Amusements in Mathematics HTML version

are necessary, from which it will be seen that one boy gets his share in two pieces and the
other three receive theirs in a single piece. I am aware that this statement "gives away"
the puzzle, but it should not destroy its interest to those who like to discover the "reason
Here is a slab of chocolate, indented at the dotted lines so that the twenty squares can be
easily separated. Make a copy of the slab in paper or cardboard and then try to cut it into
nine pieces so that they will form four perfect squares all of exactly the same size.
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 84×63. 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 615/7 (instead of 63), then the step method would apply.
For the steps would then be 102/7 in. in height and 12 in. in breadth. Note that 615/7 × 84=
the square of 72. At present no solution has been found in four pieces, and I do not
believe one possible.
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.