(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