内容来源：https://www.quantamagazine.org/seven-perfect-shuffles-randomize-a-deck-of-cards-but-how-many-sloppy-ones-20260617/

内容总结：

七次完美洗牌能打乱一副牌，那如果洗得马虎呢？

1992年，数学家们提出了一个著名结论：只需七次“鸽尾式洗牌”——即玩家将一副牌分成两叠，然后用拇指像拉链一样将两叠牌交叠在一起——就足以将牌序彻底打乱。数学家戴夫·拜尔和珀西·迪亚科尼斯在证明这一结论的同时，还揭示了一个令人惊讶的现象：在洗牌过程中，牌序起初会保持相对有序，但到了第七次洗牌时，整副牌会突然进入一种高度无序的状态。这种被称为“截止现象”的行为，其研究意义远超纸牌游戏本身，许多动力系统——包括凝聚态物理学中的“自旋玻璃”——据信也存在类似现象。

然而，拜尔和迪亚科尼斯的证明——被一些人誉为“数学奇迹”——仅在遵循严格的切牌和洗牌规则时才能成立。如果你的洗牌手法更像中学生而非魔术师，那么结论便不再适用。

如今，三位数学家终于将这一发现推广到了不那么精确的洗牌方式上。哈佛大学统计学家马克·塞尔克（目前休假在OpenAI工作）与剑桥大学研究生石嘉璐、普林斯顿大学研究生王佳敏共同证明，即使在切牌时未将牌堆分成均匀的两叠，鸽尾式洗牌中同样存在截止现象。

迪亚科尼斯对其研究成果的更新感到欣喜不已。“这是一个新颖的想法，而且令人惊叹的是，在没那么严格的条件下它依然如此有效，”他说，“这是一篇杰出的数学佳作。”

“冷区”的混合

将普通的鸽尾式洗牌称为“复杂”都远远不够。一副普通扑克牌的可能排列方式数量是52的阶乘——即52 × 51 × 50 × … × 3 × 2 × 1，约等于8后面跟67个零，接近银河系中原子数量的估算值。换个角度理解这个数字：每当你洗一次牌，你几乎肯定是在创造一种从未存在过、也永远不会再出现的排列。

然而，数学界对洗牌的兴趣并不仅仅源于其组合复杂性。早在1981年，迪亚科尼斯和梅赫达德·沙赫沙哈尼就在洗牌背景下发现了截止现象，此后数学家们开始在各个领域发现此类现象。

截止现象类似于物理学中的相变，例如零摄氏度时液态水突然结晶成固态冰。但截止现象发生在特定的数学背景——“马尔可夫链”中，这是一种用概率描述系统（如一副扑克牌）在不同状态间转换的数学模型。

正如其名，截止现象的发生方式与海明威对破产的著名描述如出一辙：先是逐渐发生，然后突然爆发。尽管截止现象无处不在——塞尔克表示，它“预计会出现在大多数大型复杂系统中”——但要证明关于它的一般性定理却十分困难。“对于大多数被认为存在截止现象的问题，”康奈尔大学数学家、曾与迪亚科尼斯合作的洛朗·萨洛夫-科斯特说，“人们并不知道如何证明它。”

这就是为什么“七次洗牌足矣”的定理如此重要。拜尔和迪亚科尼斯——后者少年时离家出走，师从一位专攻纸牌戏法的魔术师，后来成为著名数学家——不仅证明了现实系统中精确截止现象的存在，还给出了一个适用于任意大小牌组的统一公式。

但该结论也有适用条件。第一：鸽尾式洗牌必须遵循一个现实但严格的模型，即纸牌从左堆或右堆随机、逐张地落下（每张牌从左边或右边掉落的概率与该堆剩余的牌数成正比）。这意味着纸牌不会简单地左右交替，因为那会产生可预测的结构；相反，顺序可能是“左、右、右、左、右、左、左”。第二：洗牌前，牌堆必须大致切成两半。

“我们的所有分析都依赖于这些细节，”迪亚科尼斯说。

1999年，芝加哥大学数学家史蒂文·拉利试图放宽这些限制，寻求在不要求均匀切牌的情况下证明鸽尾式洗牌截止现象的存在。“在我看来，提出这个问题很自然——有些人切牌时会稍微切多一点或少一点，”他说。

这些切得不那么均匀的牌堆中，存在一些纸牌组，它们在多次洗牌后仍倾向于保持原有的相对顺序。虽然牌堆的其他部分看起来已经充分混合，但这些特定的牌组——拉利称之为“冷区”——仍然保留了其原始位置的信息。

例如，假设你给牌标上1到52号。多次洗牌后，16和17号牌虽然不再紧邻，但16号牌出现在17号牌之前的概率，可能仍比在完全随机牌堆中更高。如果原始牌堆中某个片段内的多对牌（比如15到25号）都表现出类似的偏差，那么这一组牌就构成了一个“冷区”。拉利希望证明，当这些冷区消失时，牌堆中最后的有序痕迹也会随之消失——这将成为他证明截止现象存在的方法。但他未能证明这一点。

追踪“标签”

二十年后，2019年，拉利合作者托马斯·塞尔克的儿子——当时还是斯坦福大学研究生的马克·塞尔克——在迪亚科尼斯的一堂课上学到了原始的“七次洗牌”结论。“他随口提到，如果你不把牌堆对半切，那么（证明中的）所有方法都不再适用了，”马克·塞尔克回忆道，“我当时想，‘就这？……拜托，我们肯定能做到。’”

到2021年，马克·塞尔克已经找出了在切牌远非均匀（甚至将牌堆切成两叠以上）的情况下截止现象的存在。但每次洗牌之间的切牌方式仍需保持一致。他想要一个更现实的结论，即每次洗牌后的切牌方式可以非常不同。于是，在2024年夏天，他与同样对这个课题感兴趣的施嘉璐和王佳敏联手。

三人首先为每张牌分配了一个“条形码”。从切牌开始：左边那叠的所有牌都分配数字1，右边那叠分配0。然后洗牌，将两叠牌随机逐张交叠。再次切牌：如果一张牌最终进入左堆，其标签后加1；进入右堆，则加0。随着这一过程在多次鸽尾式洗牌中重复，每张牌都会积累起越来越长的由1和0组成的条形码，编码了它在左堆和右堆之间跳跃的路径。例如，如果第17张牌在四次洗牌后条形码是0110，这意味着它最初在右堆，然后两次进入左堆，最后又回到右堆。

这些数字为牌堆中的每张牌创建了一个独特的追踪标签。如果最初相对顺序相同的两张牌（如16和17）最终获得了相同的由1和0组成的条形码，这意味着它们在洗牌过程中经历了完全相同的路径，因此仍然保持着相同的相对顺序。

要证明截止现象的存在，你必须说明：无论初始牌堆大小如何，无论切牌方式如何，在特定次数的洗牌后，这些匹配的条形码变得非常少。但逐一比较每个条形码非常耗时。幸运的是，正如拉利所期望的那样，“冷区”提供了一条捷径。由于这些是牌堆中抵制混合的区域，因此只需检查这些区域内的条形码是否匹配。

将一副有n张牌的牌堆中所有位于“冷区”的牌的条形码按升序排列，然后利用这些信息构建一个由点和线组成的“图”。将每张牌（以其条形码标记）表示为一个点，如果两张牌有相同的条形码，则用一条边连接这两个点，以表明对应的一对牌尚未被混合。对第二副有n张牌的牌堆重复这一过程。然后，对齐代表两副牌的图，找出它们重叠的部分。

这些重叠部分表明了这两副牌在多少次洗牌后仍在抵制混合。塞尔克、施嘉璐和王佳敏随后证明，在特定次数的洗牌后（次数取决于牌堆中的牌数），两副牌中未混合区域的重叠部分以指数速度消失。据塞尔克称，这一所谓的“指数拖尾”为他们提供了一个上限，即需要多少次洗牌才能溶解一副典型牌堆中最后的有序痕迹。

从“逐张”到“成叠”及未来

根据新的证明，对于一副52张的牌，如果每次洗牌都随机选择切牌位置，那么这个上限大约是14次鸽尾式洗牌。超过这个次数，牌序将被完全打乱。

“对我来说，动机非常简单，”塞尔克说，“我偶尔和朋友打扑克，我想知道我该洗多少次牌。”

“这太棒了，”拉利评价道，“我从马克还是个孩子时就认识他。这个问题搁置了26年，最终被他攻克了。”

严格来说，仍有工作要做。与拜尔和迪亚科尼斯的结论一样，这一新成果所依赖的洗牌模型仍然假设纸牌是一张接一张地落下，而不是成叠地落下。这种洗牌水平超出了大多数普通牌手的能力范围（尽管迪亚科尼斯特意指出，这完全在他的能力之内）。塞尔克表示，他对解决这一版本的问题“非常感兴趣”，并将其视为推广洗牌中截止现象研究的另一种方式。

“我真的很喜欢这个‘成叠洗牌’的问题，”他说，“我有段时间没取得进展了。也许等有人取得一些突破后，我们可以尝试将我们已经得出的理解应用到其中。”

中文翻译：

七次完美洗牌可使一副扑克牌完全随机化，但随意洗牌需要多少次？
引言
1992年，数学家们著名地证明了七次“交错洗牌”——即玩家将一副牌分成两叠，然后用拇指像拉链一样将它们交替合并——足以将牌序打乱。
当戴夫·拜尔和佩尔西·戴康尼斯提出这一证明时，他们还揭示了过程中一个令人惊讶的现象：起初，牌序相对有序。但到第七次洗牌时，牌序会突然陷入高度无序的状态。这种被称为“截止现象”的行为，其意义远不止于纸牌，许多动力系统——包括凝聚态物理中的“自旋玻璃”——被认为也会表现出类似特征。
不幸的是，拜尔和戴康尼斯的证明——被一些人称为数学奇迹——只有在遵循关于切牌和洗牌的严格约束时才成立。如果你洗牌的方式更像初中生而非魔术师，这个结果就不成立了。
现在，三位数学家终于将这一发现扩展到了不那么精确的洗牌方式。哈佛大学统计学家马克·塞尔克（目前休假在OpenAI工作），与石嘉璐和王嘉敏（分别是剑桥大学和普林斯顿大学的研究生）共同证明，即使不将牌叠分成整齐的两等份，交错洗牌也存在截止现象。
戴康尼斯对这一成果的更新赞不绝口。“这是一个新颖的想法，而且类似的东西能如此有效地运作，真是了不起，”他说，“这是一项杰出的数学成果。”
混合冷点
用“复杂”来形容普通的交错洗牌简直是大材小用。一副普通扑克牌的可能排列数量是52的阶乘——即52 × 51 × 50 × … × 3 × 2 × 1，粗略来说是一个8后面跟着67个零，接近我们银河系中估计的原子数量。另一种理解方式：每次你洗牌时，几乎可以肯定你创造了一个从未存在过、也永远不会再出现的排列。
但数学界对洗牌的兴趣远不止于其组合复杂性。早在1981年，戴康尼斯和梅赫达德·沙赫沙哈尼就在洗牌背景下发现了截止现象——此后数学家们开始在各种地方发现它们。
卡罗琳·盖茨
截止现象类似于物理学中的相变，例如液态水在零摄氏度时突然结晶成固态冰。但截止现象发生在特定的数学背景——“马尔可夫链”中，这些数学模型以概率方式描述系统（如一副牌）如何在不同的状态之间移动。
顾名思义，截止现象的发生方式类似于欧内斯特·海明威描述破产的著名说法：先是逐渐地，然后突然地。虽然截止现象无处不在——据塞尔克说，它们预计会出现在“大多数大型复杂系统”中——但证明关于它们的普遍定理也很困难。“对于大多数人们认为存在截止现象的问题，”康奈尔大学的数学家洛朗·萨洛夫-科斯特（曾与戴康尼斯合作）说，“人们不知道如何证明它。”
这就是为什么“七次洗牌就够了”这一定理如此重要。拜尔和戴康尼斯——后者少年时离家出走，师从一位专攻纸牌戏法的魔术师，后来成为著名数学家——不仅证明了现实世界系统中精确截止现象的存在，还提供了一个单一公式来确定截止点，该公式适用于任何大小的牌叠。
然而，这也有附加条件。第一：交错洗牌必须遵循一个现实但严格的模型，即纸牌从左叠或右叠一张一张随机交错落下。（每张牌从左叠或右叠掉落的概率与该叠剩余牌数成比例。这意味着纸牌不会简单地在左右之间交替，那样会产生可预测的结构；相反，顺序可能变成“左、右、右、左、右、左、左”。）
第二：洗牌前，牌叠必须大致切成两半。
“我们所有的分析都依赖于这些细节，”戴康尼斯说。
1999年，芝加哥大学的数学家史蒂文·拉利试图放宽这些限制，寻求一种针对起始牌叠并非大致均分的交错洗牌的截止现象证明。“对我来说，提出这个问题似乎很自然——有些人倾向于把牌叠切得高一点或低一点，”他说。
这些不那么均匀的牌叠中，有些牌组即使在多次洗牌后也倾向于保持相同的相对顺序。虽然牌叠的其他部分看起来混合得很好，但这些特定的牌组——拉利称之为“冷点”——仍然保留了它们原始位置的信息。
例如，假设你将牌编号为1到52。多次洗牌后，16号和17号牌不会再紧挨着出现，但在随机牌叠中，16号出现在17号之前的频率可能仍然更高。如果原始牌叠中某个区段内的许多对牌（例如15号到25号）表现出类似的偏差，那么这组牌就形成了一个冷点。
拉利希望证明，当这些冷点消失时，牌叠中最后的有序痕迹也会消失——这为他展示截止现象的存在提供了一种方法。
但他未能证明。
追踪标签
二十年后，2019年，拉利合作者托马斯·塞尔克的儿子——当时还是斯坦福大学研究生的马克——发现自己坐在戴康尼斯的一堂课上，了解到七次洗牌的原始结论。“他随口提到，如果你不把牌叠切成两半，那么（关于证明的）一切都不再适用了，”马克·塞尔克回忆道，“我当时想，‘就这？……拜托，我们肯定能做到。’”
到2021年，马克·塞尔克已经确定了针对切割程度远不如拜尔和戴康尼斯原始研究中那么均匀的牌叠的截止点——包括切分成两个以上堆的牌叠。但每次洗牌之间的切割方式仍需保持一致。他想要一个更现实的结果，即一次洗牌到下一次洗牌的切割方式可能非常不同。于是，在2024年夏天，他与同样对这个问题的感兴趣的施嘉璐和王嘉敏合作。
三人首先为每张牌分配一个条形码。从你切牌开始，左堆中的所有牌被分配数字1，右堆中的牌分配0。然后洗牌，从左堆和右堆中一张一张随机交错取出牌。再次切牌。如果一张牌最后落在左堆，就在其标签上加一个1；如果落在右堆，则加一个0。
随着这个过程在更多的交错洗牌中重复，每张牌建立起一个由1和0组成的越来越长的条形码，它编码了该牌在洗牌过程中从左到右再回来的路径。例如，如果第17张牌在四次洗牌后有一个0110的条形码，这意味着它开始时在右堆，然后两次落在左堆，最后又回到右堆。
这些数字为牌叠中的每张牌创建了唯一的追踪标签。如果两张原本相对顺序相同的牌——比如16号和17号——最终得到相同的由1和0组成的条形码，这意味着它们在洗牌过程中经历了完全相同的路径，并且仍然保持着相同的相对顺序。
为了证明截止现象的存在，你必须证明在特定次数的洗牌后，这些匹配的条形码很少存在——无论你开始时有多少张牌，或者牌叠是如何切割的。但比较每个条形码很耗时。幸运的是，正如拉利所希望的，冷点提供了一条捷径。由于这些是牌叠中往往抗拒混合的区域，它们是你唯一需要检查条形码是否匹配的地方。
从一副n张牌的牌叠开始，按升序列出牌叠冷点中所有牌的条形码。
马克·贝兰/《量子杂志》
然后利用这些信息构建一组点和线，称为图。将每张牌（以其条形码标记）表示为一个点。
如果两个点有相同的条形码，就用一条边连接它们，以反映对应的两张牌尚未混合的事实。
对第二副n张牌的牌叠重复这个过程。然后并排放置代表两副牌的图，并确定它们重叠的位置。
这些重叠表明两副牌继续抗拒混合的时间。塞尔克、施嘉璐和王嘉敏随后证明，在特定次数的洗牌后（取决于牌叠中的牌数），两副牌未混合区域的重叠以指数级速率消失。根据塞尔克的说法，这种所谓的指数尾部给了他们一个上界，该上界决定了瓦解一副典型牌叠中最后一点有序性所需的洗牌次数。
从成组洗牌到更远
根据新的证明，对于一副52张的牌叠，如果你每次洗牌时随机选择一个位置切牌，那么这个上限大约是14次交错洗牌。超过这个次数，牌将完全混合。
“对我来说，动机很简单，”塞尔克说，“我偶尔和朋友打扑克，我想知道我该洗多少次牌。”
“这太惊人了，”拉利说，“我从小就认识马克。这个问题搁置了26年，最终他解决了它。”
从技术上讲，仍有工作要做。新结果所依赖的洗牌模型，像之前的拜尔和戴康尼斯模型一样，仍然假设纸牌是一张一张交错落下，而不是成组落下。这种洗牌技巧超出了大多数普通牌手的能力（尽管戴康尼斯本人完全能做到，他特意指出这一点）。塞尔克说，他“非常感兴趣”于攻克这个版本的难题，作为推广纸牌洗牌中截止现象的另一种方式。
“我真的很喜欢这个成组洗牌的问题，”他说，“我有一段时间没有取得进展。也许等有人取得一些突破后，我们可以再次尝试运用我们已经产生的理解。”

英文来源：

Seven Perfect Shuffles Randomize a Deck of Cards. But How Many Sloppy Ones?
Introduction
In 1992, mathematicians famously proved that seven “riffle shuffles” — the kind where a player splits a deck of cards into two piles, then uses their thumbs to interleave them back together in a zipperlike motion — are enough to mix up the deck.
When Dave Bayer and Persi Diaconis came up with this proof, they also revealed something surprising about what happens along the way: At first, the cards stay relatively orderly. But with that seventh shuffle, the deck suddenly tips into a highly unstructured state. This kind of behavior, called a cutoff phenomenon, is of interest beyond cards, and many dynamical systems — including “spin glasses” in condensed matter physics — are believed to exhibit it.
Unfortunately, Bayer and Diaconis’ proof — referred to by some as a mathematical miracle — only works if you adhere to some rigid constraints about how to cut and shuffle the deck. If you shuffle more like a middle schooler than a magician, the result doesn’t hold.
Now three mathematicians have finally extended the finding to less precise shuffles. Mark Sellke, a Harvard University statistician currently on leave to work at OpenAI, along with Jialu Shi and Jiamin Wang (graduate students at the University of Cambridge and Princeton University, respectively), proved that a cutoff phenomenon exists for riffle shuffling even when you don’t cut the deck into two nice, even piles.
Diaconis was effusive about the update to his work. “It’s a fresh idea, and it’s remarkable that something like that would work as effectively as it does,” he said. “It’s a brilliant piece of mathematics.”
Mixing Cold Spots
To call the humble riffle shuffle “complicated” sells it absurdly short. The number of possible arrangements for an ordinary deck of cards is 52 factorial — that is, 52 × 51 × 50 × … × 3 × 2 × 1, or (roughly speaking) an 8 followed by 67 zeros, close to the estimated number of atoms in our galaxy. Another way to put the figure into context: Every time you shuffle a deck of cards, you produce a configuration that has almost certainly never existed before, and never will again.
But mathematical interest in card shuffling goes beyond its combinatorial complexity. Back in 1981, Diaconis and Mehrdad Shahshahani discovered cutoff phenomena in the context of card shuffling — after which mathematicians started to uncover them all over the place.
Caroline Gates
Cutoffs are similar to phase transitions in physics, such as the sudden crystallization of liquid water into solid ice at zero degrees Celsius. But cutoffs occur in the specific mathematical context of “Markov chains,” mathematical models that probabilistically describe how a system (like a deck of cards) moves between different configurations.
Cutoff phenomena, as their name suggests, happen in much the same way as Ernest Hemingway famously described going bankrupt: gradually, then suddenly. And while cutoffs are ubiquitous — they’re expected to occur in “most large, complex systems,” according to Sellke — it’s also hard to prove general theorems about them. “For most problems where one thinks there is a cutoff,” said Laurent Saloff-Coste, a mathematician at Cornell University who has collaborated with Diaconis, “one doesn’t know how to prove it.”
That’s why the “seven shuffles are enough” theorem was such a big deal. Bayer and Diaconis — who as a teenager ran away from home to apprentice with a magician specializing in card tricks, before becoming a renowned mathematician — didn’t just prove the existence of a precise cutoff in a real-world system. They provided a single formula for where that cutoff should be, and that formula worked for decks of any size.
Yet terms and conditions also apply. One: The riffle shuffle has to follow a realistic but strict model where cards are randomly interleaved from the left or right pile one by one. (Each card gets dropped from either the left or the right pile with a probability that’s proportional to the number of cards remaining in that pile. This means that the cards don’t simply alternate between left and right, which would result in a predictable structure; instead, the order might go “left, right, right, left, right, left, left.”)
Two: The deck has to be cut more or less in half before shuffling.
“All of our analysis depends on those details,” Diaconis said.
In 1999, Steven Lalley, a mathematician at the University of Chicago, attempted to loosen those constraints by seeking a cutoff proof for riffle shuffles that didn’t start with roughly evenly cut decks. “It seemed natural to me to ask — there are some people who tend to cut the deck a little higher or a little lower,” he said.
These less evenly cut decks have sets of cards that tend to stay in the same relative order even after multiple shuffles. While the rest of the deck looks well mixed, these particular sets of cards — which Lalley called “cold spots” — still retain information about their original locations in the deck.
Imagine, for instance, that you label your cards 1 through 52. After multiple shuffles, cards 16 and 17 will no longer appear right next to each other in the deck, but 16 might still tend to appear before 17 more often than it would in a random deck. If many pairs within a section of the original deck — say, cards 15 through 25 — show similar biases, then that set of cards forms a cold spot.
Lalley hoped to prove that when those cold spots disappeared, so would the last traces of order in the deck — giving him a way to show the existence of a cutoff.
But he couldn’t prove it.
Tracking Labels
Two decades later, in 2019, the son of Lalley’s collaborator Thomas Sellke — Mark, then a graduate student at Stanford University — found himself in one of Diaconis’ classes, where he learned about the original seven-shuffles result. “He mentioned offhandedly that if you don’t cut the deck in half, then nothing [about the proof] works anymore,” Mark Sellke recalled. “I was like, ‘This is it? … Come on, we must be able to do this.’”
By 2021, Mark Sellke had pinpointed the cutoff for decks cut much more unevenly than those in Bayer and Diaconis’ original work — including for decks cut into more than two piles. But the deck still had to be cut in the same way between each shuffle. He wanted a more realistic result, where the cuts from one shuffle to the next might look very different. And so in the summer of 2024, he teamed up with Shi and Wang, who had also expressed interest in the problem.
The trio first assigned each card a barcode. It starts when you cut the deck. All the cards in the left pile get assigned the number 1; those on the right, zero. Now shuffle, randomly interleaving the cards from the two piles one by one. Cut the deck again. If a card ends up in the left pile, add a 1 to its label; if it ends up in the right pile, add a zero.
As this process repeats through more riffle shuffles, each card builds up a longer and longer barcode of ones and zeros, which encodes its path through the shuffling process as it hops from left to right and back again. For instance, if the 17th card has a barcode of 0110 after four shuffles, that means it started in the right pile, ended up on the left twice, and then landed back on the right.
These numbers create a unique tracking label for every card in the deck. If two cards that started out in the same relative order — say, 16 and 17 — end up with the same barcode of ones and zeros, that means they took the exact same path through the shuffling process and are still in the same relative order.
To prove the presence of a cutoff, you have to show that very few of those matching barcodes remain after a certain number of shuffles — no matter how many cards you started with, or how the deck was cut. But comparing every barcode is time-consuming. Fortunately, the cold spots offer a shortcut, just as Lalley had hoped. Since those are the regions in the deck that tend to resist mixing, they’re the only places you have to check for barcodes that match.
Start with a deck of n cards and list the barcodes of all the cards in the deck’s cold spots in ascending order.
Mark Belan/Quanta Magazine
Then use that information to build a collection of points and lines, called a graph. Represent each card (labeled with its barcode) as a point.
Connect two points with an edge if they have the same barcode, to reflect the fact that the corresponding pair of cards has still not gotten mixed up.
Repeat this process with a second deck of n cards. Then line up the graphs representing both decks and identify where they overlap.
These overlaps indicate how long both decks continue to resist mixing. Sellke, Shi, and Wang then showed that after a certain number of shuffles (which depended on how many cards were in the deck), the overlap between the decks’ regions of unmixed cards disappeared at an exponential rate. This so-called exponential tail, according to Sellke, gave them an upper bound on the number of shuffles it would take to dissolve the last bits of orderliness in a typical deck.
To Clumping and Beyond
According to the new proof, that ceiling is roughly 14 riffle shuffles for a 52-card deck, if you cut your deck in a random place with each shuffle. Beyond that point, the cards will be fully mixed.
“For me the motivation is pretty simple,” Sellke said. “I occasionally play poker with my friends, and I want to know how many times I should be shuffling my cards.”
“It’s spectacular,” Lalley said. “I’ve known Mark ever since he was a young kid. The problem sat there for 26 years, and finally he cracked it.”
Technically speaking, there’s still work to be done. The model of shuffling that the new result depends on, like Bayer and Diaconis’ before it, still assumes that the cards riffle down one by one, rather than in clumps. That level of shuffling skill is beyond what most casual card players possess (although well within Diaconis’ own powers, as he made sure to note). Sellke said he’s “very interested” in attacking this version of the problem as another way to generalize the cutoff phenomenon in card shuffling.
“I really like this clumpy shuffle question,” he said. “I haven’t made progress for a while. Maybe after someone makes a bit of headway, then we can try again to adopt the understanding we’ve already produced.”