Suppose we have a line of 2013 banks that ends with a vault. The bank nearest to the vault has 1 sack of money, the bank second nearest has 2 sacks. The bank farthest away has 2013 sacks.
Jerry corporation and Meyer corporation are hired to take the sacks of money to the vault. In each turn a player can select a bank and take a positive integer number of sacks to the next bank (or directly to the vault if he chose the bank next to the vault). The winner is the company that makes the move that leaves all bags in the vault.
Which player has a winning strategy? Which is the strategy? Can we generalize to $n$ banks?
Hints: this is an impartial game, so must be Nim. Each bank is a nim heap. The first bank is clearly a heap of the number of bags in it. Even numbered banks don't matter, as the second player can mirror the first player's actions, moving each bag down two positions and ending in the vault.