Lecture 05 · Mathematics
Turning a packet is two operations
Video in production
This lecture is still face down. Its script is written; filming and animation are under way.
Meanwhile, read the outlineWhat you will learn
Represent a turnover as order reversal plus face complementation and use that representation to audit card-state animations.
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.
Identify the ambiguity in a familiar instruction
- Whole packet: A↓B↑ → B↓A↑
- Individual flips: A↓B↑ → A↑B↓
- Turnover = reverse + complement
Encode orientation without replacing identity
- 0 = down; 1 = up
- Identities: A B C D E F
- Faces: 0 1 0 1 1 0
- Flip: b→1−b
Compute a four-card prefix turnover exactly
- Turn top 4
- D↓ C↑ B↓ A↑ E↑ F↓
- Same bit pattern, different identities
Generalize the position and face maps
- Within prefix: i→k+1−i
- Face: b→1−b
- Outside prefix: unchanged
- Two identical turns = identity
Show local face cases underlying Hummer
- DD→UU
- DU→DU
- UD→UD
- UU→DD
- One odd slot; one even slot
Use even and odd prefix sizes to expose parity
- Even k: position parity flips
- Odd k: position parity stays
- Faces flip in both cases
Distinguish state equivalence from hidden spectator information
- Complete state →face pattern →face-up count
- Hidden does not mean undefined
- Views share one stored state
Give an animation-audit game and worked round
- Audit order AND faces
- Correct: B↓A↑C↓D↑
- Turn twice: complete reset
Companion interactive
Planned
Audit and predict exact packet turnovers at identity and orientation levels.
The interactive for this lecture is being built and will appear here.
Notes
Written notes for this lecture will appear here.
Sources
- Magical Mathematics, Chapter 1, pp. 4–7
- Mathematical Card Magic, Chapter 9, pp. 209–212
How these sources were used
- Face-up notation and exact two-card turnover.
- CATO turns two cards as a unit.