GHOST GRID

Place: every move is one pair of spooky marks. X takes odd-numbered moves and O takes even-numbered moves. Put the pair in two different squares that do not contain real marks. Spooky marks from any number of earlier moves may share a square. If only one nonclassical square remains on move nine, both marks go there.

Entangle: pairs sharing a square become entangled. If a new pair closes a loop, the entire connected group—including any stems—must collapse. Its marks are underlined while a collapse choice is pending.

Collapse: the player who did not close the loop chooses which square receives the loop-closing pair. That choice forces every linked pair into one square. The kept halves become real marks; the rejected halves disappear. Real squares are locked. Choosing the collapse does not use the chooser’s regular move.

Win: only real marks count. Three real marks in a row wins after a collapse. If one collapse produces several rows, a row’s completion number is its largest move number. Each row with the smallest completion number scores 1 point and later rows score ½ point; the higher total wins. Equal totals are a draw. If move nine produces no real three-in-a-row, the game is a cat’s game (tie).

Controls: use the right-side pad or keys 1–9. Escape cancels the first square. During a cycle, choose either destination offered for the newest pair. Each pair has a color, move number, and halves A and B. After choosing a collapse, click Next step, either highlighted square, or press Enter, Right Arrow, or Page Down to resolve each forced mark. Orange shows the departing mark; green shows its destination. Wins are checked only when the whole group is resolved. Undo cancels a selection or reverses one placement, collapse choice, or chain step. New game resets everything. Full screen is best for projection.

Rules follow Allan Goff, “Quantum tic-tac-toe: A teaching metaphor for superposition in quantum mechanics,” American Journal of Physics 74 (2006), pp. 962–973. The score calculation uses the classical move-order method described for simultaneous three-in-a-rows.