Perfectly Mixed Pairs

You cut the packet, deal some cards into a pile, and riffle the two packets together as unevenly as you like. Then the cards come off the top two at a time, and every pair holds one red card and one black card.

You need: Sixteen cards: ace to eight of hearts and of clubs
Principle: Gilbreath's first principle

Watch the lectureLecture 28 · A messy riffle, perfectly mixed pairs

First, be the spectator

Try it

You do all the mixing: where to cut, how many cards to deal, and which packet each card of the riffle comes from. The cards stay face down until the end.

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

The secret

Why it works

The only thing arranged in advance is the colours. From the top, the sixteen cards go red, black, red, black, all the way down. A cut doesn’t spoil that: the bottom card is black and the top card is red, so the alternation carries on round the join.

Dealing is the important move. When you deal cards one at a time onto a pile, the pile ends up in reverse order. So the two packets you riffle both start from the same place in the deck: the top of the dealt pile and the top of the cards in your hand were neighbours. Neighbours have opposite colours.

Now watch the next two cards of the riffle. There are four ways to take them: A then A, B then B, A then B, or B then A. Two cards in a row from one packet were neighbours in the deck, so they differ. One card from each packet means the two tops, which differ. Either way, the pair is mixed. And afterwards the two tops are opposite colours again: two cards from one side move that side along by two places, so its top keeps its colour, while one card from each side changes both tops.

After every complete pair, the two tops are opposite colours again, so the next pair can’t match either.

There is a tidier way to see this, on the deck’s original order. Lay the starting positions out as a ribbon. Because both packets read outward from the place where you stopped dealing, the cards taken so far always form one unbroken stretch of the ribbon. If that stretch runs from a to b and holds an even number of cards, b − a + 1, then the next card on its left and the next on its right are b − a + 2 places apart. That is odd, so one front has an odd label and the other an even one: opposite colours.

Step through one riffle of eight cards below, labelled 1 to 8 by where they start. It is the worked shuffle from the episode: deal three, then take A, B, B, A, B, A, B, B.

Notice exactly what is promised: positions 1 and 2, 3 and 4, and so on, counting from the top. Two neighbours that sit either side of a gap between pairs can match, and often do. That is why the trick forbids a cut after the shuffle: moving one card from top to bottom shifts every boundary.

Test your understanding

Challenge

Make a pair of two cards of the same colour. Inside the rules you won’t manage it. So switch off one rule at a time, and find which rules let a bad pair through.

Take it to the table

Perform it yourself

Set-up

Take the ace to eight of hearts and the ace to eight of clubs: sixteen cards.
Arrange them face down so that, from the top, they read A♥, A♣, 2♥, 2♣, 3♥, 3♣, and so on down to 8♥, 8♣. The colours alternate, starting with red on top and ending with black at the bottom.

Performance

Hand over the packet and invite a cut: lift a group from the top, put it on the table, and put the rest on top. They may cut again as often as they like. Complete cuts only.
Ask them to deal cards from the top, one at a time and face down, into a single pile. Each new card goes on top of the last. They stop whenever they like, with cards still in hand; somewhere around the middle (five to eleven cards) looks best.
They riffle the two packets together, once. It can be uneven, with several cards falling from one side; the one rule is that the order inside each packet stays intact.
If a riffle feels awkward, do the same thing on the table: they take the next card from the top of either packet and place it next in a row, left to right. Gather the row starting with the rightmost card, putting each next card on top, so the leftmost card finishes on top.
Leave the packet exactly as it is: no further shuffle, no cut. Make your prediction: every pair will have one red card and one black card. Not most pairs. Every pair.
Take the top two cards together, turn them over and set them down as a pair. Continue down the packet, eight pairs in all, keeping the boundaries between pairs clear.

Things to watch

Cut only before the deal. A cut after the shuffle shifts the pair boundaries, and the promise goes with them. (Cutting an even number of cards afterwards is safe: it moves whole pairs.)
The dealt pile must be dealt, not lifted off as a block. Dealing reverses it; that reversal is what makes the trick work.
Within each pair the order doesn’t matter, but which two cards share a pair does. Peel the top two off as one block.
If cards spill or the order is lost, say so and reset to the starting order. If someone asks for another shuffle, keep this result for the reveal and try another shuffle afterwards.

Further

Going deeper

Why deal the pile, instead of just cutting it off?

A cut also makes two packets, but it doesn’t reverse either of them. Label eight alternating cards 1 to 8 and split after card 4 without reversing: the packets read 1, 2, 3, 4 and 5, 6, 7, 8. Take card 1, then card 5, and the first pair is red and red. That is a perfectly legal interleaving of those two packets, so no promise about every interleaving can survive. Dealing the first four instead gives 4, 3, 2, 1 and 5, 6, 7, 8: two packets growing outward from the same boundary, which is exactly what the proof needs.

Which cuts are safe, and when?

Before the deal, any complete cut is safe, because in an even alternating packet the last card and the first card differ, so a cut keeps the alternation. In an odd packet they agree, and a cut can bring two equal colours together in the middle; without a preliminary cut, an odd packet still gives mixed pairs from the top, with one card left over.

After the shuffle, the alternation may be gone, and only the pair boundaries are protected. Cutting an even number of cards rotates whole pairs and keeps every pair intact. Cutting an odd number shifts the boundaries; in the eight-card example 3, 4, 5, 2, 6, 1, 7, 8, moving the 3 to the bottom makes the second pair 2 and 6, both black. The same physical move can be harmless at one stage and harmful at another.

What the promise does not say

It doesn’t tell you which card of a pair is red. It doesn’t need the two packets to be the same size, or the riffle to alternate sides. And it says nothing after a second shuffle. The spectator has a great deal of freedom, but not every possible order of the deck: the shuffles allowed form a family large enough to look messy and small enough to keep a structure.

A stronger fact hiding underneath

The ribbon argument proves more than the colour claim. Whatever you take, the first few cards of the shuffled packet always form one unbroken stretch of the original order. Colours only see whether a label is odd or even, but the stretch remembers much more. The later tricks in this act use the same fact with four suits, and with poker hands. Be careful, though: it is a statement about the top of the shuffled packet. A block from the middle can draw from both ends of the stretch.

Sources

The principle is due to Norman Gilbreath, who published it in 1958. This presentation follows Persi Diaconis and Ron Graham, Magical Mathematics, Chapter 5 (pp. 61–63, with the interval explanation on pp. 68–71), scaled to sixteen cards with the same guarantee; and Colm Mulcahy, Mathematical Card Magic, Chapter 6 (pp. 130–134 for aligned pairs and the proof by the two fronts; pp. 130–143 for the reverse-deal conditions).