Magic & Math with Cards

Lecture 10 · Mathematics

All the states the spectator can reach

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What you will learn

Use a finite state graph to distinguish reachable arrangements from impossible targets, and derive the identity–position–orientation constraint.

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 a complete state and count the ambient space

    • State = identity order + face bits
    • 4! × 2⁴ = 384 possible states
    • Which states can these moves reach?

  2. Construct graph edges from two primitive legal moves

    • C: cut one to bottom
    • T: turn top two as a packet
    • Different paths can reach the same state

  3. Explain complete finite exploration and its limits

    • Expand every discovered node
    • Stop only when the queue is empty
    • Four-card census: 48 reachable of 384

  4. Derive the per-card parity signature

    • qᵢ = (i + labelᵢ + faceᵢ) mod 2
    • C: every q toggles
    • T: every q stays the same
    • All q values agree

  5. Reject targets with explicit witnesses

    • Forbidden target: q = 1, 1, 0, 0
    • Balanced counts need not imply reachability
    • A witness: two unequal signatures

  6. Count candidates and separate necessity from sufficiency

    • 2 face patterns for each identity order
    • Four cards: 2 × 4! = 48
    • General sufficiency: theorem in Diaconis–Graham

  7. Use an explicit route and distinguish shortest path from any path

    • Route: C, T, C, T, C, T
    • Goal: original order, every face up
    • Path length depends on the move convention

  8. Define reachability companion and retrieval challenge

    • Match identities AND faces
    • Reject with two unequal signatures
    • A route proves possibility; an invariant can prove impossibility

Companion interactive

Planned

Reach a complete four-card target or certify that it is unreachable.

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: Exact reachable-state theorem and its orientation constraint; four-card graph and breadth-first exploration are original teaching illustrations.