1992年，数学家戴夫·拜尔和佩尔西·戴康尼斯证明了进行七次完美洗牌足以将一副扑克牌彻底打乱，并揭示了「截止现象」（cutoff phenomenon），即牌组在第七次洗牌时会突然从有序状态转变为高度无序状态。然而，这一著名定理仅适用于切牌极为精准的理想状况，若洗牌不够均匀，该结论便主动失效。

为了解决不均匀切牌的洗牌问题，哈佛大学统计学家马克·塞尔克与剑桥大学的石佳路、普林斯顿大学的王佳敏合作，利用二进位条形码系统追踪每张牌在多次洗牌中的运动路径。他们重点分析了那些难以被打乱的「冷点」区域，并透过图论与重叠区域的指数级衰减，成功证明了即使在切牌不均匀的情况下依然存在截止现象。

根据这项最新研究，若每次洗牌都随机切牌，一副52张的扑克牌大约需要进行14次洗牌才能达到完全随机的状态。这项研究成功推广了数十年来的洗牌数学理论，虽然目前的模型仍假设牌是一张张落下而非成叠落下，但研究团队已计划未来进一步探究更贴近现实的「成叠洗牌」问题。

In 1992, mathematicians Dave Bayer and Persi Diaconis proved that seven perfect riffle shuffles are sufficient to randomize a deck of cards, revealing a "cutoff phenomenon" where the deck suddenly transitions from an orderly to a highly unstructured state. However, this famous proof only holds true under the rigid constraint that the deck is cut precisely in half, failing if the shuffles are performed less accurately.

To address the issue of imprecise cuts, mathematicians Mark Sellke, Jialu Shi, and Jiamin Wang tracked each card's path using a binary barcode system during shuffles. They focused on "cold spots"—regions of the deck that resist mixing—and used graph theory to show that the overlap between unmixed regions decays exponentially, proving that a cutoff phenomenon still exists even with uneven cuts.

According to the new proof, it takes approximately 14 riffle shuffles to fully randomize a 52-card deck when the deck is cut at a random location each time. This work successfully extends a decades-old mathematical theory, and while the current model still assumes cards fall one by one rather than in clumps, the team plans to investigate more realistic "clumpy shuffles" in the future.