A game of 11 marbles wherein each player can either pick one or two marbles from the total. Starting from Player A & then Player B alternatively. This continues till all the marbles are picked. The winner is the one having odd number of marbles.

What is the strategy to be followed for Player A & B to win?. What happens for higher total number of marbles (13, 15 etc )?

How to win: if you have an even number of marbles, and there are 4K+1 marbles, pick 1 or 2; if there are 4K+2 marbles, pick 1; and in other cases pick whatever you will, but you'll lose.

If you have an odd number of marbles, and there are 4K marbles on the table, pick 1; if there are 4K+2 or 4K+3 marbles, pick 2; and if there are 4K+1... sorry, you'll lose.

I came to this strategy after much (essentially simple, but long and tedious) work with a states machine. [If there is interest I may try to write it up for a later message.]

If I am right, the first player wins for 13, 17, 21... and so on marbles, and the second wins for 15, 19, 23...