Session notes
Chapter 5 of Magical Mathematics builds the Gilbreath principle into a poker-hands effect. The construction is nearly as clean as the trick, except for one place where the book’s stated bounds are too generous.
1:02 Recap: from an alternating red–black deck, deal off any number of cards, riffle the two piles, and the cards taken two at a time are one red and one black. The principle holds for any cyclic property (the Numberphile video shows the general version).
2:49 The effect, as the spectator performs it: count 25 cards off a deck that looks random; two of the five resulting poker hands will be a straight (ace, king, queen, jack, ten of mixed suits; I kept calling it a straight flush) and a flush, the flush being the better hand.
3:51 Deal off some of the 25 one at a time (the book allows anywhere from 5 to 20), riffle the two piles. I used a riffle-shuffling machine since I riffle badly; in performance the spectator riffles.
7:01 Deal five hands of five round the table, one card at a time; show they look random; pick up the packets one at a time in any order the audience likes (keeping each packet’s internal order); deal five hands again.
7:37 One hand is the straight, another is all spades. The book says the first and second hands; in all my trials spades came first and the high values came last, so perhaps “second” means the neighbour on the other side at a round table.
10:24 Always deal off before riffling: splitting and riffling destroys the reversal the principle needs. Neeta asked about dealing five at a time; it doesn’t work because the spades land at every fifth position after the first deal.
15:33 The setup (the only preparation): the top 25 cards are five spades in any order, three copies of ace–king–queen–jack–ten (suits free), five more spades. My eight of spades and ace of hearts had gone missing and were later found on the floor.
22:01 Sunita asked whether a complete ace-to-ten block might survive the riffle intact. It doesn’t have to: the riffle guarantees values, not groupings.
24:07 Both piles end with five spades (the dealt-off pile because the top five were reversed to its bottom, the other by setup), so whatever the riffle does, the bottom five of the shuffled deck are spades. Ten together is possible with a perfect riffle, not guaranteed.
27:13 A detour to state Gilbreath more carefully with an alternating deck: splitting in half and riffling perfectly puts the ten of clubs on the two of spades and breaks the pairs; dealing off 26 cards first and then riffling gives red–black pairs throughout. The general form covers any block-of-four (suits) or block-of-thirteen (values) property, as in the Numberphile demonstration.
33:03 After the first deal, the bottom five spades land one on top of each packet; after reassembly in any order they sit at positions 1, 6, 11, 16, 21, and the second deal gathers them into the first hand.
35:11 Symmetrically, if the top five after the riffle carry the ace-to-ten values, they land at the bottom of each packet, at positions 5, 10, 15, 20, 25, and the second deal gathers them into the last hand. Sunita stated the needed claim precisely: positions must carry the top values.
41:56 Each packet therefore has a spade on top and a top-value card at the bottom. Ravi put it in the chat: the first deal spreads them out, the second brings them back. What happens in the middle three hands is irrelevant.
45:38 The gap: are the top five after the riffle really the ace-to-ten values? Sunita’s counterexample: deal off exactly five (just the spades), then a degenerate riffle that drops four of them, then all twenty, then the last spade; the top five are now corrupted, so the book’s lower bound of 5 is not mathematically safe. Dealing off between 10 and 15 cards ensures both piles have five spades at the bottom and five top-value cards on top; the argument for the top is then the argument for the bottom with the deck turned face up.
52:12 Practical rule: deal off between 10 and 15 of the 25 for a cushion of five on each side, which also makes the riffle comfortable. The book omits the proof; this is a clean instance of Gilbreath plus deal-and-regather, with no sleight beyond remembering to deal off.
55:21 My animation of the deal is unfinished; what it should show is each packet with a spade at the top and a top-value card at the bottom. Smitha and Sunita confirmed that a degenerate riffle (one pile stacked entirely on the other) still works when ten or more cards are on each side.
1:00:28 Up next: one more effect, the Mandelbrot-set connection, and the ultimate Gilbreath principle, which characterises the permutations reachable from a sorted deck by a single Gilbreath shuffle (deal off , riffle), with proofs in the appendix and a connection to polynomials.
To think about
- Write out the proof that after a Gilbreath shuffle with 10 to 15 dealt off, the top five and bottom five are as claimed.
- Find the exact set of for which the effect is guaranteed for every riffle.
- Design a version that deals four hands from 16 cards.
- Check whether the packet order in the second deal can vary.