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Amusements in Mathematics

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.
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.
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.