lacies 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 seri