A Brute-Force Solution to the 27-Queens Puzzle Using a Distributed Computation

Abstract

The N-Queens Puzzle is an intriguing mathematical riddle that provokes many interesting but hard questions albeit having an astonishingly simple problem statement. One of these questions asks for the number of non-attacking placements of N queens onto a generalized $N\times N$ chessboard. While estimates and bounds can certainly be given, the exact solution counts, so far, have not lent themselves to a reasonable closed-form solution but rather to showcasing the arts of computer programming - and of digital design - in a tedious systematic exploration of the vast solution space. Already, Donald Knuth has made it a classic example illustrating the technique of backtracking. The largest problem sizes with known solution counts are N = 26 and N = 27. Both of them were first obtained by distributed computations relying on nodes featuring solver engines built within field-programmable hardware. This presentation will briefly introduce the capabilities and opportunities of programmable hardware highlighting its great fit for exploring the N-Queens Puzzle. It will illustrate how the computations were partitioned and how symmetries were used to prune the search spaces before even starting. Finally, the distributed architectures running the actual computations over several months each will be detailed.