Solution to the July 2000 Brain Teaser

	1)     Your aim is to leave two balls in each of two rows after your move. This ensures you win. 2)     Therefore your first move must be to remove the single ball in the first row.

 

	3)       If your opponent leaves more than two balls on a line, you remove enough balls from the same row to leave just two on that row. 4)       If your opponent leaves two balls on a line, you remove all bar two balls from the other row. 5)       If your opponent leaves just one ball in a row, you remove the other row entirely and win.

So you end up with one of the following two positions:

a)                                                                           or b)

    ( )                                                           

In case a),  if your opponent removes one or both balls from the top row, you remove all or (all minus one) balls from the second row. This leaves just one ball, which your opponent has to pick up. You win. If your opponent removes a ball or balls from the bottom row, on your turn, you remove enough balls to leave just two balls on the bottom row. You therefore end up with figure b) above.

In case b)  if your opponent removes just one ball, you remove all the long row. If your opponent removes two balls, you remove just one ball from the other row, leaving one ball. You win.