Two people are playing an exciting game in which they take turns removing marbles from a bag. At the beginning of the game, this bag contains some red marbles and some blue marbles. The bag is transparent so at any time during the game, the players know exactly how many red and how many blue marbles are in the bag.

The players alternate taking turns. On a player’s turn, he or she must remove some marbles from the bag. The player chooses which marbles to remove, under the condition that he or she remove at least one marble and the marbles removed in a single turn are all the same color. The player to remove the last marble from the bag during his or her turn wins.

Assume that player 1 is playing the game with player 2, and player 1 makes the first move. If you were player 1, what optimal strategy could you use to play this game? Under what starting conditions would this optimal strategy guarantee a win, and why? What can you say about the outcome of the game if these starting conditions are not met?

(Hint: Try thinking of an invariant you could maintain during certain points of the game)