Royal Hummer

Twelve cards, five of them a royal flush in spades. You shuffle, decide nine times whether to leave or turn a group of cards, then choose a pile to turn over. The cards left face down are the ten, jack, queen, king and ace of spades.

You need: The five spade royals and seven other cards
Principle: Face and position, read together
Devised by: Steve Freeman, from Bob Hummer’s ideas

Watch the lectureLecture 11 · Royal Hummer: a royal flush emerges

First, be the spectator

Try it

Shuffle, then make every leave-or-turn choice yourself, and pick the final pile. The cards follow the real rules.

Try it a few times first. The secret will keep.

The secret

Why it works

After your shuffle, the faces were set two cards at a time, by four cases. Neither card royal: up, down. Only the first royal: down, down. Only the second royal: up, up. Both royal: down, up.

Those four cases are really one rule. At odd positions, ordinary cards face up and royal cards face down; at even positions it is the other way round. Each pair starts at an odd position, and a finished pair goes on top of other finished pairs, an even number of cards, so every card stays in the kind of position it was coded in.

Give each card three bits: p = 1 at an odd position and 0 at an even one, f = 1 face up and 0 face down, t = 1 for a royal card and 0 otherwise. The rule says f ⊕ p = t, where ⊕ means “add, and keep only odd or even”. A card’s direction and its kind of position, read together, say whether it is royal.

Now turn a group of two or four over. Because the group has even size, every card in it moves to a position of the other parity, and every face reverses. Both bits change, so f ⊕ p does not. Leaving a group alone changes neither, and stacking even-sized groups keeps every parity. All nine of your choices preserve the code.

The deal reads the code. Cards from odd positions go left, cards from even positions go right. In the left pile the royals face down and the others up; in the right pile it is the opposite. Turning either pile makes the two piles agree. Turn the right pile, and the royals are already down. Turn the left pile, and the royals are all up, so one turn of the whole packet puts them down.

Turning an even-sized group flips both a card’s face and the parity of its position, so their combination, which records “royal or not”, survives every choice you make.

Step through a run with every card’s name shown above it and the bit f ⊕ p beneath. Watch the 1s: they always sit under the five royals. Both final choices are there, and so is your own last run.

Test your understanding

Challenge: design a routine that survives testing

Six cards: three red (R1, R2, R3) and three black (B1, B2, B3). The promise: every red card ends facing one way, every black card the other. Choose a preparation, the moves the spectator may use, and a finish; then test every state the spectator can reach. When it passes, say why. Then allow single-card flips and see what breaks.

Take it to the table

Perform it yourself

Set-up

Take twelve cards: the ten, jack, queen, king and ace of spades, and seven others (the course uses 2♥ 3♦ 4♣ 5♥ 6♦ 7♣ 8♥). They need not be in any order.
You will want two spaces on the table, LEFT and RIGHT, for the final deal.

Performance

The spectator shuffles all twelve thoroughly. Then turn the whole packet face up.
Set the faces two cards at a time, working from the top. Keep each pair in order and flip single cards in place: neither royal → up, down; only the first royal → down, down; only the second royal → up, up; both royal → down, up. Put each finished pair, as a unit, on top of a growing pile. Rehearse until this is smooth; keep the table in your head, not on the table.
From now on, only whole groups move. Take the top two as a block and ask: leave or turn? Turn means turn the pair over together, like a little book: its order reverses as its faces reverse. Place the block on top of a new pile. Repeat for all six pairs.
Pick up the new pile and do the same with three groups of four. Never peel off single cards.
Deal one card at a time, alternately LEFT, RIGHT, starting LEFT, without turning any card. Each pile gets six.
The spectator chooses either pile, turns that whole pile over and places it on top of the other.
If they turned the LEFT pile, turn the complete packet over once. If they turned the RIGHT pile, do nothing more.
Spread the cards. Exactly five face down: turn them over one at a time, and arrange them as ten, jack, queen, king, ace of spades.

Do not allow a shuffle after the faces are set. If a card is turned on its own by accident, say so, set all twelve face up, recode the pairs and repeat the group procedure.

Further

Going deeper

One stray flip, one extra card

Reverse a single ordinary card after the coding, without moving it. Its face bit changes but its position bit does not, so its code now says “royal”. Every later group turn carries that false bit faithfully, and at the end six cards are face down. The failure is local and explainable: one extra reversal, one wrong membership bit.

The same mechanism separates any category you prepare for. Code the four aces instead and four aces emerge. Poker simply gives the audience a satisfying five-card ending.

Three mechanisms, three promises

In “Always five face up”, the preserved fact is a balance between odd and even positions, and the finish turns it into a count. In the Lonely Card, an alternating background carries one deliberate defect, and the finish isolates one chosen card. Here the preparation writes a category into face and position parity, and the finish reveals the whole category.

They share the same moves, but not the same starting information or the same promise. A design starts from the reveal you want, a number, a card or a group, and then chooses an invariant that can carry it.

The general theorem

For an even packet that starts in order and face down, Diaconis and Graham show that cuts and top-two turns can reach every order of the cards, and the only restriction on the faces is that f ⊕ p ⊕ (original position parity) is the same for every card. The six-card lab above checks that for one small case: 1,440 = 2 × 6! reachable states.

Sources

Persi Diaconis and Ron Graham, Magical Mathematics, Chapter 1, pp. 8–11 (Royal Hummer, credited there to Steve Freeman, building on Bob Hummer’s ideas; compare pp. 15–16) and pp. 12–14 (the signed-position constraint and its colour-separation corollary). The course uses the book’s four-case pair preparation with an explicit twelve-card packet.

The moves descend from Bob Hummer’s face-up/face-down principle. Colm Mulcahy, Mathematical Card Magic, Chapter 9, pp. 211–212, develops the related cut-and-turn routines. The six-card design lab is an original teaching exercise.