## DISSECTING A MITRE.

(

Various Dissection Puzzles)

The figure that is perplexing the carpenter in the illustration

represents a mitre. It will be seen that its proportions are those of a

square with one quarter removed. The puzzle is to cut it into five

pieces that will fit together and form a perfect square. I show an

attempt, published in America, to perform the feat in four pieces, based

on what is known as the "step principle," but it is a fallacy.

We are told first to cut oft 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 x 63. The

steps must be 101/2 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

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.

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THE JOINER'S PROBLEM.
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THE CHOCOLATE SQUARES.