Page:Amusements in mathematics.djvu/48

36 four pieces, based on what is known as the "step principle," but it is a fallacy.



We are told first to cut off the pieces 1 and 2 and pack them into the triangular space marked off by the dotted line, and so form a rectangle.



So far, so good. Now, we are directed to apply the old step principle, as shown, and, by moving down the piece 4 one step, form the required square. But, unfortunately, it does not produce a square: only an oblong. Call the three long sides of the mitre 84 in. each. Then, before cutting the steps, our rectangle in three pieces will be 84 × 63. The steps must be 10½ in. in height and 12 in. in breadth. Therefore, by moving down a step we reduce by 12 in. the side 84 in. and increase by 10½ in. the side 63 in. Hence our final rectangle must be 72 in. × 73½ in., which certainly is not a square! The fact is, the step principle can only be applied to rectangles with sides of particular relative lengths. For example, if the shorter side in this case were $61 5⁄7$ (instead of 63), then the step method would apply. For the steps would then be 10½ in. in height and 12 in. in breadth. Note that 61$5⁄7$ × 84 = the square of 72. At present no solution has been found in four pieces, and I do not believe one possible.

151.—THE JOINER'S PROBLEM.

often had occasion to remark on the practical utility of puzzles, arising out of an application to the ordinary affairs of life of the little tricks and "wrinkles" that we learn while solving recreation problems.

The joiner, in the illustration, wants to cut the piece of wood into as few pieces as possible to form a square table-top, without any waste