As a kid, passing time in back the family car on long rides, it began, for me, with conventional tic-tac-toe, but that quickly got tiresome on account of its sheer simplicity, even though its simplicity was initially an advantage since it made it easy to visualize and play simply by means of calling off coordinates when the often skimpy supplies of paper (brown paper bags or wrapping paper) were exhausted or the ride was too rough to make play convenient while scribbling on unsupported paper with a pencil stub sharpened by means of chewing it to a point when the lead broke.

Casting about for a game somewhat more challenging, next it was simple 3-d tic-tac-toe, played in a 3 x 3 x 3 cube, which in turn quickly palled on account of the fact that the game allows a trivial win on a basis of a first move advantage, just so long as one makes no mistakes in play. To make the game more interesting it was expanded to play in a 4 x 4 x 4 field, which allows of a draw if neither player makes a mistake, just as does simple 3 x 3 2-d tic-tac-toe.

Calling off the coordinates, where the 'R' number is the number of the round of the game (each player makes one move per round), P1 is player one and P2 is player two, X is the horizontal axis with squares numbered from left to right and Y is the vertical axis with squares numbered from bottom to top, a two-d game might go as follows, with the players calling off their moves by means of giving the coordinates of the square they want to mark:

R1 [P1

**,**X2 x Y2 (the center square)

**;**P2

**,**X3 x Y1 (lower right hand square)]

R2 [P1

**,**X3 x Y3

**;**P2

**,**X1 x Y1 (blocking P1 from a win on their next move)

R3 [P1

**,**X2 x Y1, (blocking P2 from a win on

*their*next move)

**;**P2

**,**X2 x Y3 (blocking P1)]

R4 [P1, X3 x Y2

**;**P2

**,**X1 x Y2]

**,**ending the game in a draw, cats' game.

Expanding the game a dimension, in a 3 x 3 x 3 manifold, Z is from front to back, a trivial first move win follows:

R1 [P1

**,**X2 x Y2 x Z2

**;**P2

**,**X1 x Y1 x Z1]

R2 [P1

**,**X1 x Y3 x Z3

**;**P2

**,**X3 x Y1 x Z1 (blocking a possible win by P1)]

R3 [P1

**,**X2 x Y1 x Z1

**;**P2

**,**X2 x Y3 x Z3]

R4 [P1

**,**X1 x Y2 x Z2

**...**(P1 has created a situation where no matter what P2 does on their next move, P1 wins. As a matter of fact P1 always wins just so long as they make no errors in play.)]

Expanding the game again, this time to a 4 x 4 x 4 manifold, reintroduces the possibility of a taut game draw.

Though a major part of the reason for the game is to build one's capacity for visualization, one shouldn't hesitate to draw diagrams as necessary.

For a 3-d game, a simple way to diagram on paper is a vertical column of tic-tac-toe boards where the lowest board is arbitrarily the one in 'front' and the highest is the board in the 'back' of the cube.

Expanding into a 4-d game, a simple way to arrange the diagram is in the form of a square with as many tic-tac-toe boards on a side as there are squares on each board. (This also works for checkers and chess, in which case each square of the checkerboard is replaced with an entire board, the flat projection making it easy to display on a tabletop.)

For 5-d, one simply takes as many 4-d boards as there are squares on the side of the board and arranges them into a column, horizontally or vertically as one likes. With even numbered dimensions higher than zero, one can always arrange the playing field projected onto a 2-d square, and with odd numbered dimensions, the field can always be arranged in columns.

For four dimensions, its simple and convenient to first specify the given board and then the specific space on that board. Four example, one might say, where 'b' represents a board and 's' represents a space on a board, " 4-d game, 5 spaces on a side of each board, 'b' X3 x Y3, 's' X3 x Y3." to indicate the center space of your 5 x 5 x 5 x 5 four dimensional playing field.

Anybody care to get up a game?