Magic & Math with Cards

Section 3 of 7

Order Matters

A deck arranged so that every short run of colours happens exactly once.

A prepared deck in which every short run of colours appears exactly once lets a few spoken colours name the cards. This section builds such decks from graphs and Euler tours, extends the idea to suits and relative order, and follows it out to locating yourself from a tiny picture and reading DNA.

Lectures
13–27
Running time
About five hours
Formats
Four performances, eleven mathematics lectures

Performance lecture, with a trick to play Mathematics lecture

  1. Five colours reveal five cards

    Tell me only the colours of five cards. I'll name all five.     Play Five Colours, Five Cards

  2. Every short word, exactly once

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

  3. One rank, one suit, three revelations

    Name one card's value and the next card's suit. All three cards are named.     Play One Rank, One Suit

  4. Your first performable colour-window routine

    Cut sixteen cards, deal four, and say only their colours. All four are named.     Play Your First Colour-Window Routine

  5. From a card window to a directed edge

    Translate overlapping binary windows into edges of a directed graph and back without confusing vertices with edges.

  6. Euler tours: use every edge once

    Prove Euler-tour existence under directed balance and connectivity and apply it to de Bruijn graphs.

  7. Build your own de Bruijn deck

    Construct, verify and export an order-four de Bruijn card stack and matching decoder.

  8. More than two colours: suits as an alphabet

    Generalize window codes to larger alphabets while checking capacity, cyclic uniqueness and physical card supply.

  9. Gold & Silver: identify cards by relative order

    Point out the highest and the lowest of three cards, and all three are named.     Play Gold & Silver

  10. Universal cycles: patterns beyond binary words

    Define universal cycles through explicit encoding rules and distinguish verified relative-order examples from unsupported generalizations.

  11. Locate yourself from a tiny picture

    Decode a position from a verified 2×2 binary patch while respecting toroidal boundaries, orientation and sensor shape.

  12. From card windows to DNA reads

    Assemble a clean toy DNA string from its exact multiset of overlapping 3-mers using a directed edge trail.

  13. When the same clues fit two sequences

    Distinguish compatible assembly, unique reconstruction and robustness using exact small counterexamples.

  14. Decode without a giant lookup table

    Decode the prepared packet using exact bit-to-card mapping and verified feedback rules, including the zero-state exception.

  15. Design review: make a decodable card miracle

    Design and exhaustively audit a complete prepared-packet observation protocol.