The six Halma problems can be solved as follows. None of the solutions is unique:

1. Counter 6 steps diagonally up and right, 8 (or 4) steps diagonally down and left, 5 jumps all counters to end at the center.

It is possible in three all-jump moves (no steps) to end on either a corner cell or a side cell of the original pattern, but when steps are not allowed, four moves are necessary (they were given earlier) to reach the center. Two moves suffice to remove eight counters but the survivor will be outside the original pattern.

2. Counter 4 steps up, 3 jumps 8, 9, 4, 1, 2, 5, 6, then 7 jumps counter 3. A Three-move solution that puts the last counter on the board's center cell is: 4 steps up, 6 steps down, and 3 jumps the remaining eight counters. A neat symmetrical solution: 1 steps up and 7 jumps 3 (or 7 steps down and 1 jumps 3), then 1 jumps the rest.

3. Counter 6 steps up, 8 (or 4) steps down, 5 jumps all counters to rest on the center cell. This pattern and its solution are equivalent to the first problem, with each diagonal move changed to vertical and each vertical move to diagonal, all horizontal moves remaining the same. There are similarly equivalent patterns and solutions on the checkerboard and the Chinese checkers board.

4. Counter 6 can jump all counters in one move, returning to its original cell at the center. The problem is equivalent to a 10-counter equilateral triangle on the Chinese checkers board.

5. Counter 11 hops diagonally up and right (eliminating counter 8), 6 jumps 10 counters and returns to its former cell, then 5 removes 6 as it leaps to the center.

6. Counter 8 steps diagonally up and right, 14 jumps 9, 1,3, 11 and returns to its former spot, then 8 jumps 11 counters to end on the cell originally occupied by 11.

Another problem, a three-by-four rectangle on a five-by-six field, can be solved in three all-jump moves, the final counter resting on any of the 12 cells of the original pattern. In two moves the board can be cleared but the last counter will be outside the original pattern.

0 0