A game of Boter-kaas-en-eieren played on this paper.

After winning the BAPC, Beatrijs decided to pursue even more demanding mind sports: first chess, then Go, and recently, she learned the game of Crosses and Circles in Germany. (Specifically: https://wcm.cpulm.org/contest/gcpc-2026/materials/9/view#page=7) She had never heard of this game before, but it turns out that this game is known across the world under different names, like Tic-tac-toe, Noughts and Crosses, or, in Dutch, Boter-kaas-en-eieren.

The rules of the game are as follows:

• Each player has a unique symbol, either crosses ('X') or circles ('O'). The starting player can choose either symbol.
• The two players take alternating turns to place their symbol in an empty cell of a $$$3 \times 3$$$ grid.

• The first player to place three of their symbols in a row (either horizontally, vertically, or diagonally) wins, and the game ends immediately.
• If all nine cells are filled and neither player has three symbols in a row, the game is a draw.

To improve her skills, Beatrijs has been watching the world championship games of crosses and circles, and one game in particular stuck with her: she remembers a board state where $$$x$$$ crosses and $$$o$$$ circles were played, but she can recall neither the exact sequence of moves nor the final board state.

On more careful thought, Beatrijs is not even sure if the game was valid at all, and she wonders if it is even possible for a valid game to have $$$x$$$ crosses and $$$o$$$ circles.