Square values of Littlewood polynomials

We study the square values of Littlewood polynomials. Using various methods we give all these values for the degrees $$n=3, 5$$ n = 3 , 5 and n≤ 24 n ≤ 24 even. Beside this, we gather computational data (by providing all solutions in a certain range) for n odd with n≤ 17 n ≤ 17 . We propose some striking problems for further research, as well.