We present an approach to the problem of maximum number of distinct squares in a string
which underlines the importance of considering as key variables both the length n and n− d
where d is the size of the alphabet. We conjecture that a string of length n and containing d
distinct symbols has no more than n− d distinct squares, show the critical role played by
strings satisfying n= 2 d, and present some properties satisfied by strings of length bounded
by a constant times the size of the alphabet.

… d-step Approach We investigate the problem of distinct squares in relationship to the
alphabet of the string. … We conjecture that the upper bound for the maximum number of
primitively rooted distinct squares is n − d. …