Let’s call an enclosed eyespace a single-eye shape if it cannot be made to yield two eyes when the opponent moves first. A surrounded group whose only eyespace is a single-eye shape cannot live.
If you play Go you already know several of these, under the name nakade (中手, “inside move”) — the throw-in that kills them, and by extension the shapes themselves.[1] There are nine single-eye shapes on a Go board, and the five larger ones have colorful names: the square four, the pyramid four, the bulky five, the crossed five, and the rabbity six. The other four are the very small eyespaces, one to three points, perhaps too basic to have earned catchy names.
The largest is six points, and a 2003 paper by Ramon Vilà and Tristan Cazenave showed there is no seven-point shape. They worked out that a Go eyespace’s fate can be read off a small amount of information about its shape.
The nine single-eye shapes of Go, grouped by eyespace size.
The vital points of these shapes are marked with red circles. Unless the attacker plays one of them next, the defender has a move that lives. Shapes with no circle are dead even if the defender moves first.
Badux Go is played on a hex board. Same stones, same captures, same two eyes to live — the one change is that every point has six neighbors instead of four. That makes the list above a fact about square boards rather than a fact about Go. So I thought it would be useful to draw up a list for people interested in Badux.
I wrote a program that plays the eyespace out properly — real captures, real liberties, no suicide, both sides playing as well as they can. A shape is dead if the attacker, moving first, can capture the group around it. To check the program was working, I ran it on an ordinary Go board, where it produced the established nine.
The 21 single-eye shapes of Badux, grouped by eyespace size.
Twenty-one shapes, sizes one through eight. Badux single-eye shapes are bigger, so a Go player’s sense of what is large enough to live may be wrong here.
Single-eye shapes by eyespace size, Badux against Go.
Vilà and Cazenave showed there is no seven-point single-eye shape in Go. My program checked all the shapes on both boards up to size nine. So any shape of size 9 or less that is missing from these charts can make two eyes. But that is not the same as the space being safely the defender’s: a large enough eyespace can hold a seki, or an attacker’s group that lives. Past size nine, a missing shape means only that I know of no published check.
All eyespace shapes by size, Badux against Go.
The code, and the shapes as machine-readable data, are at github.com/VariationalGames/nakade-analysis.
Come play at baduxgo.com and see if you can spot one of these in the wild.
[1] I am avoiding nakade as the technical term because it is used in at least two incompatible ways. Players use it for the killing move. Vilà and Cazenave use it for a status a position acquires once the vital points are filled, and separately for shapes with one or zero vital points. “Single-eye shape” sidesteps all of that. The rabbity six, meanwhile, is called that because it looks like a rabbit, and I am not going to improve on it.
