We put nine pennies in a row on the table heads up. Two players play in turns. At each step one of the players can flip one penny from heads to tails or two neighbouring pennies from heads to tails. The player who turns the last coin over to tails wins. How can the first player ensure that he wins?

Again, two players are playing with pennies. They sit at a large square-shaped table. The first player puts a penny on the table (it should be fully on the table). Then the second player puts a penny on the table. (Two pennies are not allowed to overlap.) Then the first player puts another penny on the table, then the second, etc. They continue until one of the players cannot place a coin on the table. The player who puts the last penny on the table wins. Prove that the first player can always win the game.

Two players alternate in taking coins from a table. They have two heaps of pennies. Each heap contains 10 pennies. In each step the players can take away either one penny from one of the heaps, or one penny from each of the heaps. The player who takes the last penny wins. Show that the second player has a winning strategy.

Two players are playing again with two heaps of pennies. Now the rule is that each player can take as many pennies (but at least one) from one of the heaps. The player who takes the last penny wins. Show that if each heap contains 10 pennies, then the second player has a winning strategy. Show that if the first heap contains one more penny than the second heap, then the first player has a winning strategy.

(This problem is hard, but fun!) Now we have 3 heaps for the two players. In the first heap there are 10 pennies, in the second heap there are 20 pennies, and in the third heap there are 25 pennies. A player can take as many coins from one of the heaps as he wants. The player who takes the last penny wins. Show that the first player has a winning strategy.