Mathematical_Recreations-Kraitchik-2e

CHAPTER EIGHT
GEOMETRIC RECREATIONS
1. DISSECTION OF PLANE FIGURES
OUR first problems are applications of a remarkable theorem
proved by Hilbert: If two polygons are equivalent (that is,
have the same area) then it is possible to transform either poly-
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FIGURE 69.
Three Pieces.
Two Dissections of a Rectangle to Form a Square in
gon into the other by cutting it into a finite number of polygons
and rearranging the pieces.
To save cumbrous locutions we shall state applications of
this procedure as follows: Suppose it is desired to transform
one figure into another by making a certain number of
straight cuts (as with shears) in the first figure and rearranging
the resulting polygons so that the ass~mbled pieces form
the second. Then we shall say, "Dissect the first figure into
(so many) pieces by (so many) cuts so as to form the second
figure."
1. Dissect a given rectangle into three pieces by two cuts
so as to form a square.
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