I had a bunch of checkers, but my checkerboard had been destroyed by a willful four year old child, and all I had left was a connected strip of three squares.
+----+----+----+
| | | |
| | | |
+----+----+----+
I decided that you can always make a game if you work at it, so I piled all my checkers into three stacks, placing one stack on each square. In regular checkers, when a piece is entitled to become a "king", a second piece is placed on top of it. In my game, I assumed that each of the three stacks was a "piece", and that each was entitled to become a king. However, a piece could only become a king if there were at least enough checkers on one of the other two square to double the size of the piece. As soon as a piece had been "kinged", it was time to identify another candidate (or even the same piece again) to be kinged once more.
For instance, here is a sequence of moves that occurred in one game:
| Move | Square 1 | Square 2 | Square 3 |
|---|---|---|---|
| 0 | 17 | 3 | 8 |
| 1 | 17 | 6 | 5 |
| 2 | 11 | 12 | 5 |
| 3 | 22 | 1 | 5 |
| 4 | 21 | 2 | 5 |
| 5 | 19 | 4 | 5 |
| 6 | 19 | 8 | 1 |
| 7 | 18 | 8 | 2 |
| 8 | 16 | 8 | 4 |
| 9 | 12 | 8 | 8 |
| 10 | 12 | 16 | 0 |
Since I'm making up rules as I go along, I now decide that the game is "won" if I can empty out one of the squares, that is, make one stack of checkers disappear. So in the above game, I won in 10 moves.
Can you always win this game, no matter what number of checkers are placed on each square? Is there a strategy for winning?
I give up, show me the solution.