On the problem of Molluzzo for the modulus 4

Jonathan Chappelon (1) , Shalom Eliahou (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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.

Mots clés: Molluzzo's problem; Steinhaus triangles; balanced triangles; multisets; Pascal's rule

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Jonathan Chappelon, Shalom Eliahou. On the problem of Molluzzo for the modulus 4. Integers : Electronic Journal of Combinatorial Number Theory, 2012, 12, pp.A18. ⟨10.1515/integers-2012-0001⟩. ⟨hal-00564817v2⟩