On a problem of Steinhaus concerning binary sequences
Shalom Eliahou and Delphine Hachez
A finite (+/-)1 sequence X yields a binary triangle DX whose first row is X, and whose (k+1)st row is the sequence of pairwise products of consecutive entries of its kth row, for all k>0. We say that X is balanced if its derived triangle DX contains as many +1's as -1's. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length n congruent with 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 n-4t are balanced, for t between 0 and n/4. Our strongly balanced sequences do occur in every length n congruent with 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.
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