Gaming the coordinates

15 Apr 2024

Sensei’s library [1] lists several coordinate systems for the standard 19x19 size Go board. The system shown above must rank as one of the more unusual ones.

It was inspired by a recent email from fellow games researcher Ryan Hayward [2], in which he noticed that a triangle shaped board on just 3 points, has precisely 19 legal positions, as well as 3439 legal games (under the Logical rules [3]).

Having studied Hamiltonian games in our Combinatorics of Go paper [4], I was naturally curious to see if triangular Go was also Hamiltonian. That is, whether one can play a game that visits all legal positions.

As it turns out, not only can one do so, but (up to symmetry) the game is unique and has no intermediate passes!

That makes it quite natural to use the 19 successive positions in that game as coordinates to mark the successive vertical lines of the Go board. We similarly mark successive horizontal lines of the Go board, except for flipping the triangles upside-down, so that x and y coordinates nicely align into a slanted 2x3 grid, as shown for example with tengen coordinates

Curiously, the star points are exactly those, both of whose coordinates have 2 black stones.

Obviously this is not the most practical of coordinate systems, but it does have a certain aesthetic appeal…

[1] Sensei’s Library

[2] Ryan Hayward

[3] The Logical rules of Go

[4] Combinatorics of Go