In the “order matters” card trick described in the reading, a deck of 48 cards is cut repeatedly, and then three people each receive consecutive cards. From the relative order of the three cards (lowest, medium, highest) together with their colour pattern, the performer can name all three cards.
The seemingly random process of cutting a deck preserves its cyclic order, and the relative-order and colour patterns of consecutive cards together identify where the cut was made. As discussed in the text, this trick belongs to the mathematical concept called “Products” that combines patterns to create seemingly impossible predictions.
Session notes
Chapter 4 opens with three effects that all run on cyclically ordered decks, picking up directly from chapter 2. Each one requires multiple spectators; they cannot be done solo, and the combinatorial questions they generate turn out to be mostly open.
0:00 The common structure: a deck arranged in cyclic order, which cuts leave intact. The top cards go to people, and apparently harmless questions give the performer enough information to name all cards. Unlike the colour-pattern trick, I have not tried any of these in class.
4:05 The concrete effect from pages 52–53: a 48-card deck (one card per suit removed), cut freely; three people take the top three cards and arrange themselves low, middle, high by card value (the deck is rigged so no three consecutive cards tie), then reveal who holds red cards.
8:30 Why 48: colour patterns times orderings equals 48. The puzzle is to arrange the deck so every cut yields a distinct (ordering, colour-pattern) pair.
9:31 Sanity check on the book’s arrangement: the all-red pattern occurs six times, and each of those six triples shows a different ordering, once the ace reads as low. Every colour pattern must appear exactly six times.
12:35 The probability that a random deck works is roughly 1 in 9,200 for the 32-card colour trick, since there are good arrangements out of total ( de Bruijn sequences, and the zeros and ones labelled by the black and red cards in any order). For higher orders the probability is vanishingly small.
14:48 The book credits freshman students with finding the 48-card arrangement, without saying how. For the colour-only trick there was a clean graph construction via Euler tours; I could find no analogue here, and I suspect a computer search or patient hand-work.
16:55 The chapter’s opening effect: a full 52-card deck, five people take the top five cards, and the holder of the highest card steps forward, then the second highest, then the third; the positions of these three convey the signal. Since there is some slack, and the deck is arranged with no ties. This is the version I find awkward to perform, since the five have to confer. The book gives no arrangement and says one can be found in about an hour; I left it as homework.
21:01 A combinatorial puzzle: arrange to in a circle so that every permutation of elements appears as the relative order of some window of consecutive entries (for , six numbers, where 1, 4, 6 reads LMH and 4, 6, 2 reads MHL; 120 numbers for ). Fan Chung proved such arrangements always exist; the proof is existential, and a construction is still open as far as I know.
23:11 The most performable of the three effects: 52 cards, cut, three people take the top three. The first names their card’s value, the second its suit, the third says nothing. : exactly enough signals. A hint without spoiling: if suits run in a fixed cyclic order through the deck, one suit pins down the others, and a similar argument handles values.
25:14 Ravi asked via chat: yes, in the 48-card effect you need both the colour pattern and the ordering.
26:14 Recap of the three effects: order plus colour on 48 cards; positions of the top three among five, 60 signals for a full deck, not worked out in the book; value plus suit from three cards.
29:20 A first-time participant asked about the chapter section “To the Core of Our Being.” The schedule is behind: chapter 3 took two sessions instead of four. The plan covers chapters 1–7 (card magic); the rest of the book is other magic, juggling and stories.
32:30 The same participant noticed structure in the 48-card arrangement: the colour pattern red red red black red black black black repeats six times, and that block of eight is a de Bruijn sequence of order 3. Values in the first block are 1, 12, 6 (ace low), two extremes and a middle. The arrangement likely reduces to placing values on a fixed colour skeleton, and counting such arrangements up to cyclic shift and colour complement looks like a reasonable open question.
36:36 Allowing ties changes the count: length-3 words over a 3-letter alphabet give 27 patterns rather than 6. The book shows some arrangements are then impossible; for instance, a deck with six each of the values 1, 2, 3, 4 cannot be ordered so that every group of four consecutive cards has a distinct order pattern.
38:46 The book reports as true the conjecture that for window size and a deck of cards, at most distinct values suffice to display all orderings. Counting questions around all of this seem largely open.
39:46 Up next: the rest of chapter 4, universal cycles; then the Gilbreath principle.
To think about
- Find a 52-card arrangement for the top-three-among-five effect, by program or by hand.
- Work out the mind-reading arrangement from the cyclic-suit hint.
- Place values on the repeated 8-card colour skeleton to rebuild the 48-card arrangement, and estimate how many such arrangements exist.
- Find a cyclic arrangement of 1 to 24 showing all 24 orderings of 4 consecutive entries.