Now look at Fig. 29, and you will see an elegant method for cutting a piece of wood of the shape of two squares (of any relative dimensions) into three pieces that will fit together and form a single square.  If you mark off the distance ab equal to the side cd the directions of the cuts are very evident.  From what we have just been considering, you will at once see why bc must be the length of the side of the new square.  Make the experiment as often as you like, taking different relative proportions for the two squares, and you will find the rule always come true.  If you make the two squares of exactly the same size, you will see that the diagonal of any square is always the side of a square that is twice the size.  All this, which is so simple that anybody can understand it, is very essential to the solving of cutting-out puzzles.  It is in fact the key to most of them.  And it is all so beautiful that it seems a pity that it should not be familiar to everybody.

We will now go one step further and deal with the half-square.  Take a square and cut it in half diagonally.  Now try to discover how to cut this triangle into four pieces that will form a Greek cross.  The solution is shown in Figs. 31 and 32.  In this case it will be seen that we divide two of the sides of the triangle into three equal parts and the long side into four equal parts.  Then the direction of the cuts will be easily found.  It is a pretty puzzle, and a little more difficult than some of the others that I have given.  It should be noted again that it would have been much easier to locate the cuts in the reverse puzzle of cutting the cross to form a half-square triangle.

[Illustration:  FIG. 31.]

[Illustration:  FIG. 32.]

[Illustration:  FIG. 33.]

[Illustration:  FIG. 34.]

Another ideal that the puzzle maker always keeps in mind is to contrive that there shall, if possible, be only one correct solution.  Thus, in the case of the first puzzle, if we only require that a Greek cross shall be cut into four pieces to form a square, there is, as I have shown, an infinite number of different solutions.  It makes a better puzzle to add the condition that all the four pieces shall be of the same size and shape, because it can then be solved in only one way, as in Figs. 8 and 9.  In this way, too, a puzzle that is too easy to be interesting may be improved by such an addition.  Let us take an example.  We have seen in Fig. 28 that Fig. 33 can be cut into two pieces to form a Greek cross.  I suppose an intelligent child would do it in five minutes.  But suppose we say that the puzzle has to be solved with a piece of wood that has a bad knot in the position shown in Fig. 33—­a knot that we must not attempt to cut through—­then a solution in two pieces is barred out, and it becomes a more interesting puzzle to solve it in three pieces.  I have shown in Figs. 33 and 34 one way of doing this, and it will be found entertaining to discover other ways of doing it.  Of course I could bar out all these other ways by introducing more knots, and so reduce the puzzle to a single solution, but it would then be overloaded with conditions.