heres a non-constructive solution for the medium puzzle. the commutator of a row move a and a column move b (defined [a,b]=aba^-1b^-1). is a three cycle. it is not hard two see that any three cycle is conjugate to either a commutator or a product of two commutators [a,b]*[-a,b]. the three cycles generate the alternating group, which is maximal in Sn. but the group action also contains a 4 cycle, which is not in the alternating group. so the action must be all of Sn.