Consider a variant of the game Nim, which we played in class. Recall that the rules of

the game are the following:

i. Two players take turns to remove matches from several piles.

ii. Each player, when it is his or her turn to move, can remove any number of matches (at least

one and up to the entire pile) from any pile but not from multiple piles.

iii. The player who removes the last match wins.

We know that if there are only two piles and both piles have the same number of matches, for

example, (2, 2), then the player who moves second has a winning strategy, i.e. he is guaranteed to

remove the last match.

For each of the following cases, use the above result and apply the thinking ofbackward induction

to find out which player has a winning strategy and describe the winning strategy.

(a) (3pts) three piles (10, 10, 20)

(b) (3pts) three piles A, B, C having matches (3, 2, 1) respectively.