At Most 43 Moves, At Least 29: Optimal Strategies and Bounds for Ultimate Tic-Tac-Toe

Abstract: Ultimate Tic-Tac-Toe is a variant of the well known tic-tac-toe (noughts and crosses) board game. Two players compete to win three aligned "fields", each of them being a tic-tac-toe game. Each move determines which field the next player must play in. We show that there exist a winning strategy for the first player, and therefore that there exist an optimal winning strategy taking at most 43 moves; that the second player can hold on at least 29 rounds; and identify any optimal strategy's first two moves.