A particular two-player game starts with a pile of diamonds and a pile of rubies. On your turn, you can take any number of diamonds, or any number of rubies, or an equal
number of each. You must take at least one gem on each of your turns. Whoever takes
the last gem wins the game. For example, in a game that starts with 5 diamonds and
10 rubies, a game could look like: you take 2 diamonds, then your opponent takes 7
rubies, then you take 3 diamonds and 3 rubies to win the game.
You get to choose the starting number of diamonds and rubies, and whether you go
first or second. Find all starting configurations (including who goes first) with 8 gems
where you are guaranteed to win. If you have to let your opponent go first, what are
the starting configurations of gems where you are guaranteed to win? If you can’t find
all such configurations, describe the ones you do find and any patterns you see.