Here is a trilogy of problems about mathematical games. In all the problems it is supposed that there are two players making their moves in turn, one after another. Unless otherwise specified, you must determine who wins (the player who goes first, or the other one).

1. The numbers 1 through 100 are written in a row. A move consists of inserting one of the signs "+", '−' or '×' in a free space between any two neighbouring numbers. The first player wins if the final result is odd, and loses otherwise.
2. There are two heaps of sweets on a table: 22 sweets in one of them, and 23 in the other. A move consists of either eating two sweets from one heap, or of moving one sweet from one heap to the other. The player who cannot make a move loses.
3. There are 1001 matches in a pile. A move consists of throwing away pⁿ matches from the pile, where p is any prime number and n = 0,1,2... The player who takes the last match wins the game.