Consider the following game: You and your opponent are seated in front of a pile of blocks. On your turn, you may pick up between 1 and 6 blocks. Your goal is to be the person who picks up the last block. (This game is more fun to reason about than to play --- trust me!)
Assume that there are n blocks initially in the pile; and assume that you always go first. For each value of n between 7 and 10 develop a winning strategy, or explain why no strategy will allow you to win. In addition to explaining your strategy, you must also explain why your strategy will always cause you to win. You may explain your strategy in English; but you must be very precise.
The winning strategy for a starting pile of 1 to 6 blocks is easy: On your turn, simply pick up all the blocks. You know this is guaranteed to win you the game because, by picking up all the blocks you have, according to the rules, won the game.