Magic & Math with Cards

Section 7 of 7

The Best Card Trick

Four cards, laid in a row, name a fifth that nobody has seen.

Five cards are chosen, four are shown, and the fifth is named. This section proves why the classic method always works, counts what four cards can say, and uses Hall’s theorem to reach a deck of 124 cards.

Lectures
52–60
Running time
About three hours
Formats
Three performances, six mathematics lectures

Performance lecture, with a trick to play Mathematics lecture

  1. Four cards name the fifth

    Choose five cards. My assistant shows four. I name the fifth.     Play Four Cards Name the Fifth

  2. Why the suit-and-clock strategy always works

    Prove the same-suit and short-clock-distance guarantees for every five-card hand.

  3. Six messages in three cards

    Encode and decode all six triple permutations with explicit total-order conventions.

  4. Hide the key card’s role

    Choose any five cards. Four in a row name the fifth, and no position is special.     Play Hide the Key Card

  5. Why 24 orders are not the whole story

    Resolve the 24-versus 48 paradox and derive the 124-card counting upper bound while retaining compatibility.

  6. A smaller miracle: three cards from eight

    Model three-from-eight communication as a compatible matching and inspect a complete verified small code.

  7. Hall’s theorem explains the 124-card limit

    Prove 124-card attainability using Hall’s theorem and a regular compatibility graph.

  8. Build the 124-card encoder

    Construct and prove an explicit 124-card encoder using modular selection, label compression and permutation rank.

  9. The 124-card finale

    Five cards drawn from a deck of 124. Four shown, the fifth named.     Play The 124-Card Finale