Kruskal count

The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s[when?] discussing coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total and later called Kraus principle.

Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navigation, object alignment, and others.

Card trick

The trick is performed with cards, but is more a magical-looking effect than a conventional magic trick. The magician has no access to the cards, which are manipulated by members of the audience. Thus sleight of hand is not possible. Rather the effect is based on the mathematical fact that the output of a Markov chain, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock performed by David Copperfield is as follows. A volunteer picks a number from one to twelve and does not reveal it to the magician. The volunteer is instructed to start from 12 on the clock and move clockwise by a number of spaces equal to the number of letters that the chosen number has when spelled out. This is then repeated, moving by the number of letters in the new number. The output after three or more moves does not depend on the initially chosen number and therefore the magician can predict it.

See also

• Coupling (probability)
• Discrete logarithm
• Equifinality
• Ergodic theory
• Geometric distribution
• Overlapping instructions
• Pollard's kangaroo algorithm
• Random walk
• Self-synchronizing code

Notes

Further reading

• Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович]; Uspenskii [Успе́нский], Vladimir Andreyevich [Влади́мир Андре́евич] (1963). Written at University of Moscow, Moscow, Russia. Putnam, Alfred L.; Wirszup, Izaak (eds.). . Survey of Recent East European Mathematical Literature. Vol. 3. Translated by Whaland, Jr., Norman D.; Titelbaum, Olga A. (1 ed.). Boston, Massachusetts, US: The University of Chicago / D. C. Heath and Company. LCCN . (1+9+80+9+1 pages) (NB. This is a translation of the first Russian edition published as "Математические беседы: Задачи о многоцветной раскраске / Задачи из теории чисел / Случайные блуждания" by GTTI (ГТТИ) in March 1952 as Number 6 in Library of the Mathematics Circle (Библиотека математического кружка). It is based on seminars held at the School Mathematics Circle in 1945/1946 and 1946/1947 at Moscow State University.)
• Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович] (1965) [1963-03-10, 1962-03-31]. Written at University of Moscow, Moscow, Russia. . Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. I (121). Translated by Fabius, Jaap [at Wikidata]; Greenberg, Vida Lazarus [at Wikidata]; Maitra, Ashok Prasad [at Wikidata]; Majone, Giandomenico (1 ed.). New York, US / Berlin, Germany: Springer-Verlag (Academic Press, Inc.). doi:. ISBN 978-3-662-00033-5. ISSN . LCCN . S2CID . Title-No. 5104. (xii+365+1 pages); Dynkin, Evgenii Borisovich (1965) [1963-03-10, 1962-03-31]. Written at University of Moscow, Moscow, Russia. . Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. II (122). Translated by Fabius, Jaap [at Wikidata]; Greenberg, Vida Lazarus [at Wikidata]; Maitra, Ashok Prasad [at Wikidata]; Majone, Giandomenico (1 ed.). New York, US / Berlin, Germany: Springer-Verlag. doi:. ISBN 978-3-662-23320-7. ISSN . LCCN . Title-No. 5105. (viii+274+2 pages) (NB. This was originally published in Russian as "Markovskie prot︠s︡essy" (Марковские процессы) by Fizmatgiz (Физматгиз) in 1963 and translated to English with the assistance of the author.)
• Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович]; Yushkevish [Юшкевич], Aleksandr Adol'fovich [Александр Адольфович] [in German] (1969) [1966-01-22]. Written at University of Moscow, Moscow, Russia. (PDF). Translated by Wood, James S. (1 ed.). New York, US: Plenum Press / Plenum Publishing Corporation. LCCN . (PDF) from the original on 2023-09-06. (x+237 pages) (NB. This is a corrected translation of the first Russian edition published as "Теоремы и задачи о процессах Маркова" by Nauka Press (Наука) in 1967 as part of a series on Probability Theory and Mathematical Statistics (Теория вероятностей и математическая статистика) with the assistance of the authors. It is based on lectures held at the Moscow State University in 1962/1963.)
• Marlo, Edward "Ed" (1976-12-01). Written at Chicago, Illinois, US. Hudson, Charles (ed.). "Approach & Uses for the "Kruskal Kount" / First Presentation Angle / Second Presentation Angle - Checking the Deck / Third Presentation Angle - The 100% Method / Fourth Presentation Angle - "Disaster"". Card Corner. The Linking Ring. Vol. 56, no. 12. Bluffton, Ohio, US: International Brotherhood of Magicians. pp. 82, 83, 83, 84, 85–87. ISSN .
• Hudson, Charles (1977-10-01). Written at Chicago, Illinois, US. "The Kruskal Principle". Card Corner. The Linking Ring. Vol. 57, no. 10. Bluffton, Ohio, US: International Brotherhood of Magicians. p. 85. ISSN .
• Gardner, Martin (September 1998). "Ten Amazing Mathematical Tricks". Gardner's Gatherings. Math Horizons. Vol. 6, no. 1. Mathematical Association of America / Taylor & Francis, Ltd. pp. 13–15, 26. ISSN . JSTOR . (4 pages)
• Haigh, John (1999). "7. Waiting, waiting, waiting: Packs of cards (2)". (1 ed.). Oxford, UK: Oxford University Press Inc. pp. 133–136. ISBN 978-0-19-850291-3. (4 pages); Haigh, John (2009) [2003]. . (Reprint of 2nd ed.). Oxford, UK: Oxford University Press Inc. pp. 139–142. ISBN 978-0-19-852663-6. (4 of xiv+373+17 pages)
• Bean, Gordon (2002). . In Wolfe, David; Rodgers, Tom (eds.). (1 ed.). CRC Press / Taylor & Francis Group, LLC. pp. 103–106. ISBN 978-1-43986410-4. (xvi+421 pages)
• Ching, Wai-Ki [at Wikidata]; Lee, Yiu-Fai (September 2005) [2004-05-05]. "A Random Walk on a Circular Path". Miscellany. International Journal of Mathematical Education in Science and Technology[d]. 36 (6). Taylor & Francis, Ltd.: 680–683. doi:. eISSN . ISSN . S2CID . (4 pages)
• Lee, Yiu-Fai; Ching, Wai-Ki [at Wikidata] (2006-03-07) [2005-09-29]. (PDF). Classroom notes. International Journal of Mathematical Education in Science and Technology[d]. 37 (7). Advanced Modeling and Applied Computing Laboratory and Department of Mathematics, The University of Hong Kong, Hong Kong: Taylor & Francis, Ltd.: 833–838. doi:. eISSN . ISSN . S2CID . (PDF) from the original on 2023-09-02. (6 pages)
• Humble, Steve "Dr. Maths" (July 2008). . The Montana Mathematics Enthusiast. 5 (2 & 3). Missoula, Montana, US: University of Montana: 327–336. doi:. ISSN . S2CID . Article 14. from the original on 2023-09-03. (1+10 pages)
• Montenegro, Ravi [at Wikidata]; Tetali, Prasad V. (2010-11-07) [2009-05-31]. (PDF). Proceedings of the forty-first annual ACM symposium on Theory of computing (STOC 2009). pp. 553–560. arXiv:. doi:. S2CID . (PDF) from the original on 2023-08-20.
• Grime, James [at Wikidata] (2011). (PDF). singingbanana.com. (PDF) from the original on 2023-08-19. (8 pages)
• Bosko, Lindsey R. (2011). Written at Department of Mathematics, North Carolina State University, Raleigh, North Carolina, US. (PDF). The UMAP Journal. Modules and Monographs in Undergraduate Mathematics and its Applications (UMAP) Project. 32 (3). Bedford, Massachusetts, US: Consortium For Mathematics & Its Applications, Inc. (COMAP): 199–236. UMAP Unit 808. (PDF) from the original on 2023-08-19.
• West, Bob [at Wikidata] (2011-05-26). . dlab @ EPFL. Lausanne, Switzerland: Data Science Lab, École Polytechnique Fédérale de Lausanne. from the original on 2022-05-23. [...] it turns out there is a card trick that works exactly the same way. It's called the "Kruskal Count" [...]
• Humble, Steve "Dr. Maths" (September 2012) [2012-07-02]. Written at Kraków, Poland. Behrends, Ehrhard [in German] (ed.). (PDF). EMS Newsletter. No. 85. Zürich, Switzerland: EMS Publishing House / European Mathematical Society. pp. 20–21 [21]. ISSN . (PDF) from the original on 2023-09-02. p. 21: [...] The Kruscal count [...] (2 pages)
• Andriesse, Dennis; Bos, Herbert [at Wikidata] (2014-07-10). Written at Vrije Universiteit Amsterdam, Amsterdam, Netherlands. Dietrich, Sven (ed.). (PDF). 11th International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment (DIMVA). Lecture Notes in Computer Science. Egham, UK; Switzerland: Springer International Publishing. pp. 41–50 [45]. doi:. eISSN . ISBN 978-3-31908508-1. ISSN . S2CID . LNCS 8550. (PDF) from the original on 2023-08-26. (10 pages)
• Montenegro, Ravi [at Wikidata]; Tetali, Prasad V. (2014-09-07). (PDF). (PDF) from the original on 2023-08-22. (18 pages)
• Kijima, Shuji; Montenegro, Ravi [at Wikidata] (2015-03-15) [2015-03-30/2015-04-01]. Written at Gaithersburg, Maryland, US. Katz, Jonathan (ed.). (PDF). Proceedings of the 18th IACR International Conference on Practice and Theory in Public-Key Cryptography. Lecture Notes in Computer Science. Berlin & Heidelberg, Germany: International Association for Cryptologic Research / Springer Science+Business Media. pp. 127–149. doi:. ISBN 978-3-662-46446-5. LNCS 9020. (PDF) from the original on 2023-09-03. (23 pages)
• Jose, Harish (2016-06-14) [2016-06-02]. . Lean. from the original on 2023-09-07.
• Lamprecht, Daniel; Dimitrov, Dimitar; Helic, Denis; Strohmaier, Markus (2016-08-17). "Evaluating and Improving Navigability of Wikipedia: A Comparative Study of Eight Language Editions". (PDF). OpenSym, Berlin, Germany: Association for Computing Machinery. pp. 1–10. doi:. ISBN 978-1-4503-4451-7. S2CID . (PDF) from the original on 2023-09-04.
• Jämthagen, Christopher (November 2016). (PDF) (Thesis). Lund, Sweden: Department of Electrical and Information Technology, Lund University. p. 96. ISBN 978-91-7623-942-1. ISSN . (PDF) from the original on 2023-08-26. (1+xvii+1+152 pages)
• Mannam, Pragna; Volkov, Jr., Alexander; Paolini, Robert; Chirikjian, Gregory Scott; Mason, Matthew Thomas (2019-02-06) [2018-12-04]. . Entropy. 21 (2). Basel, Switzerland: Multidisciplinary Digital Publishing Institute: 154. arXiv:. Bibcode:. doi:. ISSN . PMC . PMID . S2CID . Article 154. p. 2: [...] The phenomenon, while also reminiscent of contraction mapping, is similar to an interesting card trick called the Kruskal Count [...] so we have dubbed the phenomenon as "Kruskal effect". [...] (13 pages)
• Blackburn, Simon Robert; Esfahani, Navid Nasr; Kreher, Donald Lawson; Stinson, Douglas "Doug" Robert (2023-08-22) [2022-11-18]. "Constructions and bounds for codes with restricted overlaps". IEEE Transactions on Information Theory. arXiv:. (17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)

External links

• Humble, Steve "Dr. Maths" (2010). . YouTube (Video). Alchemist Cafe, Dublin, Ireland. [23:40]
• . Close-Up Magic. GeniiForum. 2015–2017. from the original on 2023-09-04.
• Behr, Denis, ed. (2023). . Conjuring Archive. from the original on 2023-09-10.