[Illustration:  FIG. 23.]

[Illustration:  FIG. 24.]

Now for another puzzle.  If you have cut out the five pieces indicated in Fig. 14, you will find that these can be put together so as to form the curious cross shown in Fig. 23.  So if I asked you to cut Fig. 24 into five pieces to form either a square or two equal Greek crosses you would know how to do it.  You would make the cuts as in Fig. 23, and place them together as in Figs. 14 and 15.  But I want something better than that, and it is this.  Cut Fig. 24 into only four pieces that will fit together and form a square.

[Illustration:  FIG. 25.]

[Illustration:  FIG. 26.]

The solution to the puzzle is shown in Figs. 25 and 26.  The direction of the cut dividing A and C in the first diagram is very obvious, and the second cut is made at right angles to it.  That the four pieces should fit together and form a square will surprise the novice, who will do well to study the puzzle with some care, as it is most instructive.

I will now explain the beautiful rule by which we determine the size of a square that shall have the same area as a Greek cross, for it is applicable, and necessary, to the solution of almost every dissection puzzle that we meet with.  It was first discovered by the philosopher Pythagoras, who died 500 B.C., and is the 47th proposition of Euclid.  The young reader who knows nothing of the elements of geometry will get some idea of the fascinating character of that science.  The triangle ABC in Fig. 27 is what we call a right-angled triangle, because the side BC is at right angles to the side AB.  Now if we build up a square on each side of the triangle, the squares on AB and BC will together be exactly equal to the square on the long side AC, which we call the hypotenuse.  This is proved in the case I have given by subdividing the three squares into cells of equal dimensions.

[Illustration:  FIG. 27.]

[Illustration:  FIG. 28.]

It will be seen that 9 added to 16 equals 25, the number of cells in the large square.  If you make triangles with the sides 5, 12 and 13, or with 8, 15 and 17, you will get similar arithmetical proofs, for these are all “rational” right-angled triangles, but the law is equally true for all cases.  Supposing we cut off the lower arm of a Greek cross and place it to the left of the upper arm, as in Fig. 28, then the square on EF added to the square on DE exactly equals a square on DF.  Therefore we know that the square of DF will contain the same area as the cross.  This fact we have proved practically by the solutions of the earlier puzzles of this series.  But whatever length we give to DE and EF, we can never give the exact length of DF in numbers, because the triangle is not a “rational” one.  But the law is none the less geometrically true.

[Illustration:  FIG. 29.]

[Illustration:  FIG. 30.]