Magic & Math with Cards

Lecture 49 · Mathematics

Unique sums—and where they stop being unique

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What you will learn

Prove distinct-pair uniqueness for Fibonacci-style values, expose triple ambiguity, and decode four-card complements.

Before this lecture

In this lecture

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  1. Define the correct uniqueness domain

    • Allowed: exactly two distinct cards

  2. Prove pair uniqueness inductively

    • Old max = new value; new min > new value

  3. Derive the strict decoder

    • b = largest available value strictly below s

  4. Exhibit a genuine triple collision

    • 1+2+13 = 3+5+8 =16

  5. Decode four by missing two

    • Missing-pair sum =32−S

  6. Design alternative value sets

    • Distinct-card pair sums must not collide

  7. Separate two Fibonacci theorems

    • Uniqueness depends on the representation rules

  8. Play and retrieve

    • A complete finite collision table is a proof for that set

Companion interactive

Planned

Prove, decode and redesign sum codes under explicit selection rules.

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: Pair-sum effect, triple collision, complement extension and alternative value sets. Original direct induction and domain-based exposition.