Magic & Math with Cards

Lecture 41 · Mathematics

Count, reverse, transfer

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What you will learn

Derive the COAT position map and prove the three-transfer block formula.

Before this lecture

Ideas you will use

  • Composition of operations

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 the operation before using algebra

    • COAT: count out, then transfer
    • TOP→BOTTOM

  2. Derive the position map

    • i≤k:F(i)=n+1−i
    • i>k:F(i)=i−k
    • F maps OLD position to NEW position

  3. Prove the bottom-card route

    • n→n−k→k+1→1
    • n−k≤k ⇔ 2k≥n
    • k=n:three full reversals

  4. Choose the useful three-block partition

    • |T|=|B|=n−k
    • |M|=2k−n
    • T+M and M+B each have k cards

  5. Track the first two transfers

    • After1:B Mᴿ Tᴿ
    • After2:Tᴿ M Bᴿ
    • Reverse a concatenation in BOTH senses

  6. Prove the stronger three-transfer result

    • After3:Bᴿ Mᴿ T
    • Original bottom k becomes reversed top k

  7. Check boundary cases and a counterexample

    • k=n/2:M is empty
    • k=n:whole reversal
    • n = 10,k = 4:10→6→2→9

  8. Use the proof as a diagnostic tool

    • n = 12,k = 8:T4,M4,B4
    • Predict each move, then repair
    • After3:Bᴿ Mᴿ T

Companion interactive

Planned

Predict and justify a COAT transformation using positions and blocks.

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
  • Colm Mulcahy, Mathematical Card Magic: COAT mechanism and conditions checked against source; spoken presentation, examples and companion challenges are teaching adaptations.