GHOST COIN

Solo quick play. Start at Level 2 against the computer. Pick your two cards; the game resolves the full sequence and opens the box. The challenge is to make HEADS certain. Turn off “Resolve moves together” to reveal and step through each chip move yourself. Next round keeps your settings.

Goal. The coin starts as one green chip on HEADS. Player A wins if the coin ends on HEADS. Player B wins if it ends on TAILS.

Turns. A, then B, then A each pick one card in secret. The cards stay face-down. Then reveal them and do the moves in order.

Clean up. A green and a red chip in the same box are a zero pair: remove both. If two matching chips are left in a box, keep one.

Open the box. Chips in one box only: that side wins. Chips in both boxes: toss a real coin.

Levels. 1: only Leave and Flip. 2: Player A also gets Ghost. 3: Player B also gets Sign. 4: everyone gets every card.

On a projector: type a card's letter (L, F, S, G) instead of clicking, so nobody sees the choice.

A plays Ghost: the coin is now in both boxes with the same colors. Flip just swaps two matching chips, so B's move changes nothing. A plays Ghost again: the two TAILS chips are one green and one red, a zero pair, so they cancel. Only HEADS is left.

In Level 3, B can play Sign. The coin still looks 50/50, but now the HEADS chips cancel after the second Ghost, and the coin lands on TAILS. A hidden sign you cannot "see" decides the game.

The chip rules are exact: they give the same results as the real gates, and the coin can only be HEADS, TAILS, or a 50/50 ghost. Level 2 is David Meyer's quantum penny-flip game (Physical Review Letters 82, 1052, 1999), where a quantum player always beats a classical one. Zero pairs connect to Common Core 7.NS.A.1a: opposite quantities combine to make 0.

Discussion questions. Is Level 1 fair? Why does A win every time in Level 2? The Sign card changes nothing you can see. Why does it matter? Does a ghost coin with different colors look different from one with the same colors when you open it?