"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.
If we want to cut one plane figure into parts that by readjustment will
form another figure, the first thing is to find a way of doing it at
all, and then to discover how to do it in the fewest possible pieces.
Often a dissection problem is quite easy apart from this limitation of
pieces. At the time of the publication in the _Weekly Dispatch_, in
1902, of a method of cutting an equilateral triangle into four parts
that will form a square (see No. 26, "Canterbury Puzzles"), no
geometrician would have had any difficulty in doing what is required in
five pieces: the whole point of the discovery lay in performing the
little feat in four pieces only.
Mere approximations in the case of these problems are valueless; the
solution must be geometrically exact, or it is not a solution at all.
Fallacies are cropping up now and again, and I shall have occasion to
refer to one or two of these. They are interesting merely as fallacies.
But I want to say something on two little points that are always arising
in cutting-out puzzles--the questions of "hanging by a thread" and
"turning over." These points can best be illustrated by a puzzle that is
frequently to be found in the old books, but invariably with a false
solution. The puzzle is to cut the figure shown in Fig. 1 into three
pieces that will fit together and form a half-square triangle. The
answer that is invariably given is that shown in Figs. 1 and 2. Now, it
is claimed that the four pieces marked C are really only one piece,
because they may be so cut that they are left "hanging together by a
mere thread." But no serious puzzle lover will ever admit this. If the
cut is made so as to leave the four pieces joined in one, then it cannot
result in a perfectly exact solution. If, on the other hand, the
solution is to be exact, then there will be four pieces--or six pieces
in all. It is, therefore, not a solution in three pieces.
[Illustration: Fig. 1]
[Illustration: Fig. 2]
If, however, the reader will look at the solution in Figs. 3 and 4, he
will see that no such fault can be found with it. There is no question
whatever that there are three pieces, and the solution is in this
respect quite satisfactory. But another question arises. It will be
found on inspection that the piece marked F, in Fig. 3, is turned over
in Fig. 4--that is to say, a different side has necessarily to be
presented. If the puzzle were merely to be cut out of cardboard or wood,
there might be no objection to this reversal, but it is quite possible
that the material would not admit of being reversed. There might be a
pattern, a polish, a difference of texture, that prevents it. But it is
generally understood that in dissection puzzles you are allowed to turn
pieces over unless it is distinctly stated that you may not do so. And
very often a puzzle is greatly improved by the added condition, "no
piece may be turned over." I have often made puzzles, too, in which the
diagram has a small repeated pattern, and the pieces have then so to be
cut that not only is there no turning over, but the pattern has to be
matched, which cannot be done if the pieces are turned round, even with
the proper side uppermost.
[Illustration: Fig. 3]
[Illustration: Fig. 4]
Before presenting a varied series of cutting-out puzzles, some very easy
and others difficult, I propose to consider one family alone--those
problems involving what is known as the Greek cross with the square.
This will exhibit a great variety of curious transpositions, and, by
having the solutions as we go along, the reader will be saved the
trouble of perpetually turning to another part of the book, and will
have everything under his eye. It is hoped that in this way the article
may prove somewhat instructive to the novice and interesting to others.