QUESTION
Emma and Tom are playing a game in turns. Each player stands on their own number line, which has spaces numbered 1 through n for some integer n ≥ 3. Emma goes first, and starts at position 1 on her number line, and Tom starts at position n on his. On each player's turn, that player must move to another number on their number line. The new number need not be adjacent to the old one. But no repetitions are allowed in the pair of positions: if Emma moves back to position 1, then Tom cannot move back to position n on his next move, because the ordered pair (1,n) has already occurred. The first player who cannot move loses.
1) For each n, who wins?
2) Now suppose they are on the same number line, and cannot simultaneously occupy the same spot. (So if Emma is at 1 and Tom is at 3, Emma can move to 2, 4, 5, …, n but not 1 or 3.) Furthermore, a position is considered the same if the same spots are occupied (so N=1, T=3 is a repetition of N=3, T=1). For each n, who wins?
3) Same as (b), but the players are considered distinct (so N=1, T=3 isn’t a repetition of N=3, T=1).