Splitting a Polygon into Two Congruent Pieces

Abstract

It is quite easy to find the solution to this particular splitting problem (as shown in Fig. l(b)), but there are certainly cases where the solution is far from obvious; at a graduation party in Stockholm 1994 (with a lot of mathematicians among the guests), nobody was able to solve all the problems in the column mentioned above. On such occasions an effective algorithm that solves the problem would come in handy. It is the aim of this paper to provide such an algorithm. At the end of the paper I show that a theorem on splitting convex shapes follows from the algorithm. I also give a few nice problems for the reader to practise on. According to my friend Doron Zeilberger, this kind of problem goes back to the great puzzle inventor Henry Dudeney who wrote several books with mathematical amusement problems in the first decades of this century, e.g. [1]. In a recreational math book by Fred Schuh [3], I found a puzzle of this kind, and also one variant where the polygon should be split in four congruent quarters. I do not claim to have an algorithm for variants with more than two parts, but I would love to see one! After writing the first version of this paper, I obseIved an instance of the splitting problem on one of a series of mathematical posters on the wall by the MIT math department staircase. This poster referred to an eighteen year old