435 research outputs found
A universal sequence of integers generating balanced Steinhaus figures modulo an odd number
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 , that are Steinhaus triangles containing all the elements of
with the same multiplicity. For every odd number ,
we build an orbit in , by the linear cellular automaton
generating the Pascal triangle modulo , which contains infinitely many
balanced Steinhaus triangles. This orbit, in , is
obtained from an integer sequence called the universal sequence. We show that
there exist balanced Steinhaus triangles for at least of the admissible
sizes, in the case where 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 odd. We prove the existence of balanced generalized Pascal triangles
for at least of the admissible sizes, in the case where is an odd
prime power, and the existence of balanced lozenges for all admissible sizes,
in the case where is a square-free odd number.Comment: 30 pages ; 10 figure
On the problem of Molluzzo for the modulus 4
We solve the currently smallest open case in the 1976 problem of Molluzzo on
, namely the case . This amounts to constructing,
for all positive integer congruent to or , a sequence of
integers modulo of length generating, by Pascal's rule, a Steinhaus
triangle containing with equal multiplicities.Comment: 12 pages ; 3 figures ; 3 tables, Integers : Electronic Journal of
Combinatorial Number Theory, State University of West Georgia, Charles
University, and DIMATIA, 2012, 12, pp.A1
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