Magic & Math with Cards

Section 2 of 7

Hummer Shuffles

Cut and turn two cards together as often as you like. Something refuses to mix.

Four performances built on one move: cut, and turn two cards over together. The mathematics lectures prove what that move preserves, map every arrangement a spectator can reach, and end with designing a routine of your own that survives testing.

Lectures
04–12
Running time
About three hours
Formats
Four performances, five mathematics lectures

Performance lecture, with a trick to play Mathematics lecture

  1. Baby Hummer: four cards refuse to stay mixed

    Four cards, mixed by your own hands. Your card ends up as the one facing the other way.     Play Baby Hummer

  2. Turning a packet is two operations

    Represent a turnover as order reversal plus face complementation and use that representation to audit card-state animations.

  3. Ten cards, always five face up

    Ten cards, cut and turned however you like. Turn over every second card, and exactly five end up face up.     Play Always Five Face Up

  4. A proof in two lanes

    Prove Hummer face-count balance for 2n cards, identify the even-size hypothesis, and strengthen the statement using lane difference.

  5. The Lonely Card: the selection stands alone

    Cut and turn two at a time, then secretly turn one card over. It ends up the only one facing the other way.     Play The Lonely Card

  6. An alternating background as an error detector

    Interpret alternating orientations as a two-pattern code, transport defects through legal moves, and state the limits of detection and recovery.

  7. All the states the spectator can reach

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

  8. Royal Hummer: a royal flush emerges

    Twelve shuffled cards, nine of your leave-or-turn choices, and a royal flush comes out face down.     Play Royal Hummer

  9. Design a Hummer routine that survives testing

    Design and audit a new Hummer-style colour-separation routine by stating its preparation, move family, invariant and finish.