# On We Three Kings

Recall that the Baron's most recent game involved advancing kings from the first and last ranks of a three by three chequerboard in a pawn-like manner until either he or Sir R----- reached the opposing rank or blocked all of the other's kings from moving, having the game in either eventuality.
When the Baron told me of its rules I instantly recognised it as hexapawn, a game invented by the distinguished professor G------[1] to demonstrate the principals of mechanistic strategising; in particular the consideration of the tree of configurations of pieces that grows from the possible moves that the rules afford to the Baron and Sir R-----, the one way or the others.
Now we can prune that tree by exploiting the fact that the game is symmetric with regard to the files in the sense that exchanging the first and last of them yields equivalent strategies. At the outset this means that we need only consider the Baron advancing his first and second kings since the former move is identical under reflection to his advancing his third. Indeed, I said as much to the Baron but I am uncertain as to whether he grasped its significance.

Let us begin by reckoning the outcomes of games in which the Baron advances his first king. It consequently threatens to advance to the third rank by taking Sir R-----'s second and so he must advance it or take the Baron's king. In the former case, the Baron may only advance his third king which blocks Sir R-----'s, handing him the game.

In the latter, it stands ready to advance to the first rank and so the Baron has no choice but to take it with his second king, leaving Sir R----- to advance his third, blocking the Baron's and winning the game.

If the Baron advances his second king it threatens both the first and third of Sir R-----'s and so he must take it. Exploiting the symmetry of the pieces we shall only consider his doing so with his first, which is left ready to advance to the first rank and so must be taken by the Baron. If he does so with his first king, Sir R----- may advance his third to deny the Baron any further moves and emerge victorious.

If the Baron instead takes it with his third, he threatens Sir R-----'s third and so he must take it or advance. If he takes it then the Baron may in turn take with his first, blocking Sir R-----'s remaining king and taking the game.

If he advances then he is one space closer to his goal than is the Baron and will consequently win the race to victory.

Sir R----- may consequently force the Baron into defeat and, provided he moved thoughtfully, I would have recommended that he take the Baron's wager.
$$\Box$$

### References

[1] Gardner, M., Mathematical Games, Scientific American, March 1962.