Ghost Twins

Solo or 2 players + Keeper · 4–7 minutes
Short classroom activity · SpookyQubits · 3 November 2026

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The question

Can a shared plan answer all four possible question cards successfully, using only each player’s own question?

The answers must match unless both question cards are 1. Then they must differ.

A quick classroom route

1. Plan (1–2 minutes). Show the four question cards. Each player chooses an answer, 0 or 1, for each of their own two possible questions. Talk freely. Try to cover every case.
2. Go quiet (1–2 minutes). Lock the plan. In Pass & Play, only the named player sees the phone. That player submits the already agreed answer; everyone stays quiet until the team reveal. The phone acts as Keeper. A teacher or fourth participant can manage handoffs.
3. Compare (1–2 minutes). Try the quantum model on the same four cases. Discuss which cards each team won. The ideal CHSH model wins with probability about 85.36%, so four rounds may not beat the plan. Sample at least 100 fresh rounds in the lab and include every result.
4. Ask one more question. Does winning eight of ten random rounds prove entanglement? What else would you need to know?

The four question cards

Why a fixed plan cannot cover them all

If the first three question cases all succeed, A’s two responses and B’s two responses must all agree. The final question case then demands that two of those same responses differ. One case must fail.

There are 16 fixed plans. Exactly 8 cover three cases; none cover all four. Shared randomness mixes fixed plans and cannot improve their best average under uniform question sampling and no communication.

Keep the comparisons straight

The quick challenge visits each case exactly once in a shuffled order and holds the response plan fixed for the entire deck. It is an exhaustive four-case audit. Players cannot change answers mid-deck or adapt their plan to the round number.

The classroom lab instead independently samples a uniformly random question case on every fresh round. It uses the same questions for the classical plan and the ideal model. A finite classical sample can exceed 75% by luck. Do not treat a high classroom score as evidence of entangled hardware.

The ideal model uses (|00⟩ + |11⟩)/√2, with Alice’s Z/X measurements and Bob’s (Z + X)/√2 and (Z − X)/√2 measurements. Each winning outcome pair has probability cos²(π/8)/2. The average win probability is cos²(π/8), about 85.36%.

One phone, an honest model

The ordinary phone centrally sees all questions and samples the model’s joint outcomes. It does not contain entangled ghosts and is not a physical Bell test. Every individual modeled outcome is a fair random bit, independent of the other question settings. Comparing outcomes needs ordinary communication; correlations cannot send a chosen instant message.

Shared-phone privacy relies on participants looking away. The game is not synchronized across separate phones. Teachers may project Solo mode for a class discussion. No personal information is collected or uploaded.

Class tally

Record the four-card audit and random-batch results separately. Include every run rather than selecting the luckiest one.

Sources and scope

IBM Quantum Learning: CHSH game. The protocol, local bound, and ideal joint probabilities determine the model.

The game’s story, interface, and locked-plan classroom sequence are educational design choices. No improved learning outcome or novelty claim has been established.

Revisit: Ghost Trio → · The collection download includes both self-contained games and these printable guides.