1. ## Pebble Challenge

There are 100 pebbles on the table. There are two players, A and B, who move alternatively. Player A moves first. The rules of the game are the same for both players: at each move they can remove one, two, three, four of five pebbles. The winner is the player who takes the last pebble. Who is guaranteed to win provided that he plays properly? Convince me why you think this. Same question if the one who takes the last pebble loses.

2. ## Re: Pebble Challenge

This is a variant of the ancient game of NIm. See this link Nim - Wikipedia, the free encyclopedia for the general solution of Nim and in particular, your variant.

3. ## Re: Pebble Challenge

2) Leave $6n + 1$ pebbles (for some integer $n$) after your go. Again, the first player can always win.