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116 competition/games/connect.four.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.

116 competition/games/connect.four.p

Is there a winning strategy for Connect Four?


An AI program has solved Connect Four for the standard 7 x 6 board.
The conclusion: White wins, was confirmed by the brute force check made by
James D. Allen, which has been published in rec.games.programmer.

The program called VICTOR consists of a pure knowledge-based evaluation
function which can give three values to a position:
1 won by white,
0 still unclear.
-1 at least a draw for Black,

This evaluation function is based on 9 strategic rules concerning the game,
which all nine have been (mathematically) proven to be correct.
This means that a claim made about the game-theoretical value of a position
by VICTOR, is correct, although no search tree is built.
If the result 1 or -1 is given, the program outputs a set of rules applied,
indicating the way the result can be achieved.
This way one evaluation can be used to play the game to the end without any
extra calculation (unless the position was still unclear, of course).

Using the evaluation function alone, it has been shown that Black can at least
draw the game on any 6 x (2n) board. VICTOR found an easy strategy for
these boardsizes, which can be taught to anyone within 5 minutes. Nevertheless,
this strategy had not been encountered before by any humans, as far as I know.

For 7 x (2n) boards a similar strategy was found, in case White does not
start the game in the middle column. In these cases Black can therefore at
least draw the game.

Furthermore, VICTOR needed only to check a few dozen positions to show
that Black can at least draw the game on the 7 x 4 board.

Evaluation of a position on a 7 x 4 or 7 x 6 board costs between 0.01 and 10
CPU seconds on a Sun4.

For the 7 x 6 board too many positions were unclear. For that reason a
combination of Conspiracy-Number Search and Depth First Search was used
to determine the game-theoretical value. This took several hundreds of hours
on a Sun4.

The main reason for the large amount of search needed, was the fact that in
many variations, the win for White was very difficult to achieve.
This caused many positions to be unclear for the evaluation function.

Using the results of the search, a database will be constructed
of roughly 500.000 positions with their game-theoretical value.
Using this datebase, VICTOR can play against humans or other programs,
winning all the time (playing White). The average move takes less
than a second of calculation (search in the database or evaluation
of the position by the evaluation function).

Some variations are given below (columns and rows are numbered as is customary
in chess):

1. d1, .. The only winning move.

After 1. .., a1 wins 2. e1. Other second moves for White has not been
checked yet.
After 1. .., b1 wins 2. f1. Other second moves for White has not been
checked yet.
After 1. .., c1 wins 2. f1. Only 2 g1 has not been checked yet. All other
second moves for White give Black at least a draw.
After 1. .., d2 wins 2. d3. All other second moves for White give black
at least a draw.

A nice example of the difficulty White has to win:

1. d1, d2
2. d3, d4
3. d5, b1
4. b2!

The first three moves for White are forced, while alternatives at the
fourth moves of White are not checked yet.

A variation which took much time to check and eventually turned out
to be at least a draw for Black, was:

1. d1, c1
2. c2?, .. f1 wins, while c2 does not.
2. .., c3 Only move which gives Black the draw.
3. c4, .. White's best chance.
3. .., g1!! Only 3 .., d2 has not been checked completely, while all
other third moves for Black have been shown to lose.

The project has been described in my 'doctoraalscriptie' (Master thesis)
which has been supervised by Prof.Dr H.J. van den Herik of the
Rijksuniversiteit Limburg (The Netherlands).

I will give more details if requested.

Victor Allis.
Vrije Universiteit van Amsterdam.
The Netherlands.


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