Magic & Math with Cards

Lecture 44 · Mathematics

Unequal counts can still cancel

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What you will learn

Prove restoration for alternating counts and construct a protected off-center region.

Before this lecture

In this lecture

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  1. Replace one count with a repeating pair

    • Counts: s,t,s,t
    • 1≤s,t≤n
    • s+t≥n

  2. Derive the asymmetric partition

    • |T|=n−t
    • |M|=s+t−n
    • |B|=n−s
    • T+M has s; M+B has t

  3. Follow the first pair

    • After s: B Mᴿ Tᴿ
    • After s,t: Tᴿ M Bᴿ
    • M returns after one PAIR

  4. Repeat the pair to restore

    • After s,t,s: Bᴿ Mᴿ T
    • After s,t,s,t: T M B
    • s=t recovers the earlier theorem

  5. Interpret a pair as an involution

    • G = F_t ∘ F_s
    • G²=identity
    • Ordered pair matters
    • 6,9 protects5–6; 9,6 protects8–9

  6. Design a protected region backwards

    • Protect positions a..b
    • s=b
    • t=n−a+1
    • Example n = 12,a4,b8: s8,t9

  7. Test the boundary and an exception

    • s+t=n: empty M is valid
    • s+t<n: general guarantee unavailable
    • n = 9,s3,t5: partial restoration only

  8. Explore the valid region and build a design

    • Predict blocks before dealing
    • Protect a region by choosing counts
    • Guarantee ≠ every possible case

Companion interactive

Planned

Predict restoration and design a preserved consecutive region.

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.