A bar game: you and I take turns claiming the numbers 1–9, and the first to hold three that add to fifteen wins. Feels like arithmetic. Peel it and it is TIC-TAC-TOE wearing a disguise — the nine numbers are a magic square, and every three-that-make-fifteen is a straight line on the board.
The nine cells are the Lo Shu magic square (2 7 6 / 9 5 1 / 4 3 8): every row, column and diagonal sums to exactly 15 — verified live, 8/8 lines. 'Hold three numbers that sum to 15' is then identical to 'own a straight line' — so the number game and tic-tac-toe are the same game, move for move. The eight 15-triples ARE the eight tic-tac-toe lines.
The numbers game 'Fifteen', the coins-and-corners patter, and casting the machine as an opponent are the wrapper; a real magic square is used, but which cultural game you call it is costume.
A game is not a proof — this shows an isomorphism you can check by hand, it does not make tic-tac-toe 'about' arithmetic; the machine plays a simple heuristic, not perfect adversary theory.