âSmallest amount of score a player needs to have a a guaranteed victoryâ does not exist for Japanese rules (since the opponents score can be arbitrarily large thanks to captures). For Chinese rules, the answer is as @Groin said above: for a 19x19 board with 7.5 komi this means having an area of 185 points on the board.
Itâs sligthly more interesting to consider what is the minimal score that could theoretically win. Letâs start with Japanese rules.
If we count prisoners as negative points as is usually done at the end of OTB games, arbitrarily large negative scores can win. So letâs not do that.
If we donât use komi, a winning score must then be larger than 0. Itâs easy to make a player have exactly 2 points - simply fill the board with a single group having just two eyes. But how do we make just one point?
This would work if Japanese rules counted territory in seki, but they donât:
If Japanese rules used superko, we could have black play the first move in the middle of the board and then set up a position like this where white is alive with just one eye:
(but Japanese rules donât use superko)
(also this actually requires white to make a capture, increasing her score)
For some reason I considered both these solutions before realizing that we can just give one capture to one of the players!
So the answer for Japanese rules without komi is exactly 1 point (from a capture), and no territory on the board (like in the seki example above). If we give white 0.5 komi, we can skip the capture and bring this down to 0.5 points. And if we allow any komi value, we could pick an arbitrarily small positive real number.
Now, whatâs the smallest score that can win in Chinese rules, no komi? The trick here is to maximize the number of neutral points on the board. For instance, the area score on this board is W 151, B 150:
This could certainly be improved further - I stole this seki example from
Senseiâs and just rearranged the lower left corner to make the score more even.
If we donât require the game to be âcorrectly scoredâ we can do this:
Black has 2 points, white has 1 point, rest of the board is neutral.
In summary, the question is pretty meaningless, and the answers are not very clever