Pop Your Balloon

Strategies and Hints

To win every time you must let your opponent start first and then make one of the following numbers of balloons on each round: 18, 15, 12, 9, 6, 3. These numbers are computed by the equation n = O (mod 3), where n = safe number, and the 3 was used because it is one more than the maximum number of balloons which may be popped. If you allowed 1, 2, 3, or 4 balloons to be popped, the equation would be n = O (mod 5).

Extension:

1. You would win if you started by taking 3 balloons and then made 8 and 4 balloons on each of the following rounds. These numbers are computed by the equation
   n = O (mod 4).

2. If the number of balloons you can pop is 1 or 2, then you would win if you leave n balloons, where n = 1 (mod 3).