Intro: We know that if there are two symbols, e.g. a and b, giving two spaces to place these two types of symbols at random, there will be $2^2 = 4$ distinct ways if the order matters: {aa, bb, ab, ba}. The minimum length of a circular arrangement that includes each of the four two-place patterns at least once is abba or baab.
I am stuck at the following question: If we have n symbols, with m spaces, then there will be $n^m$ possible distinct ways to arrange these symbols. What is the minimum length of the circular arrangement that contains each of the $n^m$ patterns at least once?
