# My question: how to prove or disprove the following two conjectures?

### Conjecture 1: (Gomoku large conjecture) there is no draw on infinite board for Gomoku with any initial opening with finite stones, if both players play optimally.

It has been widely believed among Gomoku players. But I have never seen researches on "something cannot draw" for $N$-in-a-row.

Maybe we can start from 3-in-a-row? (2-in-a-row is trivial for any opening)

### Conjecture 2: (One famous open problem) $6$-in-a-row on infinite board is a draw.

[Edited May.16.2024] Today I found many researches on this, without solving even $7$-in-a-row. So proving it may be hopeless in a short time.

### Definition of $\mathbf{M},\mathbf{N}, \mathbf{K}$-games and Gomoku ($\mathbf{K}=5$): relevant Wikipedia entry

## The following statements are the reasons why I proposed these conjectures.

### Background of Gomoku

Gomoku, or Five-in-a-row, is a popular board game worldwidely. Black and white alternatively play one stone. Who first connects five stones straight or diagonally wins.

It is widely known that black (the first player) has a winning strategy for Gomoku (Five in a row). So the professional players are using swap rules: these allow the forming of a "balanced opening", in which both players think neither black nor white has advantage. Then two players start the game from this opening.

### Draw-rate vs board-size for Gomoku

For $15\times15$ boards started with balanced openings, the draw rate(the probability of draw) is about $50\%$ for top players or AIs. However, the draw rate drops exponentially when board size increases, $\sim 3\%$ for $19\times 19$ board and $\sim 0.1\%$ for $25\times 25$.

In addition, I found that more and more openings on bigger boards can be "almost-solved" by AI. For example, $F2$, $G2$, and $H2$ openings (openings with just one black stone initially) are three balanced openings for "swap after first move" rule on $15\times 15$ board. However, all of these have big advantages for black with $19\times 19$ board, means there is no balanced openings for "swap after first move" rule.

From this, I have the following conjecture：

### Conjecture 1: there is no draw on infinite board for Gomoku, if both players play optimally and the number of the stones of the initial opening is finite.

Which means, either black or white has a winning strategy.

### $6$ or more in a row, and the Critical Point of no-draw and draw

If there is no draw for $5$-in-a-row, then if we increase the difficulty of winning, there might be a critical point of no-draw and draw.

Simply increase to 6-in-a-row might be too much. It's very easy for human players to have a draw for 6-in-a-row. However it has not been proved yet. $8$-in-a-row has been proved to draw by Zetters, which is still very far from $6$.

Alphazero-like AI (Katago) can help finding the critical point. Some results for different rules for large boards are listed below

- Gomoku/Renju:
*no draw*. - $6$ in a row:
*very easy draw*. - $5$ straight and $6$ diagonal/$6$ straight and $5$ diagonal:
*easy draw*. - Gomoku on Hex board (equal to blocking one diagonal direction):
*easy draw* - Caro(not win if two teriminals of the five stones are blocked):
*no draw*. - Terminal-five(the last move must be the terminal of the five stones):
*no draw*. - Terminal-five + Caro:
.**near critical point**

Winning Waysor the variousGames of No Chancecollections? They're perhaps the best starting point for this. AFAIK youmaybe able to show that 5-in-a-row is a win on a sufficiently large board 'conceptually', but I would expect that to be well explored already and (given the lack of results) to not readily yield to any techniques that we could offer you. Finding the smallest board on which it's a win seems almost guaranteed to be a product of long computer exploration if at all. $\endgroup$Winning Ways.' $\endgroup$5more comments