Yes, with Nim being the best-known example --- You ask for a strongly solved game (presumably referring to the term in combinatorial game theory). [According to Wikipedia][1], games that are strongly solved > Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides. It goes on to note that in many cases, this is determined by brute force: > By contrast, "strong" proofs often proceed by brute force—using a computer to exhaustively search a game tree to figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board. For instance, we can strongly solve 3x3 tic-tac-toe because we can brute force every possible combination and determine perfect moves. Of course, brute force is impossible for an infinite board size, and some games are only strongly solved up to a certain board size. For instance, Hex is only strongly solved up to a 6x6 board. Still, the hope is that this brute force on games like these will lead to an algorithm for perfect play that works for an infinite size (e.g. an *N*x*N* board). The best studied strongly solved game is [Nim][3], where an algorithm has been found that produces perfect play from any position of any size heaps and objects. This game can be played on a board (or just by drawing on a piece of paper), so it counts as a board game. Most other combinatorial games are played on a fixed board and are not typically studied at *N*x*N* sizes. In some cases, an algorithm for those sizes may not be possible. As described above Hex has been strongly solved for a 6x6 board. However, an algorithm that works for *N*x*N* boards is unlikely to be found because the problem is [PSPACE-complete][2]. [1]: https://en.wikipedia.org/wiki/Solved_game [2]: https://en.wikipedia.org/wiki/PSPACE-complete [3]: https://en.wikipedia.org/wiki/Nim