On the problem of Molluzzo for the modulus 4

TL;DR

This paper solves the longstanding Molluzzo problem for modulus 4 by constructing sequences that generate Steinhaus triangles with equal element multiplicities for specific lengths.

Contribution

It provides the first solution to the Molluzzo problem for m=4, completing the classification for all positive integers n congruent to 0 or 7 mod 8.

Findings

Constructed sequences for all n ≡ 0 or 7 mod 8

Generated Steinhaus triangles with equal element multiplicities

Solved the smallest open case of Molluzzo's problem

Abstract

We solve the currently smallest open case in the 1976 problem of Molluzzo on $\mathbb{Z}/m\mathbb{Z}$, namely the case $m=4$. This amounts to constructing, for all positive integer $n$ congruent to $0$ or $7 \bmod{8}$, a sequence of integers modulo $4$ of length $n$ generating, by Pascal's rule, a Steinhaus triangle containing $0,1,2,3$ with equal multiplicities.