AFAIK, the most complex board game solved to date is checkers http://en.wikipedia.org/wiki/Chinook_(draughts_player). Chess is also solvable, but researchers are far from solving it yet
This means that Chinook can play perfectly from the start position against any set of opposing moves, but if you play a move on Chinook's side that it wouldn't play for itself, the resulting position is probably not solved by Chinook.
An arbitrary position is in fact very unlikely to have been solved.