How does revealing all other cards prove a hidden card is red without exposing its number?

Conditions

• The deck contains exactly 26 red and 26 black cards.
• The target card is drawn uniformly at random.
• The prover reveals the identities of the remaining 51 cards.

Reasoning, step by step

1. Start with a standard 52-card deck containing 26 red and 26 black cards.
2. Draw one card at random and keep it hidden.
3. Reveal the remaining 51 cards to the verifier.
4. Count the number of black cards among the revealed cards.
5. Deduce that if exactly 26 black cards are revealed, the single hidden card must be red.

Example

The speaker demonstrates proving a card is red without revealing its number by showing all 26 black cards from the rest of the deck. Since the deck contains precisely 26 black cards total, the absence of any black card among the revealed set logically necessitates that the single hidden card belongs to the red subset.

Common misconceptions

• Assuming the verifier learns the card's identity: The verifier confirms the color property while gaining zero information about the card's identity.
• Thinking the protocol is probabilistic: Unlike the pen game, this protocol achieves certainty in a single interaction.

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Explore next

• What are the three formal criteria that define a valid zero-knowledge proof protocol?
• Why must the pen swapping experiment be repeated multiple times to constitute a valid zero-knowledge proof?