ABSTRACT

This chapter ventures into the familiar world of three-dimensional space. Some of the eleven puzzles presented concern space directly; some seem to be about the plane, but benefit by lifting into space; and others don't seem to be about space at all. Simple geometric considerations lead the reader to split a cubical cake neatly into three pieces, each with the same amount of cake and the same amount of icing. The reader also learns how to draw the same curve on two potatoes, cover a manhole with boards, and minimize the cost of shipping a carton. At the end is a classical theorem of plane geometry whose standard proof moves into the third dimension, but is flawed; another proof is given which also uses three dimensions, but is correct. The theorem says that given three circles, the three points obtained by intersecting the outside tangents of each pair of circles lie on a line.