How many stones for small go boards?

I know the full-size game of Go is played on a 19x19 board and comes with 181 black stones and 180 white stones. This is to accommodate filling every spot on the board with a stone. I want to know if there are suggestions for (much) smaller boards.

While for other typical board sizes (9x9 and 13x13), I'm guessing we could adopt a similar rule for the number of stones to include (41/41 and 85/84, respectively), but what about very small boards?

I made this 3x3 novelty board for a friend, and I'm not sure how many stones to include with it. Because of the size, I am certain many games could involve a lot of capturing, and I wouldn't want him to run out of stones. He doesn't own a full-size game, so he has no pieces to use with this little board (we have been playing on my full-size board).

For folks who have played on small boards, how many pieces do you think I should include?

Extended Question: Do you have suggestions for other small sizes -- say, the 5x5 "capture go", or some irregular sizes?

Edit: I'm not looking for a theoretical upper/lower bound on the number of stones, but rather a "from a practical standpoint, how many might I include to prevent players from having to borrow captured stones to continue playing" kind of number.

• You include exactly a single black stone for a 3x3 board. After that one stone is placed, it's game-over for white... Commented Dec 15, 2018 at 17:10
• Note that I included a "star point" for the handicap game :) Commented Dec 15, 2018 at 19:22

The number of stones for a board is meant to be effectively infinite. The typical procedure if players run out is to exchange captured stones in equal quantities and continue with them.

This means that enough stones so that either player could completely fill the board is guaranteed to be enough stones.

• "...enough stones so that either player could completely fill the board..." This is not how the large game (19x19) allots stones, preferring to split the board size between black and white, and giving the last stone to black. Commented Dec 13, 2018 at 21:31
• @user138719 that is true, because while it's the absolute maximum that could be placed, that board state isn't really logical, so in practice you don't need quite that many. It's merely an upper bound. It means that 9 stones per player is guaranteed to be more than you need for a 3x3 grid. Commented Dec 14, 2018 at 0:17
• @user138719 In a full size board it isn't likely that a game will be so one sided that they need more than half the board in any color, here though that's probably closer to 2/3 of the board as a reasonably safe number. Commented Dec 14, 2018 at 17:39
• Perhaps “effectively infinite” is better (less worrying to pedants like me). Commented Dec 15, 2018 at 18:16
• @user138719: I also believe that experience shows that, on a larger board, once a mildly competent player has occupied half the points they have won. Commented Dec 15, 2018 at 18:21

I think you could follow the same formula you've listed for other boards and you will be fine. That is black gets half the square of the number of lines rounded up and white gets half the square of the number of lines rounded down.

In a regular game you will not need more stones. That may not be true for go variants.

I would recommend to use Chinese scoring, where prisoners are given back to your opponent.

Go is played on a 19x19 board and comes with 181 black stones and 180 white stones. This is to accommodate filling every spot on the board with a stone.

Think about this: you'll never fill every spot in a game of go. Also, technically it is possible that one player might end up with more than 181 occupied spots. 180 is just a reasonable number that accomodates most scenarios.