Design a Tic-Tac-Toe Engine

Implement an `n x n` Tic-Tac-Toe game engine for two players. Support a method `move(row, col, player)` that records a move and returns whether that move causes the player to win. Discuss two versions: 1. A straightforward implementation that stores the full board and checks rows, columns, and diagonals after each move. 2. An optimized implementation that avoids storing the entire board and updates the winner status in constant time per move. Assume all moves are valid and no move is made after the game has already been won.