Is there a positive integer n ≥ 2 and words u1, u2, ..., un such that both equalities

(u1u2 ⋅⋅⋅ un)2 = u12u22 ⋅⋅⋅ un2,

(u1u2 ⋅⋅⋅ un)3 = u13u23⋅⋅⋅ un3

hold simultaneously and the words ui, i = 1, ..., n, do not pairwise commute (that is, uiuj ≠ ujui for at least one pair of indices i,j∈{1,2,...,n })?

For the solution an award of 200 (100 upped during WORDS 2011 in Prague) is promised. Details here.

Recently, the problem has been solved by Aleksi Saarela.

-- StepanHolub - 01 Aug 2011

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Topic revision: r4 - 2018-02-22 - JeffreyShallit
 
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