Efficient Generation of Binary Magic Squares

TL;DR

This paper introduces an efficient algorithm for generating Binary Magic Squares, providing theoretical guarantees, extending to non-square cases, and releasing GPU-accelerated implementations for practical use.

Contribution

The paper presents a simple, provably correct algorithm for generating Binary Magic Squares, including non-square variants, with publicly available GPU-accelerated Python implementations.

Findings

Algorithm always produces valid BMS with optimal complexity

Extension to non-square BMS with formal conditions

GPU-accelerated implementation for parallel generation

Abstract

We propose a simple algorithm for generating Binary Magic Squares (BMS), i.e., square binary matrices where the sum of all rows and all columns are equal. We show by induction that our algorithm always returns valid BMS with optimal theoretical complexity. We then extend our study to non-square Binary Magic Squares, formalize conditions on the sum of rows and columns for these BMS to exist, and show that a slight variant of our first algorithm can generate provably generate them. Finally, we publicly release two implementations of our algorithm as Python packages, including one that can generate several BMS in parallel using GPU acceleration.