Reasoning Contest-147

[HELP]

A and B play a game in which they alternate calling out positive integers less than or equal to n, according to the following rules:

• A goes first and always calls out an odd number.
• B always calls out an even number.
• Each player must call out a number which is greater than the previous number. (Except for A's first turn.)
• The game ends when one player cannot call out a number.

Some example games (for n = 8):

• 1, 8
• 3, 4, 5, 8
• 1, 2, 3, 4, 5, 6, 7, 8

The length of a game is defined as the number of numbers called out. For example, the game 1, 8, above, has length 2.

1. How many different possible games are there?
2. How many different possible games of length k are there?