The Gilbreath Principle — the shuffle that fails | Sleightless
The Gilbreath principle
Alternate the colors of a deck, deal any number of cards off the top,<br>and riffle them back in. Every pair still comes out one red and one black.
It is not only colors. Arrange the deck in repeating suits — spades, hearts,<br>diamonds, clubs, over and over — and every group of four comes out holding<br>one of every suit . Colors are the same rule with a pattern two cards long.
Do it yourself
1 Choose how the deck starts out
Alternating colors — check it in pairs<br>Repeating suits — check it in fours
Slowly, so I can follow<br>Watchable<br>Quickly, I know this
2 Take some cards off the top
dealt off<br>cut off
Take them off
3 Riffle them back into the deck
Riffle them<br>together
cards taken off the top
— of 26<br>pairs holding one of each color
Start again<br>Try it ten thousand times
Green edges are cards that fell from the packet you took off,<br>red edges came from the rest of the deck. Watch the two packets go down while the<br>riffled deck builds underneath in pairs, each one ticked as it completes.
failures in 10,000 deals · measured, not claimed
Most card tricks that survive a shuffle survive it by making sure no real shuffle<br>happens. This one is the exception: the spectator genuinely riffles, as badly as<br>they like, and the order they destroy is not the order the trick needs.
Why it cannot fail
The mechanism is the deal, and it is worth being precise about why. Dealing cards one<br>at a time into a pile reverses them . That is the whole trick; a cut,<br>which does not reverse, behaves quite differently, and there is a button above to watch<br>it misbehave.
Number the cards 1 to 52 from the top, so card 1 is red and card 52 is black. Deal<br>k of them off. Now read the finished deck from the bottom up ,<br>which is the order the cards fell in during the riffle. It is an interleaving of two<br>sequences:
the dealt pile, from its bottom → card 1, card 2, card 3, …
the rest of the deck, from its bottom → card 52, card 51, card 50, …
The first begins on card 1 and the second on card 52. In a deck of an even size whose<br>colors alternate, those two are always opposite colors. That single fact is the whole<br>proof, and the rest is bookkeeping:
At the start of a pair, the two piles are offering opposite colors — say red and<br>black. One of them falls. If the red one fell, that pile now offers black, and the other<br>pile was already offering black, so the next card is black whichever hand it comes<br>from . The pair is one of each.
And now count what has been used. Either both cards came from the same pile, which has<br>advanced two and so offers its original color again while the other pile has not moved; or<br>one came from each, and both have advanced one. Either way the two piles are offering<br>opposite colors again, which is exactly where the pair started. So it holds for the next<br>pair, and the one after that, all the way down.
Why a cut only works half the time
A cut takes the same cards off the top but does not turn them over. So the two<br>sequences read from their bottoms are:
the cut-off packet, from its bottom → card k, card<br>k−1, …
the rest of the deck, from its bottom → card 52, card 51, …
Now the first begins on card k rather than card 1. Card 52 is black, so the<br>two start on opposite colors only when card k is red — which is to say only<br>when k is odd . Cut an odd number and the principle holds, every<br>time, for the same reason as before.
Cut an even number and it almost always fails. Almost, rather than<br>always, and the gap is worth being honest about: across 50,000 even cuts it failed 96.1%<br>of the time, so about one in twenty-five survived anyway. Those survivors are not a<br>second principle hiding underneath. They are riffles that barely riffled.
Cards cut offPairs survived anyway<br>232.2%<br>411.1%<br>63.8%<br>81.1%<br>100.7%
Cut two cards off and a third of riffles leave the pairs intact, because two cards<br>very often fall as a single clump — and a packet that falls in one clump has not<br>been interleaved at all, it has simply been put back. Cut ten and it is under one in a<br>hundred. The tail of the deck behaves the same way by symmetry: cutting 50 survives 32%<br>of the time as well.
So the ten-thousand-trial button reports no failures at all when the cards are dealt,<br>and failures roughly half the time when they are cut — about half the numbers from<br>1 to 51 are even, and an even cut fails 96% of the time.
The second principle
Gilbreath published the general case eight years after the first. Arrange the deck in a<br>repeating run of n cards — spades, hearts, diamonds, clubs, thirteen times<br>over — deal any number off the top, riffle, and every consecutive group of<br>four holds one of every suit . The alternating colors are simply the case where<br>n is 2.
The proof is the same argument counted modulo n rather than modulo 2, and the<br>demonstration above will run it: change the arrangement to repeating suits and press the<br>same buttons.
Where...