Posts Tagged ‘combinatorial game theory’

Duke and Pawn Endgames

September 3, 2007

Note: to understand this post, you don’t need to know anything about chess; I’ll explain everything necessary right here.

One of the best ways for a chess beginner to improve is to study king and pawn endgames, and one of the best ways to do that is to play a game with only kings and pawns. Just remove all the rest of the pieces, and have at it.

Unfortunately, while Chess with just kings and pawns is not trivial, as you improve you will eventually outgrow it, because between two good chess players, the game is very likely to end in a draw.

As I mentioned in an earlier post, my son Adam likes to design games, and he designed this very clever variant of the King and Pawn endgame that eliminates the possibility of a draw.



Set up the pieces as above, with four pawns plus a King for both White and Black.

The players alternate moves, starting with White.

The pawns move as in chess: one square forward, or optionally two squares forward if they haven’t moved yet, and they capture a piece one square ahead of them diagonally.

Pawns may still capture en passant as in Chess, which means the following. If an enemy pawn moves two squares forward, and one of your pawns could have captured it if it had only moved one square forward, you may capture that pawn as if it had only moved one square forward, but only if you do so with your pawn on the turn immediately after the enemy pawn moves.

The Kings can only move one square up or down, or left or right; they cannot move diagonally. (Such limited versions of Kings are sometimes called “Dukes;” hence the title of this post.)

You win the game immediately if you capture the opponent’s King, or if one of your pawns reaches the other side of the board.

You must always move; if you cannot make a legal move, you lose the game. (This is different from Chess, where stalemate is a draw.)

You may not make a move which recreates a position that has previously occurred during the game. (This is also different from Chess, but such a rule exists in Shogi, the Japanese version of Chess.)

That’s all the rules. In this game, draws are impossible, even if only two Kings remain on the board. For example, imagine that the White King is on a1, and the Black King is on b2. If White is to move, he loses, because he must move to a2 or b1, when Black can capture his King. But if Black is to move, he must retreat to the north-east, when White will follow him, until Black eventually reaches h8 and has no more retreat.

So arranging that your opponent is to move if the two Kings are placed on squares of the same color (if there are no pawn moves) is the key to victory. This concept, known as the “opposition,” is also very important in ordinary Chess King and pawn endgames, which is why skill at chess will translate into skill in this variant, and why improving your play at this variant will improve your Chess. Of course, the pawns are there, and they complicate things enormously!

Naturally, one can consider other starting positions, with different numbers of pawns.

This game can be analyzed using the methods of Combinatorial Game Theory (CGT). Noam Elkies, the Harvard mathematician, has written a superb article on the application of CGT to ordinary chess endgames, but it required great cleverness for him to find positions for which CGT could be applied; with this variant, the application of CGT should be much easier.


Combinatorial Game Theory

August 29, 2007

Combinatorial Game Theory (CGT) is a mathematical discipline that studies two-player, zero-sum games, where the players play sequentially, and there is perfect information and no chance. The aim of CGT is to build mathematical theories that will help players win such games! In practice, the ideas from CGT are of minimal help to the serious player in more complicated games like Chess or Go, but the theories are extremely helpful for simpler but still non-trivial games like Dots-and-Boxes.


The CGT bible is Winning Ways for Your Mathematical Plays, a four-volume set of books (for the 2nd edition from the early 2000’s; the first edition from 1982 was in two volumes), by the eminent mathematicians Elwyn Berlekamp, John Horton Conway, and Richard Guy.

The basic idea of CGT is to develop a calculus, which lets you assign a value to a position in the game. One normally considers games with the condition that a player who cannot move loses. If both players are without a move in a position (whoever moves loses), the position is assigned a value of 0. If the player on the right can make one move that transforms the position to one of value 0, and the player on the left cannot move at all, the position is assigned a value of 1 (if the roles are reversed, the position has value -1).

Now what happens if the player on the left has a move to a position of value 0, and the player on the right has a move to a position of value 1? What is the value of that position? It turns out to be worth 1/2, as can be seen by considering the position obtained by adding such two such positions together with a position of value -1, when one gets back to a position of value 0 where whoever moves loses (I’ll let you work out the details).

There are more different kinds of values than just fractions. Berlekamp, Conway, and Guy consider a multitude of different interesting games, which give rise to ever deeper mathematics, and game positions with strange values along with complex rules for combining those values.

The writing is informal, with constant punning, but the mathematics is serious. Still, one charm of the field is that there is no pre-requisite mathematical material needed to understand these books–bright high school students can launch right in.

After finishing “Winning Ways,” the ambitious student can move on to the two volumes “Games of No Chance” and “More Games of No Chance” (note that nearly all the papers in these books are available online), and then move onto the rest of the papers collected at David Eppstein‘s CGT page.

Also, take a look at Aaron Siegel‘s Combinatorial Game Suite software to help you do CGT computations.

There’s still much more to say, but I’ll leave that for other posts.