Magic & Math with Cards

Lecture 33 · Mathematics

The general Gilbreath principle

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What you will learn

Prove the general Gilbreath aligned-block inventory theorem using prefix intervals and cumulative counts.

Before this lecture

In this lecture

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  1. State the general phenomenon

    • One repeated cycle per aligned block

  2. Prove the prefix interval property

    • Every output prefix is an interval as a set

  3. Count periodic labels in intervals

    • Prefix km contains k of each label

  4. Derive the block theorem

    • Block k = prefix km minus prefix (k−1)m

  5. Inspect the hypotheses

    • Reversal / source order / alignment

  6. Verify exact and counterexamples

    • Blocks may be scattered in original positions

  7. Separate proof from computation

    • Computation checks examples; proof covers all cases

  8. Play the proof

    • Nested prefixes make inventory subtraction valid

Companion interactive

Planned

Choose cycle, split and source word, then justify each aligned block by prefix inventories.

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Notes

Written notes for this lecture will appear here.

Sources

How these sources were used
  • Persi Diaconis and Ron Graham, Magical Mathematics: General principle derived through the consecutive-prefix property; repeated-label multiset extension stated explicitly.