Magic & Math with Cards

Lecture 03 · Mathematics

The magician’s question

What cannot change?

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What you will learn

Formulate an invariant with an initial condition and legal moves, and distinguish proof, testing and counterexamples.

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. Ask which reliable information survives change

    • State + allowed moves + claim
    • 12 cards? 6 red? red on top?
    • Invariant: unchanged under specified moves

  2. Clarify the object being measured

    • Used imbalance + remaining imbalance =0
    • After RR: +2 + (−2) =0
    • Name the object you measure

  3. Teach initial condition and local preservation

    • Start: 0 face up
    • Two-card flip changes count by−2, 0,+2
    • Even stays even

  4. Explain that an invariant can be too weak for the reveal

    • Even total does not imply exactly half
    • UUDD → UDDU
    • UDUD → UUUU
    • Ask what forces the finish

  5. Give a counterexample and distinguish illegal changes

    • Counterexample = valid start + legal moves + failure
    • RBRB → BRBR
    • Changing the rules changes the claim

  6. Separate evidence from an exhaustive check and a proof

    • Sampled trials: evidence
    • Exhaustive finite check: specified state space
    • Proof: covers stated family

  7. Design a claim backward from the desired finish

    • Before: odd=a, even=b
    • After flipping even positions: a+(3−b)
    • Exactly 3 if a=b

  8. Provide an adversarial invariant workshop

    • Start? Preserve? Conclude?
    • Counterexample: cut 1 moves A from 1 to 4
    • Repair the claim or repair the rules

Companion interactive

Planned

Write a valid invariant claim and defend it against legal adversarial moves.

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
  • Persi Diaconis and Ron Graham, Magical Mathematics: Model of initial-condition and preservation proof.
  • Colm Mulcahy, Mathematical Card Magic: Elementary mathematical vocabulary.