breadth.  Therefore, by moving down a step we reduce by 12 in. the side 84 in. and increase by 101/2 in. the side 63 in.  Hence our final rectangle must be 72 in. x 731/2 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 10+2/7 in. in height and 12 in. in breadth.  Note that 61+5/7 x 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.

I have 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.

[Illustration]

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 of material.  How should he go to work?  How many pieces would you require?

152.—­ANOTHER JOINER’S PROBLEM.

[Illustration]

A joiner had two pieces of wood of the shapes and relative proportions shown in the diagram.  He wished to cut them into as few pieces as possible so that they could be fitted together, without waste, to form a perfectly square table-top.  How should he have done it?  There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or a little too small it will not affect the method of solution.

153—­A CUTTING-OUT PUZZLE.

Here is a little cutting-out poser.  I take a strip of paper, measuring five inches by one inch, and, by cutting it into five pieces, the parts fit together and form a square, as shown in the illustration.  Now, it is quite an interesting puzzle to discover how we can do this in only four pieces.

[Illustration]

154.—­MRS. HOBSON’S HEARTHRUG.

[Illustration]

Mrs. Hobson’s boy had an accident when playing with the fire, and burnt two of the corners of a pretty hearthrug.  The damaged corners have been cut away, and it now has the appearance and proportions shown in my diagram.  How is Mrs. Hobson to cut the rug into the fewest possible pieces that will fit together and form a perfectly square rug?  It will be seen that the rug is in the proportions 36 x 27 (it does not matter whether we say inches or yards), and each piece cut away measured 12 and 6 on the outside.

155.—­THE PENTAGON AND SQUARE.