Limit Plus Peel
Some observations and assertions of a mathematical nature are easy, or at least routine, to generalize from two to three or beyond. Others are trickier (e.g., cake cutting), and a few turn out to be impossible (e.g., angle trisection using only a straightedge and compass). In the August 2011 Card Colm mentioned above, we made an appeal for a new kind of generalization of the ever-popular Gilbreath shuffle principle, askingand expect some order to remain, without assuming perfect interweaving?
The good news is that this month we have something to report along these lines. We can now sleep at night, although a really impressive magic application is still proving elusive.
Treeless Shutoff
First it pays to switch our focus from riffle shuffling—which in the Gilbreath context follows the dealing out of some cards from a packet to form two piles—to something equivalent to that, which has a more obvious generalization to three or more piles. We're referring to Lennart Green's rosette shuffle as depicted below. Two side-by-side piles are twirled into rosettes with the fingers before being pushed together and squared up.When we say that this is equivalent to riffle shuffling we really mean that the following internal coherence or solitaire principle applies.
- If two piles of cards, running Ace to King of Hearts and Ace to King of Spades, respectively, are riffle or rosette shuffled together, and the cards in the resulting packet are dealt one at a time into two piles separated by suit, then the original piles will be reformed, in reverse order.
Sheaf Bullfighters
In the first August edition of Card Colm, back in 2005, we explored the first (or basic) Gilbreath principle from 1958, which concerns the structure retained when two piles of cards are riffle or rosette shuffled, assuming that we start with an alternating packet of some kind and then deal "about half" of the cards to the table to give rise to the two piles. Generalizing that to situations where the packet started by repeating a cyclic pattern of length greater than two leads easily and naturally to the second (or generalized) Gilbreath principle from 1966, which we covered here in August 2006.In a nutshell, the Gilbreath Principle says that
- If we start with a packet of sm cards that consists of m repeated stacks of a particular set of s cards, each set being in the same fixed order with respect to some characteristic of interest (such as color, suit, or value), and we count out some of these to the table—thereby reversing their order—and then riffle shuffle together the two resulting piles, we're sure to get a packet in which each stack of s cards pulled off the top contains exactly one card of each of the key types, in some order.
For the case s = 3, we could start with a packet consisting of trios of Spades, Diamonds, and Hearts, in that order, over and over. The shuffled packet then has the property that the cards in positions 1-3, 4-6, 7-9, etc, always consist of one each of those three suits, again in some order. When s = 13, starting with a packet consisting of four stacks of the thirteen card values, in the same fixed order, we end up with four clumps (the cards 1-13, 14-26, 27-39 and 40-52) consisting of the thirteen values in some mixed order.
Lovely though all of this is—and it has spawned a ton of magic over the last half century—it's well-explored territory. What more is there to say? Persi Diaconis & Ron Graham give a complete characterization of such matters in what they call The Ultimate Gilbreath Principle on page 70 of their beautiful, inspirational and prize-winning book Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks (Princeton University Press, 2011). Some Gilbreath variations were considered here in subsequent Augusts, in The Bligreath Principle (2009) and Gilbreeath Shuuffling (2012). Just six months ago we couldn't resist sharing a new twist on all of this due to John Hostler in Flushed with Embarrassment (February 2013). Normal Gilbreath's own long-awaited book is coming soon, from L&L Publishing.
But there is another direction in which to turn, in search of a different kind of thrill. What if we rosette shuffle three or more piles with some simple structure?
Certainly, the earlier solitaire principle extends in the obvious way:
- If k piles of cards, running 1-n, representing k "suits," are rosette shuffled together, and the cards in the resulting packet are dealt one at a time into k piles separated by suit, then the original piles will be reformed, in reverse order.
Canonical Indecencies Pix
Now we highlight something Karl Fulves pointed out in his hard-to-find book Riffle Shuffle Set-Ups (1968), which saw distribution only in magic circles. The rosette shuffle wasn't on his radar screen, of course, he was thinking in the context of riffle shuffling multiple piles, say, two at a time, until he had a single packet again. Even for the case of two piles, this fact has languished in obscurity for decades.- If k piles of cards, running 1-n, representing k "suits," are rosette shuffled together, and the cards in the resulting packet are peeled off k at a time, then if the jth set matches in value, they have value j.
In practice, this is better performed with the suits arranged in some less obvious but memorized order. For instance, suppose we use the starting arrangement 3, 5, 10, Ace, 9, Jack, 2, 8, 7, Queen, 6, 4, King, in each suit. (This is much favored by John H. Conway, and is based on the mnemonic "The Five Tenacious Boys Nicely Joke To Hated Servant Girls Sick For Absent Kings"—we've omitted Jokers.) Have an audience member twirl the four face-down suit piles and mush them together, then turn away and request that cards be peeled off four at a time and inspected. If any such quadlet matches, you are informed. If the fifth and eighth quadlets match, say, "Coincidences I can explain." Confidently announce that they are the Jacks and 8s, respectively.
Incidentally, Fulves also has something to say about the values in any situation where the jth set of k cards contains of one of each suit (with none repeated).
Arrange Wall Modish
A year ago—and a full year after we issued the Call For A New Shuffle Principle (Need Rot Sextet?)—Ron Graham graciously responded to repeated prodding by coming up with the following key breakthrough, shared here with his kind permission.- Arrange a packet in cyclic order 0, 1, 2, 0, 1, 2, .... Now split into three piles that have as their top cards, 0, 0, and 1. In other words, the cards in the three piles are in the order 0, 1, 2, 0, 1, 2, ..., 0, 1, 2, 0, 1, 2, ... and 1, 2, 0, 1, 2, 0, ... . Rosette shuffle them together, so that in the result, the relative order of the three piles is preserved. Start removing sets of three consecutive cards at a time. Then none of these three-card sets will have all values the same.
- Start with a packet arranged cyclically as 0, 1, ..., , s-1, 0, 1, ..., s-1, and then break it into s piles, and rosette shuffle those together. If we now remove sets of s cards at a time from the top, and the top card of the original ith pile is denoted by a_{i}, then assuming the sum of the a_{i} is not 0 (mod s), we can be sure that none of the sets of s cards will have all values the same.
For s = 3, start with a packet cycling Clubs, Hearts, and Diamonds, and split it into three piles that have as their top cards Clubs, Clubs, and Hearts. Rosette shuffle, and pull cards off the top three at a time. No such triplet will consist of three cards of the same suit. The same conclusion also holds whenever the top cards of the three piles account for exactly two of the three suits in play. The principle is not saying anything when the top cards of the three piles are all the same, or account for one of each suit; but perhaps those cases yield to a different kind of analysis.
More details—and a preliminary magic application—can be found in Chapter 8 of Mathematical Card Magic: Fifty-Two New Effects (AK Peters/CRC Press, 2013).
"Limit Plus Peel" is an anagram of "Multiple Piles," "Treeless Shutoff" is an anagram of "Rosette Shuffles," "Sheaf Bullfighters" is an anagram of "Gilbreath Shuffles," "Canonical Indecencies Pix" is an anagram of "Coincidences I Can Explain," and "Arrange Wall Modish" is an anagram of "Ronald Lewis Graham."