160.—­THE TWO HORSESHOES.

The puzzle was to cut the two shoes (including the hoof contained within the outlines) into four pieces, two pieces each, that would fit together and form a perfect circle.  It was also stipulated that all four pieces should be different in shape.  As a matter of fact, it is a puzzle based on the principle contained in that curious Chinese symbol the Monad.  (See No. 158.)

[Illustration]

The above diagrams give the correct solution to the problem.  It will be noticed that 1 and 2 are cut into the required four pieces, all different in shape, that fit together and form the perfect circle shown in Diagram 3.  It will further be observed that the two pieces A and B of one shoe and the two pieces C and D of the other form two exactly similar halves of the circle—­the Yin and the Yan of the great Monad.  It will be seen that the shape of the horseshoe is more easily determined from the circle than the dimensions of the circle from the horseshoe, though the latter presents no difficulty when you know that the curve of the long side of the shoe is part of the circumference of your circle.  The difference between B and D is instructive, and the idea is useful in all such cases where it is a condition that the pieces must be different in shape.  In forming D we simply add on a symmetrical piece, a curvilinear square, to the piece B. Therefore, in giving either B or D a quarter turn before placing in the new position, a precisely similar effect must be produced.

161.—­THE BETSY ROSS PUZZLE.

Fold the circular piece of paper in half along the dotted line shown in Fig. 1, and divide the upper half into five equal parts as indicated.  Now fold the paper along the lines, and it will have the appearance shown in Fig. 2.  If you want a star like Fig. 3, cut from A to B; if you wish one like Fig. 4, cut from A to C. Thus, the nearer you cut to the point at the bottom the longer will be the points of the star, and the farther off from the point that you cut the shorter will be the points of the star.

[Illustration]

162.—­THE CARDBOARD CHAIN.

The reader will probably feel rewarded for any care and patience that he may bestow on cutting out the cardboard chain.  We will suppose that he has a piece of cardboard measuring 8 in. by 21/2 in., though the dimensions are of no importance.  Yet if you want a long chain you must, of course, take a long strip of cardboard.  First rule pencil lines B B and C C, half an inch from the edges, and also the short perpendicular lines half an inch apart. (See next page.) Rule lines on the other side in just the same way, and in order that they shall coincide it is well to prick through the card with a needle the points where the short lines end.  Now take your penknife and split the card from A A down to B B, and from D D up to C C. Then cut right through the card along all the short perpendicular lines, and half