You've probably come across games involving removing counters from piles before - here's a take inspired by KCLMS student Jason F, who posed part 1 to me today. 

(1) Players A and B have a pile of counters in front of them. A goes first, and they take turns to either (a) remove 3 counters or (b) add 1 counter. A player wins if they remove the last counter. (To clarify: if there are 2 counters left, a player cannot remove 3 counters.)

Who wins the game?

(You might like to start by analysing specific, small starting numbers of counters. For example, who wins if there are 3 counters to begin? What if there are 2 counters?)

(2) The game is the same, except now the available moves are (a) remove 3 counters or (b) add 2 counters.

Who wins?

(3) The game is the same, except now the available moves are (a) remove 5 counters or (b) add 2 counters.

Who wins?

If you enjoyed this, you might like to look at some past challenges - 93, 111 and 223 all share some DNA with this problem.