Take Away

There is (supposedly) a parlor game called Take Away (TA). It goes like this: there are 21 pennies on a table and 2 players who alternate in taking a turn. During a turn, each player may remove 1,2, or 3 pennies. The loser is the one who removes the last penny. I was thinking about this game and how one would go about winning it. Obviously, you want to leave your opponent with the last penny (because if there are 2 left (or 3 or 4), then they will just leave you with the last penny.

As I was reading in my Jehle & Reny text, the second player should always win the game (kind of like tic-tac-toe...except that there IS a winner)...From my above logic, it would be good to leave your opponent with 5 pennies when they make their move, because then, no matter what they do, you can leave only one after your move. J&R call 5 a "losing position." In fact, they extend this to say that the following are losing positions: 1,5,9,13,17,21...all others are winning positions. So, be careful of just taking away 3 at the beginning to get the game over fast, because even if you're the second player, you could lose...player one takes 3, you take 3, he takes only 2, then you are in a losing position!

I thought that it was cool that I thought about the structure of the game using backward induction without even realizing what I was doing until later on.