Magic & Math with Cards

Lecture 56 · Mathematics

Why 24 orders are not the whole story

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What you will learn

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

Before this lecture

Ideas you will use

  • Combinations and permutations introduced here

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. Pose the 24-versus 48 apparent contradiction

    • Fixed visible set: 24 orders
    • Assistant chooses visible set
    • Count full hands and full messages

  2. Explain why complete messages must distinguish hands

    • Message + decoded target = original hand
    • Different hands need different messages
    • Containment remains required

  3. Count hands using combinations

    • H=C(N,5)
    • Ordered five-card lists divided by 5!
    • 5!=120

  4. Count ordered messages without dividing by 4!

    • M=N(N−1)(N−2)(N−3)
    • M=24C(N,4)
    • For 52: H=2,598,960; M=6,497,400

  5. Derive 124 bound by cancellation

    • C(N,5)≤P(N,4)
    • (N−4)/120≤1
    • N≤124

  6. Examine equality and containment obstruction

    • At 124: H=M=225,150,024
    • Equal totals do not prevent local shortage
    • Compatibility graph

  7. Specify count explorer and meaningful feedback

    • Counting permits ≠ existence proved
    • Ratio=(N−4)/120
    • Check legal containment separately

  8. Retrieve resolution and announce small matching model

    • Message identifies the whole hand
    • 124 is an upper bound so far
    • Next: three cards from eight

Companion interactive

Planned

Count the correct objects and distinguish necessary capacity from compatibility.

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
  • Norman Do, “A Mathemagical Card Trick”: Counting bound and compatibility formulation; derivation and examples fully written here.