Squaring the square is the problem of cutting up a square entirely into square pieces, all different sizes, where all sizes are integers. Such decompositions are always complicated: the smallest known example of a squared square is 112 units wide and is cut into 21 smaller squares. The problem has a good many variations, and in particular, some solutions divide a square into a number of rectangles, each of which is ultimately cut up into squares again subject to the restriction that all the squares must have different sizes. The Wikipedia article describes a number of related problems, and a journal article presents two algorithms: one incomplete but practical and the other "complete, but computationally prohibitive". There are a lot of possibilities for implementing these algorithms and possibly designing new ones, improving their performance, etc.
Last revised: Thursday, 6 October 2022