Abstract
When black stones, white stones and empty points are placed on a Go board, the arrangement is a position that can exist on the board (a legal position) if every group of stones is adjacent to an empty point. This paper counts legal arrangements on small boards in two ways, exhaustively and row by row, and finds how their share of all arrangements changes with board size and width. No new theorem or law is claimed.
First, the two counts agreed on five boards: $57$ on $2\times2$, $12675$ on $3\times3$, $321689$ on $3\times4$, $299681$ on $2\times6$ and $141209$ on $1\times12$ (Section 2).
Second, the legal share falls as points are added, and differs with board shape for the same number of points: the share of all arrangements was $0.703704$ on $2\times2$, $0.643957$ on $3\times3$ and $0.605315$ on $3\times4$; with the same $12$ points, $2\times6$ gave $0.563903$ and $1\times12$ gave $\mathbf{0.265710}$ (Section 2).
Third, the growth from adding a row settles to a fixed ratio: for widths $1$, $2$, $3$ and $4$, the legal count grew by factors of $2.769292354$, $8.533652512$, $25.445014706$ and $75.709341135$ per added row, agreeing with the previous row to $9$ digits (Section 3).
Fourth, the per-point ratio is closer to 1 on wider boards: the per-point share (the factor by which the legal share changes when one point is added) was $\mathbf{0.923097}$ for width $1$, $0.973747$ for width $2$, $0.980422$ for width $3$ and $\mathbf{0.983255}$ for width $4$ (Section 3).
Fifth, the separator is how many neighbours a point has: on a one-row board a point has at most two neighbours and groups rarely get an empty neighbour. As the board widens, neighbours increase and the same rule removes a smaller share per point (Section 4).
Keywords: Go, legal positions, liberties, transfer matrix, enumeration, board width.
Scope (what is not claimed)
No new theorem or law is claimed: the definition of a legal Go position, counting row by row with a state, and the exponential growth of the number of legal positions with the number of points are all known. Only small boards are counted: only boards up to width $4$ and exhaustive counts up to $12$ points; the number for $19\times19$ and limiting values for wide boards are not given. The rules of play are not entered: suicide, ko, the ban on repeated positions, and which positions appear in real games are not treated. Only arrangements are counted. Strength is not discussed: winning and losing and the quality of moves are not treated. Relation to earlier papers: Paper 846 showed that whether you can win is settled by painting backwards from the end, and how many moves it takes is not in that label. Here not winning or losing but the arrangements that can exist on the board are counted. What is added: checking the two counts against each other on five boards, listing the legal share by board size and shape, confirming that the ratio per added row settles, finding the per-point ratio for each width, and placing the separator on “how many neighbours a point has”.
1. Notation and the Question
Legal: each point of the board holds black, white or empty. Stones of the same colour adjacent vertically or horizontally join into groups. An arrangement is called legal if every group is adjacent to at least one empty point.
Counting: (1) Exhaustive: all arrangements, $3$ to the power of the number of points, are checked (up to $12$ points). (2) Row by row: rows are added one at a time with a state holding the colours of the last row and, for its stones, which group each belongs to and whether that group already has an empty neighbour (a transfer matrix). An arrangement is discarded if a group does not continue into the next row and has no empty neighbour. The number of states was $5$, $17$, $67$ and $277$ for widths $1$, $2$, $3$ and $4$.
Question: what share of all arrangements is legal, and how does that share change with board size and width?
2. Counts on Small Boards
Computation 2.1: exhaustive and row-by-row counts.
| board | exhaustive | row by row | share of all arrangements |
|---|---|---|---|
| $2\times2$ | $57$ | $57$ | $0.703704$ |
| $3\times3$ | $12675$ | $12675$ | $0.643957$ |
| $3\times4$ | $321689$ | $321689$ | $0.605315$ |
| $2\times6$ | $299681$ | $299681$ | $0.563903$ |
| $1\times12$ | $141209$ | $141209$ | $\mathbf{0.265710}$ |
The row-by-row count also agreed for boards turned sideways ($2\times3$ and $3\times2$, $3\times4$ and $4\times3$).
Observation 2.2 the more points, the lower the legal share, and for the same number of points the narrower board is lower. Of the three $12$-point boards, $3\times4$ gave $0.605315$, $2\times6$ gave $0.563903$ and $1\times12$ gave $0.265710$: on a one-row board far more arrangements leave a group without an empty neighbour.
3. Ratio per Added Row and per Point
Computation 3.1: for width $w$, the ratio when the height grows by one (the last two of heights counted up to $40$, $30$, $24$ and $18$) and its $w$-th root (per point).
| width | ratio per added row | ratio per point | share relative to $3$ |
|---|---|---|---|
| $1$ | $2.769292354$ | $2.769292$ | $\mathbf{0.923097}$ |
| $2$ | $8.533652512$ | $2.921242$ | $0.973747$ |
| $3$ | $25.445014706$ | $2.941266$ | $0.980422$ |
| $4$ | $75.709341135$ | $2.949765$ | $\mathbf{0.983255}$ |
For all four widths, the ratio per added row agreed with that of the previous row within a relative $10^{-8}$ (it has settled as the height grows).
Observation 3.2 the legal share falls by a nearly fixed ratio per point, and that ratio is closer to 1 on wider boards. The number of arrangements triples per point, but the legal count grows only by $2.769292$ per point for width $1$: the legal share becomes $0.923097$ times as large per added point for width $1$ and $0.983255$ times for width $4$. In the order of widths $1$, $2$, $3$ and $4$ the ratio approached $1$.
4. The Separator
Observation 4.1 the separator is how many neighbours a point has. The legality rule is only “every group has an empty neighbour”, the same on every board: on a one-row board a point has at most two neighbours, groups rarely get an empty neighbour, and a larger share is removed per added point. As the board widens, inner points have four neighbours and the same rule removes a smaller share per point.
Observation 4.2 what is not said. The number for $19\times19$, the limiting ratio for wide boards, rules of play such as suicide and ko, and playing strength are not treated. What is shown is that on boards up to width $4$ the legal share falls by a fixed ratio per point and that ratio approaches $1$ as the width grows.
5. Roll Call and Closing
- legality rule and two ways of counting (Section 1): $5$, $17$, $67$ and $277$ states.
- agreement on five boards (Computation 2.1): $12675$ on $3\times3$.
- $\mathbf{0.265710}$ for $1\times12$ with the same $12$ points (Computation 2.1): narrower is lower.
- ratio per added row settles (Computation 3.1): within a relative $10^{-8}$.
- per point from $\mathbf{0.923097}$ to $\mathbf{0.983255}$ (Computation 3.1): closer to $1$ on wider boards.
- number of neighbours of a point (Observation 4.1): the separator; Paper 846.
Closing: The share of legal Go positions shrinks by a fixed ratio per point, and less so on wider boards. The exhaustive and row-by-row counts agreed on five boards, and $3\times3$ had $12675$ legal arrangements, a share of $0.643957$: per added point the legal share became $0.923097$ times as large for width $1$ and $0.983255$ times for width $4$, and with the same $12$ points $1\times12$ gave $0.265710$ and $3\times4$ gave $0.605315$. The separator is how many neighbours a point has. Placed among the earlier papers: before the winning and losing that Paper 846 settled by painting backwards, the arrangements that can exist on the board were counted. To be honest: values for wide boards are not given. How much the legality rule removes from the arrangements on small boards, and how that changes with width, and no further.
References
- Tromp, J., & Farnebäck, G. (2007). Combinatorics of Go. Computers and Games. Springer.
- Yamagishi, Y. (2026). Whether You Can Win Is Settled by Painting Backwards; How Many Moves It Takes Is Not in That Label [Paper 846]. YOI: https://www.yamagishiyuuki.com/yoi/846
Reference
Yamagishi, Y. (2026). The Share of Legal Go Positions Shrinks by a Fixed Ratio per Point, and Less So on Wider Boards [Paper 945]. YOI: https://www.yamagishiyuuki.com/yoi/945
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 945. This paper is a preprint and has not been peer reviewed.