Piece Destiny

When you play crumble, the goal is to create a chain of your pieces that touches all four sides of the board. To force your way to that outcome, you generally create chains of impervious and redundant pieces, to stop your opponent from interfering with your plans. But which pieces are susceptible to becoming impervious? How can you tell whether a given piece is worth fighting for? How can you identify the future path of your winning chain?

Piece destiny refers to the expected outcome of the local battle for a given piece, when both players are forced to play out that battle.

Consider the situation where your opponent is threatening to win, and you can’t directly block the winning move, but you can break their piece-chain somewhere else along its length, by occupying a certain piece with your color. In that situation you have no choice but to make that move, or you’ll lose the game. But then your opponent, since they want to win, has no choice but to re-occupy that piece with their color on the very next turn, to prevent you from cementing your hold on that piece, and making a permanent hole in their position.

So the two of you go back and forth, each occupying that one piece in turn, until you run out of pieces that are able to occupy it. At that point, whatever color the piece is, is the color it will remain. The piece has become impervious.

Of course, if both players have renewables aimed at that piece, the game could be a draw right then and there. But, in the general case, the outcome will be that the piece becomes impervious as a certain color, and the game proceeds from there.

The situation doesn’t have to be as dire as winning and losing. Sometimes there is just a very good piece in the middle of the board, and both players want to occupy it if possible, and where a single tempo would be enough to render the piece impervious in whatever color it was.

In that type of situation, where both players will occupy a given piece on their turn, the destiny of that piece — its final color — can be calculated fairly easily. Then you can see whether that piece is worth fighting for right now, or if you need to prepare to fight for it, by changing its destiny from your opponent’s color (or from having an unknown destiny) to your own.

To determine who will be the last player to occupy a given piece, you simply calculate who will be the last player to swap through each of the four sides of that piece. If the same player will be the last one to swap through each side, then the piece’s destiny is that player’s color. If one player will be the last to swap through certain sides, and the other player will be the last to swap through other sides of the same piece, then the piece’s destiny is unknown.

To take a simple example, in the starting position of a game of crumble, the destiny of each piece is the opposite color from what it is. Consider the following example:

In the above diagram, the destiny of the highlighted white piece is black. This is because, when the players play out the local battle for that piece, after black finishes swapping the four neighboring pieces into the target piece, only four white rectangles will border the piece, unable to do any more splits and swaps. The final black square will split, and swap into the target piece, leaving only a white rectangle that cannot split or swap into that piece. Thus black’s final move closes the door on any further swaps by the white pieces.

In the above diagram, the north and south edges of the target piece are already impervious, so we’re only concerned with the remaining two sides. From the west, Black can swap into the target piece. From the east, White’s rectangle can split and swap into the target piece, leaving a black square behind in its wake. Then that black square can swap into the target piece, leaving an unswappable white rectangle behind. Thus both east and west sides of the target piece show the same player — Black — performing the final swap into the target piece. The destiny of the target piece is therefore black.

Here are the rules for determining the destiny of a given piece:

• • Ignore impervious edges

  • A black square on a swappable edge indicates a black destiny across that edge; a white square indicates a white destiny across that edge

  • A black rectangle on a swappable edge indicates a white destiny across that edge; a white rectangle indicates a black destiny across that edge

  • If the destinies across all swappable edges are the same, then that is the destiny of the piece. Otherwise the destiny of the piece is unknown

Why do we care?

To win, we need to create long chains of impervious or redundant pieces. We get no value from making a piece impervious, if its final state will be the opponent’s color. A key piece that is destined to be our color should be strengthened by connecting our pieces to it as we lead it towards imperviousness. A key piece that is destined to be our opponent’s color needs to be interfered with — its destiny must be rendered unknown, or else its destiny must be changed to our own color.