Connect-Four is a tic-tac-toe-like two-player game in which players alternately place pieces on a vertical
board 7 columns across and 6 rows high. Each player uses pieces of a particular color
(commonly black and red, or sometimes yellow and red), and the object is to be the
first to obtain four pieces in a horizontal, vertical, or diagonal line. Because
the board is vertical, pieces inserted in a given column always drop to the lowest
unoccupied row of that column. As soon as a column contains 6 pieces, it is full
and no other piece can be placed in the column.
Both players begin with 21 identical pieces. If all 42 pieces are played and no player
has placed four pieces in a row, the game is drawn.
The first player has a winning strategy on the standard
board (Allis 1988, p. 81).
The numbers of board arrangements with , 1, 2, ..., 42 pieces, satisfying
gravity and having equal numbers of each color for even or one extra piece of the first player's
color for odd, are 1, 7, 56, 252, 1260, 4620,
18480, 59815, 206780, ... (OEIS A090224; Allis
1988, p. 91). These counts include arrangements unreachable by alternating play
and arrangements in which play has continued after a win (Allis 1988, pp. 10-11).
At , seven of the 56 arrangements have the second player's
piece below the first player's piece in a single column, leaving 49 legal positions.
, 1, 2, ..., 42 pieces, satisfying
gravity and having equal numbers of each color for even 
or one extra piece of the first player's
color for odd 
, are 1, 7, 56, 252, 1260, 4620,
18480, 59815, 206780, ... (OEIS A090224; Allis
1988, p. 91). These counts include arrangements unreachable by alternating play
and arrangements in which play has continued after a win (Allis 1988, pp. 10-11).
At 
, seven of the 56 arrangements have the second player's
piece below the first player's piece in a single column, leaving 49 legal positions.
