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114 competition/games/chess/size.of.game.tree.p


This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

114 competition/games/chess/size.of.game.tree.p

How many different positions are there in the game tree of chess?


Consider the following assignment of bit strings to square states:

	Square State		Bit String
	------ -----		--- ------
	Empty			0
	White Pawn		100
	Black Pawn		101
	White Rook		11111
	Black Rook		11110
	White Knight		11101
	Black Knight		11100
	White Bishop		11011
	Black Bishop		11010
	White Queen		110011
	Black Queen		110010
	White King		110001
	Black King		110000

Record a position by listing the bit string for each of the 64 squares.
For a position with all the pieces still on the board, this will take
164 bits. As pieces are captured, the number of bits needed goes down.
As pawns promote, the number of bits go up. For positions where a King
and Rook are in position to castle if castling is legal, we will need
a bit to indicate if in fact castling is legal. Same for positions
where an en-passant capture may be possible. I'm going to ignore these
on the grounds that a more clever encoding of a position than the one
that I am proposing could probably save as many bits as I need for these
considerations, and thus conjecture that 164 bits is enough to encode a
chess position.

This gives an upper bound of 2^164 positions, or 2.3x10^49 positions.

Jurg Nievergelt, of ETH Zurich, quoted the number 2^70 (or about 10^21) in
e-mail, and referred to his paper "Information content of chess positions",
ACM SIGART Newsletter 62, 13-14, April 1977, to be reprinted in "Machine
Intelligence" (ed Michie), to appear 1990.

Note that this latest estimate, 10^21, is not too intractable:
10^7 computers running at 10^7 positions per second could scan those
in 10^7 seconds, which is less than 6 months.

In fact, suppose there is a winning strategy in chess for white.
Suppose further that the strategy starts from a strong book opening,
proceeds through middle game with only moves that Deep Thought (DT)
would pick using the singular extension technique, and finally ends in
an endgame that DT can analyze completely. The book opening might take
you ten moves into the game and DT has demonstrated its ability to
analyze mates-in-20, so how many nodes would DT really have to visit?
I suggest that by using external storage such a optical WORM memory,
you could easily build up a transposition table for such a midgame. If
DT did not find a mate, you could progressively expand the width of the
search window and add to the table until it did. Of course there would
be no guarantee of success, but the table built would be useful
regardless. Also, you could change the book opening and add to the
table. This project could continue indefinitely until finally it must
solve the game (possibly using denser and denser storage media as
technology advances).

What do you think?


I think you are a little bit too optimistic about the feasibility. Solving
mate-in-19 when the moves are forcing is one thing, but solving mate-in-19
when the moves are not forcing is another. Of course, human beings are no
better at the latter task. But to solve the game in the way you described
would seem to require the ability to handle the latter task. Anyway, we
cannot really think about doing the sort of thing you described; DT is just a
poor man's chess machine project (relatively speaking).

i dont think that you understand the numbers involved.
the size of the tree is still VERY large compared to all
the advances that you cite. (speed of DT, size of worms,
endgame projects, etc) even starting a project will probably
be a waste of time since the next advance will overtake it
rather than augment it. (if you start on a journey to the
stars today, you will be met there by humans)


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