Playing Backwards

If you have a solitaire position, the complement of that position is what you get when you interchange the holes and the pegs. We've encountered this idea before. For example, in the one-peg complement problems the finish is the complement of the start. Here, we are going to look at the connection between complements and "playing backwards."

The earliest known description of Peg Solitaire is due to the philosopher Gottfried Wilhelm Leibniz, in a letter dated 1710. Leibniz liked to play the game backwards--starting with one peg on the board and adding pegs by the same rule that one usually removes pegs. The usual rule jumps a peg over a peg, leaving one peg and two holes. Leibniz would jump a hole over a hole, leaving one hole and two pegs. Here's how Leibniz would have done the Vee problem.

But Leibniz did not seem to realize that pegs and holes are just arbitrary markers, and they can be interchanged. If we take the complement of each of the pictures above, here is what we get.

The jumps h4-f2, e5-g3, f2-h4, i5-g3 are a solution to the Vee problem. If we take these jumps in backwards order-- i5-g3, f2-h4, e5-g3, h4-f2 --we get the solution to a different problem. We will call this the anti-problem.

Problem with solution is h4-f2, e5-g3, f2-h4, i5-g3

Anti-Problem with Anti-Solution i5-g3, f2-h4, e5-g3, h4-f2

Here's how you go from a problem to the anti-problem:
the anti-problem start is the complement of the problem finish, and
the anti-problem finish is the complement of the problem start.

The main thing to remember is:
If you have a solution to a solitaire problem, then taking the jumps in reverse order will give you a solution to the anti-problem.

Some people like to say that you have a solution to an anti-problem, its an anti-solution. Then all you have to say is

an anti-solution is a solution with the jumps taken in reverse order.

If you feel you fully understand the relation between problems and anti-problems, then you may want to go ahead and look at some problems. If you're not ready for the problems yet, go ahead and look at some more examples.

Playing Backwards--Examples

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