The 7x7 Go board game appears to be "practically solved", in that we have a good idea of optimal play, but not "solved" in a mathematical sense.
This market resolves based on 7x7 Go with a komi of 9. This is believed to be a draw. An ultraweak solution is sufficient. A solution for any mainstream Go rules is sufficient, eg Tromp-Taylor.
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I'm confident sure the analysis described here is good and can convince ago player that the correct komi for 7x7 go is 9. https://senseis.xmp.net/?7x7ArticleByJDavies
The thing is, often a move is obviously bad for a go player, or even obviously not the best one. Sometime it may be possible to prove it, like filling an eye or more generally playing inside an already living group. However, in most cases there is no possible way to prove this mathematically other than brute force exploration.
John Tromp has solved 2x2 go : http://tromp.github.io/java/go/twoxtwo.html
I'm quite sure if 3x3 go was solved, it would be somewhere on this site.