Two players, X and Y, play a game on a board which consists of a narrow strip which is one square wide and n squares long. They take turns at placing counters, which are one square wide and two squares long, on unoccupied squares on the board. The first player who cannot go loses. X always plays first and both players always make the best available move. The board: A counter:
a) Who wins the game on a 4 x 1 board. Explain how they must play to win and why they are then certain to win.
b) Whowinsthegameona5x1board? Explainwhy. Whowinsona7x1boardandwhy?
c) Whowinsthegameona6x1board? How?
d) Whowinsthegameona8x1board? How?
12. X and Y play another game in which X starts by choosing a number, which must be 1 or 2.
Y then adds either 1 or 2 and states the total of the two numbers chosen so far.
X does likewise, adding either 1 or 2 and stating the total, and so on. The winner is the first player to make the total reach (or exceed) 20.
a) Explain how X can always win.
b) The game is now modified so that at each stage the number chosen must be 1, 2 or 4. Which of
X or Y can now always win and how