Wednesday, September 7, 2011

Knights are Weird

While daydreaming about topology the other day, I was taken by a singular question.

"How do the chess pieces view the chess board?"

If each move is taken as a single step, if each square that a piece can move to is considered adjacent to the square the piece is on, each piece lives in a very different world from the others.  Bishops do not even see half of the board. On an empty board 14 squares are adjacent to the Rook, and 49 others 50 are two steps away, no matter what square they are on. This is not new ground of course, it is covered often when discussing the mobility of the pieces, but it is another step closer to verbalizing and understanding a vague idea I have had for a while.

When I was teaching my sons how to play chess, I tried to help them visualize how the pieces interact, by calling the squares that a piece can move to the shape of the move. I was reaching for a Euclidian geometric description.

The chessboard appears to be a Euclidean place, 64 squares assembled into a flat 8x8 grid. The Go board is such a place, but the chessboard isn't. It is a network of 64 points, but the connectivity of that graph is different for the different pieces. The graph changes for the pieces as other pieces move. No point is reachable for a piece if a ally piece occupies that point. For the pawn, the graph changes radically as the pieces move, because diagonal moves are dependent on an enemy piece occupying a point.

And the Knight views the board in a wholly different way:
1-red, 2-orange, 3-yellow, 4-green, 5-blue, 6-purple
I have colored the squares based on how many moves it takes for a Knight to reach them. But beyond the red squares (one step) coloring is not enough, because the path to those squares matters. Lets place a white pawn on f2, so the knight can only go through g3.
1-red, 2-orange, 3-yellow, 4-green, 5-blue, 6-purple
Now it takes two more moves to reach e1, e5, d1, d3, h3,g4, b2, and b4 .

I think it is well known that Knights are tricky, but how do we get better at using our Knights? I do not think that this is so clear. I am going to try working on knight endgames. They are not very common (Silman does not even include the N+B mate in his Complete Endgame Course), but I think the clarity of endgame positions will help. OTOH, here is a fun puzzle game called Black Knight, that you maneuver a knight around different board topologys.

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