It'* really not that hard to figure out. Just pay attention to what planets you can't go to on each move. Half of them you can't go to, so he just takes one of those away. I used to know a card trick like that, but I forgot how to do it.
For most of the game it switches back and forth between an odd number of moves and an even number of moves. There'* always at least one planet that cannot be "landed on" given the number of moves, so that one is taken away each time... until only one (Mars) remains.