The game of f's and g's is, basically, just the game of noughts and crosses or tic-tac-toe. The only difference is that it is set out in a way that makes it harder to keep track of what is going on and so is slightly more complicated due to the resulting confusion. It is also set out to seem quite mathematical, but is really quite simple to understand.
The variant was developed in England by a group of students while bored during a maths lecture. For the most part the game is two player, although more players can be introduced in some of the variations. The game is played on a 3x3 grid identical to that of noughts and crosses except that cartesian axes for x and y may be drawn next to it.
Method of play
One player will be f(x,y) and the other g(x,y). The players will take it in turns to write in the squares on the grid such things as f(x,y), f(x+1,y), f(x,y+1), f(x+2,y-1). If a player writes, for example, f(x,y) on a square then they claim that square. If they write f(x+1,y) they claim the square one to the right of the square they have written in. For f(x,y+1) they claim the square one square above they have written in. For f(x+2,y-1) they claim the square two to the right and one below the square they have written in. A player cannot claim a square already claimed by the other, but may write in such a square. A player wins when they have claimed three squares in a row.
As the squares a person has claimed are not neccesarily the squares they have written in the game becomes more complicated, and so confusing an opponent can become an important tactic.
Examples
Below is an example of a game of f's and g's.
Variations
One main variation of the game is to make it three dimensional, using f(x,y,z) and g(x,y,z). This is done by having three 3x3 grids, each representing a different value of z. The aim is still to get three in a row, and this can be done across z as well as across x and y. The game can similarly be extended to higher dimensions. It is possible to have more than one player in these variations, with the extra players taking subsequent letters of the alphabet, h, i, j etc.
Some have proposed that a version of the game be created using polar co-ordinates. This variation, however, has never been fully created.