Magic & Math with Cards

Lecture 57 · Mathematics

A smaller miracle

Three cards from eight

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What you will learn

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

Before this lecture

In this lecture

An outline of the lecture, scene by scene, with the statements that appear on screen. It follows the draft script, so details may change in production.

  1. Define miniature channel precisely

    • Deck 1…8
    • Choose 3; show 2 in order
    • Hand→compatible ordered pair

  2. Count 56 hands 56 messages and six options per hand

    • C(8,3)=56
    • 8×7=56
    • Each hand has 3×2=6 legal messages

  3. Count six compatible hands per message

    • Every message has 6 compatible hands
    • Both sides degree 6
    • 258→52 means hide 8

  4. Demonstrate a tempting greedy rule fails

    • Legal choices can still collide
    • 123→12 and 124→12 fails
    • Local repair ≠ full solution

  5. Explain alternating reassignment with concrete partial graph

    • 123→12; 124→24
    • Move 124→14, then 245→24
    • An alternating reassignment frees a message

  6. Present complete checked table with one exact round

    • 56 hands; 56 distinct messages
    • Check containment and uniqueness
    • 258→52→hidden 8

  7. Specify matching puzzle and counterexample feedback

    • One edge per hand
    • At most one hand per message
    • Success covers all 56 hands

  8. Retrieve graph vocabulary and preview Hall

    • Local choices versus global assignment
    • A subset needs enough neighbours
    • Next: Hall’s theorem

Companion interactive

Planned

Assign distinct compatible ordered pairs to all hands.

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
  • An original, exhaustively checked teaching example: A finite certificate supporting the small puzzle; not a proof for larger instances.
  • Norman Do, “A Mathemagical Card Trick”: Compatibility matching framework motivates this smaller example.