Magic & Math with Cards

Lecture 14 · Mathematics

Every short word, exactly once

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What you will learn

Define and verify a binary de Bruijn cycle, distinguish capacity from construction, and explain cyclic cuts and truncation failures.

Before this lecture

Ideas you will use

  • Colour-address mechanism

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. Pose the packing problem with eight words

    • 8 words of length 3
    • 8 positions in a circle
    • Overlap saves space—and imposes constraints

  2. Read and verify one concrete cycle

    • 00011101 cyclic
    • Windows: 000,001,011,111,110,101,010,100
    • Order is part of the observation

  3. Prove capacity and clarify exact length

    • 2×2×2=8
    • Binary order k: 2ᵏ positions
    • Counting is necessary; construction remains

  4. Explain cut invariance using rotations

    • Cut=rotation
    • Same cyclic windows; different first position
    • Decoder includes a direction

  5. Show a verified truncation failure

    • Delete 1 →0001110
    • 000 occurs at starts 0 and 6
    • Always inspect the new join

  6. Separate object labels from observed bits

    • Position address ≠card identity
    • 32 reports cannot distinguish 52 starts
    • Allowed states →observation →answer

  7. Define exhaustive checking and meaningful companion feedback

    • For each start: read 3 bits cyclically
    • Duplicate →two explicit starts
    • 8 distinct words →all 8 words

  8. Give challenge and retrieve definition

    • Build
    • Diagnose
    • Rotate without reversing

Companion interactive

Planned

Build and certify an eight-position binary cycle with all eight cyclic triples distinct.

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
  • Persi Diaconis and Ron Graham, Magical Mathematics: Binary windows, count, cyclic distinction; truncation counterexample is a checked teaching example.
  • Colm Mulcahy, Mathematical Card Magic: Small-cycle examples and card adaptations; no unverified linear-feedback generalization used.