Ghost Trio

Solo or 3 players + Keeper · 4–6 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?

All X questions require an even sum of answers. Exactly two Y questions require an odd sum.

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 GHZ model wins every allowed card. Open the parity explanation after students have tried to improve their plan.
4. Ask one more question. Can one player use these correlations to send a chosen message instantly? Why not?

The four question cards

Why a fixed plan cannot cover them all

Each player’s X answer appears twice across the four cards, as does their Y answer. Adding all those answers therefore gives an even total. The requested pattern, one even total and three odd totals, adds to an odd number: a contradiction.

There are 64 fixed plans. Exactly 32 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 (|000⟩ + |111⟩)/√2 and X/Y measurements. Outcome bit 0 denotes eigenvalue +1 and bit 1 denotes −1. For each allowed setting, the four outcomes with the required parity each have probability 1/4.

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

Girish, Holmgren, Mittal, Raz, and Zhan (2021), Parallel Repetition for the GHZ Game: A Simpler Proof. The GHZ definition and values motivate this classroom version.

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

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