Balanced Steinhaus triangles

TL;DR

This paper proves that for any positive integer m, there are infinitely many balanced Steinhaus triangles modulo m, constructed from periodic arithmetic progressions, resolving a long-standing open problem for even m ≥ 12.

Contribution

It demonstrates the existence of infinitely many balanced Steinhaus triangles modulo m for all positive integers m, using periodic constructions from interlaced arithmetic progressions.

Findings

Existence of infinitely many balanced Steinhaus triangles for all m

Construction method using periodic triangles from arithmetic progressions

Addresses a problem posed by Molluzzo in 1978 for even m ≥ 12

Abstract

A Steinhaus triangle modulo $m$ is a finite down-pointing triangle of elements in the finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ satisfying the same local rule as the standard Pascal triangle modulo $m$. A Steinhaus triangle modulo $m$ is said to be balanced if it contains all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. In this paper, the existence of infinitely many balanced Steinhaus triangles modulo $m$, for any positive integer $m$, is shown. This is achieved by considering periodic triangles generated from interlaced arithmetic progressions. This positively answers a weak version of a problem, due to John C. Molluzzo in 1978, that has remained unsolved to date for the even values of $m\geqslant 12$.