Magic & Math with Cards

Lecture 47 · Mathematics

A card position written in base three

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What you will learn

Derive the ternary target code, prove convergence after three adaptive rounds, and distinguish the 21-card version.

Before this lecture

In this lecture

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  1. Derive the position update

    • p = 3r + c
    • p' = 9k + floor(p/3)

  2. Interpret one round in base three

    • a₂a₁a₀ → k₁a₂a₁

  3. Prove three-round target encoding

    • Final p = 9k₃ + 3k₂ + k₁

  4. Check asymmetric examples

    • Target 6: bottom, middle, top

  5. Relate position control and information

    • 27 → 9 → 3 → 1 candidates

  6. Distinguish the twenty-one-card contraction

    • 21 cards: p' = 7k + floor(p/3)
    • Middle three times → position 11

  7. Audit physical and software semantics

    • Reversed columns change the recurrence

  8. Play and retrieve

    • First inserted digit becomes the units digit

Companion interactive

Planned

Encode and decode target positions, compare packet sizes and diagnose the first incorrect operation.

The interactive for this lecture is being built and will appear here.

Notes

Written notes for this lecture will appear here.

Sources

How these sources were used
  • Numberphile, “Beautiful Card Trick, and the 21-card trick”: Mechanisms motivate original position-update and ternary exposition, with explicit checked collection convention.