What is Toroidal Go?
Toroidal Go is Go played on the surface of a doughnut, a shape mathematicians call a torus. The board looks like a normal Go board, but its edges wrap around: the left edge joins the right edge, and the top edge joins the bottom. A stone on the left edge is next to the stone directly opposite it on the right edge, and the same goes for the top and bottom.
With no edges there are no corners and no sides. Every point has exactly four neighbours, so a stone in the corner of the drawn board is as hard to capture as one in the middle. Groups can stretch across an edge and come back from the other side, and territory can run across the edges too.
The usual opening plan of taking the corners first, then the sides, then the centre no longer applies. Every point is equal at the start, and making two eyes is much harder, so fighting decides most games.
Rule changes
| Rule | Standard Go | Toroidal Go |
|---|---|---|
| Board edges | Standard GoThe board has four edges, four corners and a centre. | Toroidal GoThe left and right edges are joined, and so are the top and bottom. There are no corners or sides. |
| Liberties | Standard GoA stone has 4 liberties in the centre, 3 on an edge and 2 in a corner. | Toroidal GoEvery stone has 4 neighbouring points, counted across the edges where the board wraps. |
| Captures, suicide and ko | Standard GoGroups without liberties are captured, suicide is not allowed and ko forbids repeating the position. | Toroidal GoThe same rules, applied across the joined edges. |
| Territory | Standard GoEmpty points surrounded by one colour, with the board edges as natural walls. | Toroidal GoEmpty points surrounded by one colour, counted across the joined edges. There are no edges to use as walls. |
| How to win | Standard GoHave more points than the opponent when both players pass. | Toroidal GoThe same: the higher score after two passes wins, with the same scoring and komi as standard Go. |
History
Toroidal Go belongs to the family of board games moved onto a torus, a classic idea of recreational mathematics that has also been applied to chess and other grid games. Its exact origin is not recorded; it has long appeared in collections of Go variants and has been played by hobbyists and computer Go programmers curious to see how the game changes without edges.
The variant shows how much of Go's strategy comes from the edges. Corners are the easiest places to make living groups in standard Go, which is why games open there. On a torus that advantage disappears, and players have to rethink the opening, the value of territory and how groups live.