A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

TL;DR

This paper constructs a universal integer sequence generating balanced Steinhaus figures modulo odd numbers, partially solving a longstanding problem by demonstrating their existence for many sizes and various figures.

Contribution

It introduces a universal sequence that produces balanced Steinhaus figures modulo odd integers, extending the understanding of their existence and properties.

Findings

Balanced Steinhaus triangles exist for at least 2/3 of sizes when n is an odd prime power.

Balanced lozenges exist for all admissible sizes when n is a square-free odd number.

The orbit of the universal sequence generates various balanced figures, including triangles, trapezoids, and lozenges.

Abstract

In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo $n$ odd. We prove the existence of balanced generalized Pascal triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where $n$ is a square-free odd number.