What is meant by : 'the solutions being done in the Gray Binary code number system' is, for example, starting off by typing in 7 ones in a row as representing the rings all threaded onto the handle . . . as at the beginning, before any move is started . The first move calls for dropping ring #1 -- this being represented as a zero (all moves being made are colored in red) .
You'll notice at the far left (all the way down alongside the 'moves' column) are references relating to the extent of the puzzle's completion . The references in red refer to when a specific major numbered ring has finally reached the  'dropped' position . Those in blue refer to when that point is reached where all the rings before the 'dropped' ring have finally been all threaded back onto the handle in preparation for the next major ring drop !
For this 7-ring puzzle, it takes a total of 64 moves to fully release the handle from the 'assembly' .