[Illustration:  FIG. 18.]

[Illustration:  FIG. 19.]

[Illustration:  FIG. 20.]

The solution is very beautiful.  If you divide by points the sides of the square into three equal parts, the directions of the lines in Fig. 18 will be quite obvious.  If you cut along these lines, the pieces A and B will form the cross in Fig. 19 and the pieces C and D the similar cross in Fig. 20.  In this square we have another form of Swastika.

The reader will here appreciate the truth of my remark to the effect that it is easier to find the directions of the cuts when transforming a cross to a square than when converting a square into a cross.  Thus, in Figs. 6, 8, and 10 the directions of the cuts are more obvious than in Fig. 14, where we had first to divide the sides of the square into six equal parts, and in Fig. 18, where we divide them into three equal parts.  Then, supposing you were required to cut two equal Greek crosses, each into two pieces, to form a square, a glance at Figs. 19 and 20 will show how absurdly more easy this is than the reverse puzzle of cutting the square to make two crosses.

Referring to my remarks on “fallacies,” I will now give a little example of these “solutions” that are not solutions.  Some years ago a young correspondent sent me what he evidently thought was a brilliant new discovery—­the transforming of a square into a Greek cross in four pieces by cuts all parallel to the sides of the square.  I give his attempt in Figs. 21 and 22, where it will be seen that the four pieces do not form a symmetrical Greek cross, because the four arms are not really squares but oblongs.  To make it a true Greek cross we should require the additions that I have indicated with dotted lines.  Of course his solution produces a cross, but it is not the symmetrical Greek variety required by the conditions of the puzzle.  My young friend thought his attempt was “near enough” to be correct; but if he bought a penny apple with a sixpence he probably would not have thought it “near enough” if he had been given only fourpence change.  As the reader advances he will realize the importance of this question of exactitude.

[Illustration:  FIG. 21.]

[Illustration:  FIG. 22.]

In these cutting-out puzzles it is necessary not only to get the directions of the cutting lines as correct as possible, but to remember that these lines have no width.  If after cutting up one of the crosses in a manner indicated in these articles you find that the pieces do not exactly fit to form a square, you may be certain that the fault is entirely your own.  Either your cross was not exactly drawn, or your cuts were not made quite in the right directions, or (if you used wood and a fret-saw) your saw was not sufficiently fine.  If you cut out the puzzles in paper with scissors, or in cardboard with a penknife, no material is lost; but with a saw, however fine, there is a certain loss.  In the case of most puzzles this slight loss is not sufficient to be appreciable, if the puzzle is cut out on a large scale, but there have been instances where I have found it desirable to draw and cut out each part separately—­not from one diagram—­in order to produce a perfect result.