man for untold ages is no doubt due in some measure to the appeal they
make to the eye as well as to the brain. Sometimes an algebraical formula or theorem
seems to give pleasure to the mathematician's eye, but it is probably only an intellectual
pleasure. But there can be no doubt that in the case of certain geometrical problems,
notably dissection or superposition puzzles, the æsthetic faculty in man contributes to the
delight. For example, there are probably few readers who will examine the various
cuttings of the Greek cross in the following pages without being in some degree stirred
by a sense of beauty. Law and order in Nature are always pleasing to contemplate, but
when they come under the very eye they seem to make a specially strong appeal. Even
the person with no geometrical knowledge whatever is induced after the inspection of
such things to exclaim, "How very pretty!" In fact, I have known more than one person
led on to a study of geometry by the fascination of cutting-out puzzles. I have, therefore,
thought it well to keep these dissection puzzles distinct from the geometrical problems on
more general lines.
"Take him and cut him out in little stars."
Romeo and Juliet, iii. 2.
Puzzles have infinite variety, but perhaps there is no class more ancient than dissection,
cutting-out, or superposition puzzles. They were certainly known to the Chinese several
thousand years before the Christian era. And they are just as fascinating to-day as they
can have been at any period of their history. It is supposed by those who have
investigated the matter that the ancient Chinese philosophers used these puzzles as a sort
of kindergarten method of imparting the principles of geometry. Whether this was so or
not, it is certain that all good dissection puzzles (for the nursery type of jig-saw puzzle,
which merely consists in cutting up a picture into pieces to be put together again, is not
worthy of serious consideration) are really based on geometrical laws. This statement
need not, however, frighten off the novice, for it means little more than this, that
geometry will give us the "reason why," if we are interested in knowing it, though the
solutions may often be discovered by any intelligent person after the exercise of patience,
ingenuity, and common sagacity.