On a Problem of Steinhaus Concerning Binary Sequences

Abstract

A finite {\small $\pm 1$} sequence X yields a binary triangle {\small $\Delta X$} whose first row is X, and whose {\small $(k+1)$}th row is the sequence of pairwise products of consecutive entries of its kth row, for all {\small $k \geq 1$}. We say that X is balanced if its derived triangle {\small $\Delta X$} contains as many +1s as {\small $-$}1s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length {\small $n \equiv$} 0 or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing strongly balanced binary sequences, i.e., binary sequences of length n all of whose initial segments of length {\small $n-4t$} are balanced, for {\small $0 \leq t \leq n/4$}. Our strongly balanced sequences do occur in every length {\small $n \equiv$} 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.