Magic & Math with Cards

Lecture 55 · Performance

Hide the key card’s role

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What you will learn

Perform and explain Mulcahy’s variable anchor-position protocol with an invariant visible-rank sum.

The trick in this lecture Hide the Key Card Choose any five cards. Four in a row name the fifth, and no position is special. Play the trick

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. Introduce a genuinely variable anchor protocol

    At the table
    • Any five distinct cards
    • Anchor position varies
    • Only identities and order

  2. Encode variable anchor during genuine performance

    At the table
    • No separate anchor marker
    • Same four visible identities before and after arranging
    • Read remaining slots left to right

  3. Decode computed position and reveal

    At the table
    • Compute position, then decode offset
    • One prediction before reveal
    • Variable Suit-Giver Principle

  4. Derive invariant position computation

    • 11+10+2+8=31≡3 mod 4
    • Anchor at position 3
    • 8♦,10♠, J♣,2♥

  5. Decode row and demonstrate residue-zero case

    • Zero remainder→position 4
    • 2♦,3♥,6♠, A♣
    • 12≡0 mod 4; offset 1

  6. Explain why variable role remains constructible

    • Visible sum is arrangement-invariant
    • Reserve one slot; six orders remain
    • No circular dependency

  7. Specify variable-position trainer

    • Sum only visible cards
    • Compute position before reading triple
    • Fixed/computed mode must match

  8. Retrieve dependencies and preview counting limit

    • Visible sum locates role
    • Permutation still supplies offset
    • Next: how large can the deck be?

Companion interactive

Encode and decode with computed anchor position.

The trick page is this lecture’s companion: be the spectator, lift the curtain to step through why it works, try the challenge, then learn the handling.

Open Hide the Key Card

Sources

How these sources were used
  • Colm Mulcahy, Mathematical Card Magic: Variable Suit-Giver Principle. The position rule is retained exactly; first-position base version and rank-first card order are course conventions.