Faiz and Sara play a game as follows.

• They are given N counters, where N is at least 5.
• The counters are initially arranged in three piles with at least one counter in each pile.
• Each player takes it in turn to move. Faiz always has the first move.
• In a move, the player must decide, for each pile, whether to take a single counter from that pile, or to leave the pile alone.
• In a move, the player must take at least one counter: they cannot decide to leave all the piles alone.
• The winner of the game is the player who takes the last counter from the table.

a        Show that, if N = 8 and the counters are arranged in two piles of 3 and one pile of 2, Faiz can always win, whatever Sara does.

b       If N = 8 but Sara can choose how to make the initial arrangement of the counters in piles, explain what arrangement she should choose, and show that she can then win, whatever Faiz does.

c        For which values of N can Faiz always win, even if Sara chooses the initial arrangement in piles? Explain your answer.