The Second Book of Mathematical Puzzles and Diversions

Henry Ernest Dudeney: England's Greatest Puzzlist 35
According to a theorem first proved by the great German
mathematician David Hilbert, any polygon can be trans-
formed into any other polygon of equal area by cutting it
into a fihite number of pieces. The proof is lengthy but not
difficult. It rests on two facts: (1) any polygon can be cut
by diagonals into a finite number of triangles, and (2) any
triangle can be dissected into a finite number of parts that
can be rearranged to form a rectangle of a given base. This
means that we can change any polygon, however weird its
shape, into a rectangle of a given base simply by chopping
it first into triangles, changing the triangles to rectangles
with the given base, then piling the rectangles in a column.
The column can then be used, by reversing the procedure,
for producing any other polygon with an area equal to that
of the original one.
Unexpectedly, the analogous theorem does not hold for
polyhedrons: solids bounded by plane polygons. There is no
general method for dissecting any polyhedron by plane cuts
to form any different polyhedron of equal volume, though of
course it can be done in special cases. Hope for a general
method was abandoned in 1900 when it was proved impos-
sible to dissect a prism into a regular tetrahedron.
Although Hilbert's procedure guarantees the transforma-
tion of one polygon into another by means of a finite num-
ber of cuts, the number of pieces required may be very
large. To be elegant, a dissection must require the fewest
possible pieces. This is often extremely difficult to determine.
Dudeney was spectacularly successful in this odd geometri-
cal art, often bettering long-established records. For exam-
ple, although the regular hexagon can be cut into as few as
five pieces that will make a square, the regular pentagon
was for many years believed to require at least seven. Dude-
ney succeeded in reducing the number to six, the present
record. Figure 13 shows how a pentagon can be squared by
Dudeney's method. For an explanation of how Dudeney ar-