*solves* a basic stage , if

= . Assume that G and S are such that for each basic state there is a process

that solves (technically this means that the group of operations on G operates transitively on the set G of basic states). The goal of the puzzle is to find such a process for each basic state. If a process

solves a basic state , then the inverse process

', that we get by reversing the sequence and replacing each simple operation by its inverse, transforms into , and I say

under the permutation ( ) by first computing the image of

under and then computing the image of that point under . For this order of multiplication it is usual to write

^ for the image of a point

under a permutation (instead of writing (

), which would be better for the other order). For this order of multiplication we must define conjugation of by as ^ := ^-1 (instead of ^ := ^-1). In this notation, it is certainly true that d(,) = d(,). This is because each process that transforms to the state , will also transform to , and likewise each process that transforms to will also transform to . In a certain sense we don't need this though. What you are looking for is a process

that effects the state , i.e.,