What's really misleading is when people try to push the question aside by saying "AI techniques should work." It's dodging the question. What I ask is whether there is an efficient algorithm to determine if one player can force a win, given a position on an arbitrary size board. Forget whether the game is imbalanced. It's just a mathematical question.
You say this game is less complicated than chess, but what measure of complexity are you talking about? Might it be computational complexity? Computer scientists of all people know that simple rules can provide arbitrary complexity.
Take chess as an example. It's an EXP-TIME complete game with a branching factor of 35, and AI techniques win pretty much all of the time. In this case you're looking at a much simpler game (if you treat it as an expectiminimax problem, you alternate between turns with a branching factor of less than or equal to 4 and ones with a branching factor of less than or equal to 32). The game is much less complicated than chess. The game tree is really small, meaning that it should be tractable to most AI techniques.
Even though it is reasonable to 'solve' the problem, there are board setups that are unwinnable. Since the player has a lot more power over the game in placing tiles than the agent does in moving them, it is easier for the player to force the game to one of the unwinnable states than it is for the agent to prevent these things. Even if the agent is acting optimally, the player should still be able to win.