SVSM 1999, Combinatorial Games

R. Tyler Thomas

The purpose of this paper is to explain how one may go about perfecting the combinatorial game of Connect Four and how one wins every time they play the game. One may think that Connect Four is a kid’s game, which in most cases it is. Even though children love to play Connect Four, they do not see the deep thought and many maneuvers that one can make at any given point in the game. That is why this game takes time and skill to perfect.

The rules for playing Connect Four are as follows:

1) Two people take turns dropping chips (black or red) into a 6x7 grid.

2) The player may only drop one chip at a time into the grid.

3) Chips must go all the way to bottom of grid until another chip stops its fall.

4) First person to get four chips in a row horizontally, vertically, or diagonally, wins the game. (Unless the players fill the grid up, which results in a tie.)

A combinatorial game is a two person game that follows these two conditions:

1. players alternate removing (or in some cases adding) counters from a finite collection or graph according to set rules and
2. The last player to remove (or add) a counter wins.

Most combinatorial games are impartial games or have a definite winner, and either player may make each move. However, Connect Four is a partisan game. This means that there may be ties and each player can only make moves from his/her own pile or pieces they must play with.

In most combinatorial games, one usually wants to achieve symmetry. However, in Connect Four one never wants to make symmetry and if one does make symmetry never replace symmetry. If one player does that then they are guaranteed a loss.

Let’s take a look at some situations of different games.

Diagram 1

a b c d e f g

In Connect Four there is a chart called "Joseki." Joseki comes from the Japanese language, and means expert opening move or perfect play. In order to play Connect Four perfect one would need to memorize the "Joseki" chart. The chart below is not the "Joseki" chart, but it is a chart that shows what would happen with perfect play if each one of the chips fell the way that it was supposed too. Such that if X (first player) dropped a chip into a1, b1, f1, or g1 then O (second player) would win with perfect play. If X moves to c1 or e1 then with best play the game will be a tie. However, if X places a chip into d1 then the first player would win. The chart follows the same pattern all the way to the end.

"X" --> X (Eques, First Player) Wins after the indicated Move

"O" --> O (Knott, Second Player) Wins after the indicated Move

"=" --> Drawn Game after the indicated Move

- - - - Move - - - -

a b c d e f g

-------------------------

X's First Move: O O = X = O O

O's Response to

1 A1 X O O O X O X

1 B1 X = O = X X X

1 C1 X X = = = = X

1 D1 X X X X X X X

X's Response to

1 A1 a2 O = = X O X O

1 A1 b1 O O O O O O O

1 A1 c1 O O O O O O O

1 A1 d1 O O O O O O O

1 A1 e1 O O O X O O O

1 A1 f1 O O O O O O O

1 A1 g1 O O O X O O O

1 B1 a1 = X = O X O O

1 B1 b2 O O O O = O O

1 B1 c1 O O O O O O O

1 B1 d1 O = O = O O O

1 B1 e1 O O O X O O O

1 B1 f1 O X = X X = O

1 B1 g1 O X = X = O O

1 C1 a1 = = = X = = O

1 C1 b1 O = O O = X O

1 C1 c2 O = O = O O O

1 C1 d1 O O = = O O O

1 C1 e1 O O = O = O O

1 C1 f1 O = = = = = O

1 C1 g1 O = X X = X O

1 D1 a1 = O X X X X X

1 D1 b1 O X O = = X O

1 D1 d2 O O O X O O O

X's Response to

1 D1 d2 D3 b1 O X O X X X O

1 D1 d2 D3 c1 O O X X O X X

Now let’s take a look at some terms that are used in Connect Four.

A cell is one of the 42 different locations on the board or grid that stones will eventually be placed in to try and connect four in a row. An Atari is a move that threatens to make Four-in-a-Row on the next move. Double Atari however, is when a win is guaranteed in the next move. In other words, no matter where the other player goes one can win. Finally, the whole objective of the game, Four-in-a-Row, or four chips of the same color in a continuous line horizontally, vertically, or diagonally.

Here is what Double Atari looks like.

a b c d e f g

6 ?

5 o ? o o

4 x o x x

3 o x o o

2 o x o x x

1 x o x o x x

No matter where (x) goes next, (o) is going to win.

From all of these graphs and different variations of playing methods one can conclude that Connect Four is a lot more than a child’s game. Connect Four is a very elaborate drawn out process, that would take years to master and perfect every move.

However, when children play Connect Four, they will never know the complicated process that one must learn in order to perfect this game.

I would like to thank James Allen for posting information on a game that I knew how to play, but never knew how to play it to the best of my ability.