Finding magic squares with Douglas—Rachford algorithm

In this expository paper, we show how to use the Douglas—Rachford algorithm as a successful heuristic for finding magic squares. The Douglas—Rachford algorithm is an iterative projection method for solving feasibility problems. Although its convergence is only guaranteed in the convex setting, the algorithm has been successfully applied to a number of similar nonconvex problems, such as solving Sudoku puzzles. We present two formulations of the nonconvex feasibility problem of finding magic squares, which are inspired by those of Sudoku, and test the Douglas—Rachford algorithm on them.The first author was supported by MINECO of Spain and ERDF of EU, as part of the Ramón y Cajal program (RYC-2013-13327) and Grants MTM2014-59179-C2-1-P and PGC2018-097960-B-C22