Kruskal count

"Kruskal's principle" redirects here. For Kruskal's method, see Kruskal's algorithm.

"Sum Total" redirects here. For the statistical quantity, see Total sum of squares.

"Dynkin's card trick" redirects here. For optimal stopping problems, see Dynkin games.

The Kruskal count^[1][2] (also known as Kruskal's principle,^{[3][4][5][6][7]} Dynkin–Kruskal count,^[8] Dynkin's counting trick,^[9] Dynkin's card trick,^{[10][11][12][13]} coupling card trick^[14][15][16] or shift coupling^{[10][11][12][13]}) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s^[when?] discussing coupling effects^{[14][15][9][16]} and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s^{[17][nb 1]} as a side-product while working on another problem.^[18] It was published by Kruskal's friend^[19] Martin Gardner^[20][1] and magician Karl Fulves in 1975.^[21] This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total^{[22][23][24][25]} and later called Kraus principle.^{[2][7][25][18]}

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

 

Explanation of Kruskal count

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.^{[26][27][28][29][6]} A simplified version using the hands of a clock is as follows.^[30] 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^[18]
• Geometric distribution^[18]
• Overlapping instructions^[27][28][31]
• Pollard's kangaroo algorithm^[4][5][6]
• Random walk
• Self-synchronizing code
• Wikipedia:Getting to Philosophy

Notes

1. According to Diaconis & Graham (2012), Martin Kruskal explained the trick, which later became known as Kruskal's principle, to Martin Gardner in a reply to a letter Gardner had sent him to recommend Persi W. Diaconis for graduate school. Diaconis graduated in 1971, earned a M.S. in mathematical statistics at Harvard University in 1972, and a Ph.D. from Harvard in 1974, so Kruskal's reply must have been between 1971 and 1974 the latest. Gardner published the trick in Gardner (1975).

References

1. Gardner, Martin (February 1978). "On checker jumping, the Amazon game, weird dice, card tricks and other playful pastimes". Scientific American. Mathematical Games. Vol. 238, no. 2. Scientific American, Inc. pp. 19–32. ISSN 0036-8733. JSTOR 24955629.
2. Gardner, Martin (1989) [1988]. "Chapter 19". Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix (1 ed.). W. H. Freeman. p. 274; Gardner, Martin (1997). "Chapter 19. Sicherman Dice, the Kruskal Count and Other Curiosities". Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix (PDF). Spectrum Series (Revised ed.). Washington DC, US: Mathematical Association of America. pp. 265–280 [280]. ISBN 0-88385-521-6. LCCN 97-70505. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (1+ix+319 pages)
3. Haga, Wayne; Robins, Sinai [at Wikidata] (June 1997) [1995-12-12]. "On Kruskal's Principle". Written at Simon Fraser University, Burnaby, British Columbia, Canada. In Borwein, Jonathan; Borwein, Peter; Jörgenson, Loki; Corless, Robert "Rob" M. (eds.). Organic Mathematics. Canadian Mathematical Society Conference Proceedings. Vol. 20. Providence, Rhode Island, US: American Mathematical Society. pp. 407–411. ISBN 978-0-8218-0668-5. ISSN 0731-1036. LCCN 97-179. ISBN 0-8218-0668-8. Retrieved 2023-08-19. (5 pages)
4. Pollard, John M. (July 1978) [1977-05-01, 1977-11-18]. "Monte Carlo Methods for Index Computation (mod p)" (PDF). Mathematics of Computation. 32 (143). Mathematics Department, Plessey Telecommunications Research, Taplow Court, Maidenhead, Berkshire, UK: American Mathematical Society: 918–924. ISSN 0025-5718. (PDF) from the original on 2013-05-03. Retrieved 2023-08-19. (7 pages)
5. Pollard, John M. (2000-08-10) [1998-01-23, 1999-09-27]. "Kangaroos, Monopoly and Discrete Logarithms" (PDF). Journal of Cryptology. 13 (4). Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK: International Association for Cryptologic Research: 437–447. doi:10.1007/s001450010010. ISSN 0933-2790. S2CID 5279098. (PDF) from the original on 2023-08-18. Retrieved 2023-08-19. (11 pages)
6. Pollard, John M. (July 2000). "Kruskal's Card Trick" (PDF). The Mathematical Gazette. 84 (500). Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK: The Mathematical Association: 265–267. doi:10.2307/3621657. ISSN 0025-5572. JSTOR 3621657. S2CID 125115379. 84.29. (PDF) from the original on 2023-08-18. Retrieved 2023-08-19. (1+3 pages)
7. MacTier, Arthur F. (2000). "Chapter 6: Kruskal Principle (Extraordinary Coincidence) / Chapter 7: Kraus Principle (The Magic of 52, Magical Coincidence II)". Card Concepts - An Anthology of Numerical & Sequential Principles Within Card Magic (1 ed.). London, UK: Lewis Davenport Limited. pp. 34–38, 39–46. (vi+301 pages)
8. Artymowicz, Pawel [in Polish] (2020-01-29) [2020-01-26]. "Codes for PHYD57 Advanced Computing in Physics, UTSC: Dynkin–Kruskal count - convergent Markov chains". from the original on 2023-08-20. Retrieved 2023-08-20. [...] We looked at the Markov chains, where a given random sequence of cards or numbers is traversed in a linked-list manner, that is when you see a value in a list of integers, you use it to determine the position of the next number in a sequence, and you repeat that until the list ends. This is the basis of a card trick with a magician correctly guessing the final number in a seemingly hidden/random sequence computed by a spectator in his/her mind (but using a given well-shuffled deck of 52 cards. [...] Random sequences that converge when the length of element is used to create a jump to the next element are called Dynkin–Kruskal sequences, after Eugene Dynkin (1924–2014), a Russian-American mathematician, who mentioned them in his work, and American mathematician Martin David Kruskal (1925–2006). The nature of these Kruskal sequences is that they converge exponentially fast, and for N=52 there is already more than 90% chance that the two randomly started sequences converge at the end of the deck, that is the magician and the spectator independently arrive at the same last key card. I saw the trick demonstrated [...] at one conference, but didn't know that these convergent, linked list-like, series are so common. Almost any books can be used to show that. Skip a number of words equal to the number of letters in a key word. By the end of the third line you normally converge to the same sequence forever after, no matter which word in the top line you start with. [...]
9. Jiang, Jiming [at Wikidata] (2010). "Chapter 10 Stochastic Processes; 10.1 Introduction". Written at University of California, Davis, California, US. Large Sample Techniques for Statistics. Springer Texts in Statistics (1 ed.). New York, US: Springer Science+Business Media, LLC. pp. 317–319. doi:10.1007/978-1-4419-6827-2. ISBN 978-1-4419-6826-5. ISSN 1431-875X. LCCN 2010930134. S2CID 118271573. Retrieved 2023-09-02. (xvii+610 pages); Jiang, Jiming [at Wikidata] (2022) [2010]. "Chapter 10 Stochastic Processes; 10.1 Introduction". Written at University of California, Davis, California, US. Large Sample Techniques for Statistics. Springer Texts in Statistics (2 ed.). Cham, Switzerland: Springer Nature Switzerland AG. pp. 339–341. doi:10.1007/978-3-030-91695-4. eISSN 2197-4136. ISBN 978-3-030-91694-7. ISSN 1431-875X. Retrieved 2023-09-02. p. 339: [...] During the author's time as a graduate student, one of the classroom examples that struck him the most was given by Professor David Aldous in his lectures on Probability Theory. The example was taken from Durrett (1991, p. 275). A modified (and expanded) version is given below. [...] Example 10.1. Professor E. B. Dynkin used to entertain the students in his probability class with the following counting trick. A professor asks a student to write 100 random digits from 0 to 9 on the blackboard. Table 10.1 shows 100 such digits generated by a computer. The professor then asks another student to choose one of the first 10 digits without telling him. Here, we use the computer to generate a random number from 1 to 10. The generated number is 7, and the 7th number of the first 10 digits in the table is also 7. Suppose that this is the number that the second student picks. She then counts 7 places along the list, starting from the number next to 7. The count stops at (another) 7. She then counts 7 places along the list, again. This time the count stops at 3. She then counts 3 places along the list, and so on. In the case that the count stops at 0, the student then counts 10 places on the list. The student's counts are underlined in Table 10.1. The trick is that these are all secretly done behind the professor, who then turns around and points out where the student's counts finally ends, which is the last 9 in the table. [...] (xv+685 pages)
10. Barthe, Gilles [at Wikidata] (2016). "Probabilistic couplings for cryptography and privacy" (PDF). Madrid, Spain: IMDEA Software Institute. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (66 pages); Barthe, Gilles [at Wikidata] (2016-09-13). "Probabilistic couplings for cryptography and privacy" (PDF). Madrid, Spain: IMDEA Software Institute. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (49 pages)
11. Barthe, Gilles [at Wikidata]; Grégoire, Benjamin [at Wikidata]; Hsu, Justin; Strub, Pierre-Yves (2016-11-07) [2016-09-21]. "Coupling Proofs are Probabilistic Product Programs". Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages. pp. 161–174. arXiv:1607.03455v5. doi:10.1145/3009837.3009896. ISBN 978-1-45034660-3. S2CID 3931131. from the original on 2023-08-19. Retrieved 2023-08-19. (14 pages)
12. Barthe, Gilles [at Wikidata]; Espitau, Thomas; Grégoire, Benjamin [at Wikidata]; Hsu, Justin; Stefanesco, Léo; Strub, Pierre-Yves (2017-07-12) [2015]. "Relational Reasoning via Probabilistic Coupling". Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science. Vol. 9450. Suva, France: LPAR. pp. 387–401. arXiv:1509.03476. doi:10.1007/978-3-662-48899-7_27. ISBN 978-3-662-48898-0. S2CID 3518579. hal-01246719v2. from the original on 2023-08-19. Retrieved 2023-08-19. (17 pages)
13. Hsu, Justin (2018) [2017-11-01]. "Probabilistic Couplings for Probabilistic Reasoning" (PDF) (Thesis). p. 34. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (147 pages)
14. Durrett, Richard "Rick" Timothy (1991) [1989]. Probability: Theory and Examples. The Wadsworth & Brooks/Cole Statistics/Probability Series (1 ed.). Pacific Grove, California, US: Wadsworth & Brooks/Cole Advanced Books & Software. p. 275. ISBN 0-534-13206-5. MR 1068527. (x+453 pages) (NB. This can be found quoted in Jiang (2010).); Durrett, Richard "Rick" Timothy (2005). "Example 5.2. A coupling card trick.". Probability: Theory and Examples. The Duxbury Advanced Series in Statistics and Decision Sciences (3 ed.). Thomson Brooks/Cole Publishing. p. 312. ISBN 0-534-42441-4. ISBN 978-0-534-42441-1. (497 pages) (NB. This can be found quoted in Kovchegov (2007).)
15. Kovchegov, Yevgeniy V. [at Wikidata] (2007-10-06). "From Markov Chains to Gibbs Fields" (PDF). Corvallis, Oregon, US: Department of Mathematics, Oregon State University. p. 22. (PDF) from the original on 2023-09-01. Retrieved 2023-09-01. p. 22: Here we will quote [R. Durrett, "Probability: Theory and Examples."]: "Example. A coupling card trick. The following demonstration used by E. B. Dynkin in his probability class is a variation of a card trick that appeared in Scientific American. The instructor asks a student to write 100 random digits from 0 to 9 on the blackboard. Another student chooses one of the first 10 numbers and does not tell the instructor. If that digit is 7 say she counts 7 places along the list, notes the digit at that location, and continues the process. If the digit is 0 she counts 10. A possible sequence is underlined on the list below: 3 4 7 8 2 3 7 5 6 1 6 4 6 5 7 8 3 1 5 3 0 7 9 2 3 . . . The trick is that, without knowing the student's first digit, the instructor can point to her final stopping position. To this end, he picks the first digit, and forms his own sequence in the same manner as the student and announces his stopping position. He makes an error if the coupling time is larger than 100. Numerical computation done by one of Dynkin's graduate students show that the probability of error is approximately [0].026. (45 pages) (NB. This can be found quoted in Weinhold (2011).)
16. Weinhold, Leonie (2011-05-13). "Vorstellung der Kopplung bei Markovketten" (PDF) (in German). Ulm, Germany: University of Ulm. p. 7. (PDF) from the original on 2023-09-01. Retrieved 2023-09-01. (1+9 pages) (NB. This work quotes Kovchegov (2007).)
17. Diaconis, Persi Warren; Graham, Ronald "Ron" Lewis (2016) [2012]. "Chapter 10. Stars Of Mathematical Magic (And Some Of The Best Tricks In The Book): Martin Gardner". Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks (4th printing of 1st ed.). Princeton, New Jersey, US & Woodstock, Oxfordshire, UK: Princeton University Press. pp. 211–219 [211–212]. ISBN 978-0-691-16977-4. LCCN 2011014755. ISBN 978-0-691-15164-9. Retrieved 2023-09-06. pp. 211–212: [...] A blurb that appears on one of his books says: [...] Warning: Martin Gardner has turned dozens of innocent youngsters into math professors and thousands of math professors into innocent youngsters. [...] We are living proof; Martin nurtured a runaway fourteen-year-old, published some of our mathematical findings to give a first publication (in Scientific American), found time to occasionally help with homework, and, when the time came to apply for graduate school, Martin was one of our letter writers. There are heart-warming stories here. Martin's letter of recommendation said something like: "I don't know a lot about mathematics but this kid invented two of the best card tricks of the past ten years. You ought to give him a chance." Fred Mosteller, a Harvard statistics professor and keen amateur magician, was on the admissions committee and let the kid into Harvard. Fred became the kid's thesis advisor and, after graduation, the kid eventually returned to Harvard as a professor. [...] One other tale about Martin's letter. It was sent to a long list of graduate schools. He got a reply from Martin Kruskal at Princeton (a major mathematician who was most well-known for his discovery of solitons) that went roughly: "It's true, Martin. You don't know about mathematics. No one with this kid's limited background could ever make it through a serious math department." Kruskal went on to explain what has come to be known as the Kruskal principle. This is a broadly useful new principle in card magic. A few years later, the kid lectured at the Institute for Defense Analyses, a kind of cryptography think tank in Princeton. Kruskal came up afterwards, full of enthusiasm for the lecture, and asked: "How come I never heard of you? That was wonderful!" The kid tried to remind Kruskal of their history. Kruskal denied it but the kid still has the letter. This was one of the few times that Martin Kruskal's keen insight led him astray! [...] (2+xii+2+244+4 pages)
18. Nishiyama, Yutaka (July 2013) [2012-12-10]. "The Kruskal principle" (PDF). International Journal of Pure and Applied Mathematics [d]. 85 (6). Department of Business Information, Faculty of Information Management, Osaka University of Economics, Osaka, Japan: Academic Publications, Ltd.: 983–992. doi:10.12732/ijpam.v85i6.1. eISSN 1314-3395. ISSN 1311-8080. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (10 pages)
19. Farrell, Jeremiah (2010). "Foshee Magically Interpreted". Indianapolis, Indiana, US. p. 316. from the original on 2023-08-19. Retrieved 2023-08-19. p. 316: Kruskal had two mathematically inclined brothers, William at the University of Chicago and Joseph of Bell Labs. All three were friends of Martin Gardner who had earlier written about their mother, Lillian Oppenheimer, a remarkable origamist. (1 page)
20. Gardner, Martin (June 1975). "The Kruskal Principle". The Pallbearers Review. Vol. 10, no. 8. Teaneck, New Jersey, US: L & L Publishing. pp. 967–970 (4 pages); Fulves, Karl, ed. (July 1975). "Cross-Cut Force". The Pallbearers Review. Vol. 10, no. 9. Teaneck, New Jersey, US: L & L Publishing. p. 985 (1 page); Gardner, Martin (1993) [June 1975]. "The Kruskal Principle". In Fulves, Karl (ed.). The Pallbearers Review: Volumes 9–10. Vol. 3. Tahoma, California, US: L & L Publishing - Quality Magical Literature. pp. 967–970, 985. from the original on 2023-09-10. Retrieved 2023-09-10 (381 pages) (NB. Volume 3 of a three-volume hardcover reprint of The Pallbearers Review magazine volumes 9 (November 1973) – 10 (1977).); Braunmüller, Rudolf, ed. (January 1984). "Das Kruskal-Prinzip" [The Kruskal Principle]. intermagic - Ein Magisches Journal (in German). Vol. 10, no. 3 & 4. Munich, Germany. pp. 125–.
21. Fulves, Karl (June 1975). "Kruskal Phone Effect". The Pallbearers Review. Vol. 10, no. 8. Teaneck, New Jersey, US: L & L Publishing. pp. 970–; Fulves, Karl (1993) [June 1975]. "Kruskal Phone Effect". The Pallbearers Review: Volumes 9–10. Vol. 3. Tahoma, California, US: L & L Publishing - Quality Magical Literature. pp. 970–. from the original on 2023-09-10. Retrieved 2023-09-10. (381 pages) (NB. Volume 3 of a three-volume hardcover reprint of The Pallbearers Review magazine volumes 9 (November 1973) – 10 (1977).)
22. Kraus, Alexander F. (December 1957). Lyons, Philip Howard (ed.). "Sum Total". ibidem. No. 12. Toronto, Ontario, Canada. p. 7. Part 1 (Problem). (1 page) (NB. The second part can be found in Kraus (1958).); Kraus, Alexander F. (1993). "Sum Total (Problem)". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.). ibidem - P. Howard Lyons. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US: Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.). p. 232. (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
23. Kraus, Alexander F. (March 1958). Lyons, Philip Howard (ed.). "Sum Total". ibidem. No. 13. Toronto, Ontario, Canada. pp. 13–16. Part 2 (Solution). (4 pages) (NB. The first part can be found in Kraus (1957).); Kraus, Alexander F. (1993). "Sum Total (Solution)". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.). ibidem - P. Howard Lyons. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US: Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.). pp. 255–258. (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
24. Ransom, Tom; Katz, Max (March 1958). Lyons, Philip Howard (ed.). "Sum More". ibidem. No. 13. Toronto, Ontario, Canada. pp. 17–18. (2 pages); Ransom, Tom; Katz, Max (1993). "Sum More". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.). ibidem - P. Howard Lyons. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US: Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.). pp. 258–259. (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
25. Havil, Julian R. [in German] (2008). "Chapter 12: Two Card Tricks". Impossible? Surprising Solutions to Counterintuitive Conundrums (1 ed.). Princeton, New Jersey, US: Princeton University Press. pp. 131–140. ISBN 978-0-691-13131-3. JSTOR j.ctt7rnph. LCCN 2007051792. Retrieved 2023-08-19. [7] (xii+235 pages) (NB. The book contains a significant number of typographical errors: ASIN 0691150028); Havil, Julian R. [in German] (2009) [2008]. "Kapitel 12 - Numerologie und Kartentricks: Das Kruskal-Prinzip". Das gibts doch nicht – Mathematische Rätsel [Impossible? Surprising Solutions to Counterintuitive Conundrums] (in German). Translated by Zillgitt, Michael (1 ed.). Spektrum Akademischer Verlag Heidelberg / Springer Science+Business Media. pp. 128–135. ISBN 978-3-8274-2306-1. ISBN 978-3-8274-2306-1. (xiv+234 pages)
26. Lagarias, Jeffrey "Jeff" Clark; Vanderbei, Robert J. (1988). The Kruskal Count. Murray Hill, New Jersey, US: AT&T Bell Laboratories.
27. Lagarias, Jeffrey "Jeff" Clark; Rains, Eric Michael; Vanderbei, Robert J. (2009) [2001-10-13]. "The Kruskal Count". In Brams, Stephen; Gehrlein, William V.; Roberts, Fred S. (eds.). The Mathematics of Preference, Choice and Order. Essays in Honor of Peter J. Fishburn. Studies in Choice and Welfare. Berlin / Heidelberg, Germany: Springer-Verlag. pp. 371–391. arXiv:math/0110143. doi:10.1007/978-3-540-79128-7_23. ISBN 978-3-540-79127-0. S2CID 18273053. (22 pages)
28. Jacob, Matthias; Jakubowski, Mariusz H.; Venkatesan, Ramarathnam [at Wikidata] (20–21 September 2007). Towards Integral Binary Execution: Implementing Oblivious Hashing Using Overlapped Instruction Encodings (PDF). Proceedings of the 9th workshop on Multimedia & Security (MM&Sec '07). Dallas, Texas, US: Association for Computing Machinery. pp. 129–140. CiteSeerX 10.1.1.69.5258. doi:10.1145/1288869.1288887. ISBN 978-1-59593-857-2. S2CID 14174680. (PDF) from the original on 2018-09-04. Retrieved 2021-12-25. (12 pages)
29. Paulos, John Allen (November 1998). . Once Upon a Number - The Hidden Mathematical Logic of Stories (1 ed.). Basic Books. p. 64. ISBN 978-0-46505159-5. Archived from the original on 2015-04-01.
30. Delbert, Caroline (2020-02-27). "How to Do the Math Magic Trick That Will Impress Everyone You Know - Here's the secret". Science. Popular Mechanics. Hearst Magazine Media, Inc. ISSN 0032-4558. from the original on 2021-10-19. Retrieved 2021-12-25.
31. Jakubowski, Mariusz H. (February 2016). "Graph Based Model for Software Tamper Protection". Microsoft. from the original on 2019-10-31. Retrieved 2023-08-19.

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.). Random Walks (Mathematical Conversations Part 3). 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 63-19838. Retrieved 2023-09-03. (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. Markov Processes-I. 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:10.1007/978-3-662-00031-1. ISBN 978-3-662-00033-5. ISSN 0072-7830. LCCN 64-24812. S2CID 251691119. Title-No. 5104. Retrieved 2023-09-02. [10] (xii+365+1 pages); Dynkin, Evgenii Borisovich (1965) [1963-03-10, 1962-03-31]. Written at University of Moscow, Moscow, Russia. Markov Processes-II. 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:10.1007/978-3-662-25360-1. ISBN 978-3-662-23320-7. ISSN 0072-7830. LCCN 64-24812. Title-No. 5105. Retrieved 2023-09-02. (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. Markov Processes: Theorems and Problems (PDF). Translated by Wood, James S. (1 ed.). New York, US: Plenum Press / Plenum Publishing Corporation. LCCN 69-12529. (PDF) from the original on 2023-09-06. Retrieved 2023-09-03. (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 0024-4023.
• 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 0024-4023.
• 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 1072-4117. JSTOR 25678174. (4 pages)
• Haigh, John (1999). "7. Waiting, waiting, waiting: Packs of cards (2)". Taking Chances: Winning with Probability (1 ed.). Oxford, UK: Oxford University Press Inc. pp. 133–136. ISBN 978-0-19-850291-3. Retrieved 2023-09-06. (4 pages); Haigh, John (2009) [2003]. "7. Waiting, waiting, waiting: Packs of cards (2)". Taking Chances: Winning with Probability (Reprint of 2nd ed.). Oxford, UK: Oxford University Press Inc. pp. 139–142. ISBN 978-0-19-852663-6. Retrieved 2023-09-03. (4 of xiv+373+17 pages)
• Bean, Gordon (2002). "A Labyrinth in a Labyrinth". In Wolfe, David; Rodgers, Tom (eds.). Puzzlers' Tribute: A Feast for the Mind (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:10.1080/00207390500064254. eISSN 1464-5211. ISSN 0020-739X. S2CID 121692834. (4 pages)
• Lee, Yiu-Fai; Ching, Wai-Ki [at Wikidata] (2006-03-07) [2005-09-29]. "On Convergent Probability of a Random Walk" (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:10.1080/00207390600712299. eISSN 1464-5211. ISSN 0020-739X. S2CID 121242696. (PDF) from the original on 2023-09-02. Retrieved 2023-09-02. (6 pages)
• Humble, Steve "Dr. Maths" (July 2008). "Magic Card Maths". The Montana Mathematics Enthusiast. 5 (2 & 3). Missoula, Montana, US: University of Montana: 327–336. doi:10.54870/1551-3440.1111. ISSN 1551-3440. S2CID 117632058. Article 14. from the original on 2023-09-03. Retrieved 2023-09-02. (1+10 pages)
• Montenegro, Ravi [at Wikidata]; Tetali, Prasad V. (2010-11-07) [2009-05-31]. How Long Does it Take to Catch a Wild Kangaroo? (PDF). Proceedings of the forty-first annual ACM symposium on Theory of computing (STOC 2009). pp. 553–560. arXiv:0812.0789. doi:10.1145/1536414.1536490. S2CID 12797847. (PDF) from the original on 2023-08-20. Retrieved 2023-08-20.
• Grime, James [at Wikidata] (2011). "Kruskal's Count" (PDF). singingbanana.com. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19. (8 pages)
• Bosko, Lindsey R. (2011). Written at Department of Mathematics, North Carolina State University, Raleigh, North Carolina, US. "Cards, Codes, and Kangaroos" (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. Retrieved 2023-08-19.
• West, Bob [at Wikidata] (2011-05-26). "Wikipedia's fixed point". dlab @ EPFL. Lausanne, Switzerland: Data Science Lab, École Polytechnique Fédérale de Lausanne. from the original on 2022-05-23. Retrieved 2023-09-04. [...] 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.). "Mathematics in the Streets of Kraków" (PDF). EMS Newsletter. No. 85. Zürich, Switzerland: EMS Publishing House / European Mathematical Society. pp. 20–21 [21]. ISSN 1027-488X. (PDF) from the original on 2023-09-02. Retrieved 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.). Instruction-Level Steganography for Covert Trigger-Based Malware (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:10.1007/978-3-319-08509-8_3. eISSN 1611-3349. ISBN 978-3-31908508-1. ISSN 0302-9743. S2CID 4634611. LNCS 8550. (PDF) from the original on 2023-08-26. Retrieved 2023-08-26. (10 pages)
• Montenegro, Ravi [at Wikidata]; Tetali, Prasad V. (2014-09-07). Kruskal's Principle and Collision Time for Monotone Transitive Walks on the Integers (PDF). (PDF) from the original on 2023-08-22. Retrieved 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.). Collision of Random Walks and a Refined Analysis of Attacks on the Discrete Logarithm Problem (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:10.1007/978-3-662-46447-2_6. ISBN 978-3-662-46446-5. LNCS 9020. (PDF) from the original on 2023-09-03. Retrieved 2023-09-03. (23 pages)
• Jose, Harish (2016-06-14) [2016-06-02]. "PDCA and the Roads to Rome: Can a lean purist and a Six Sigma purist reach the same answer to a problem?". Lean. from the original on 2023-09-07. Retrieved 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". Proceedings of the 12th International Symposium on Open Collaboration (PDF). OpenSym, Berlin, Germany: Association for Computing Machinery. pp. 1–10. doi:10.1145/2957792.2957813. ISBN 978-1-4503-4451-7. S2CID 13244770. (PDF) from the original on 2023-09-04. Retrieved 2021-03-17.
• Jämthagen, Christopher (November 2016). On Offensive and Defensive Methods in Software Security (PDF) (Thesis). Lund, Sweden: Department of Electrical and Information Technology, Lund University. p. 96. ISBN 978-91-7623-942-1. ISSN 1654-790X. (PDF) from the original on 2023-08-26. Retrieved 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]. "Sensorless Pose Determination Using Randomized Action Sequences". Entropy. 21 (2). Basel, Switzerland: Multidisciplinary Digital Publishing Institute: 154. arXiv:1812.01195. Bibcode:2019Entrp..21..154M. doi:10.3390/e21020154. ISSN 1099-4300. PMC 7514636. PMID 33266870. S2CID 54444590. 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:2211.10309. (17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)

External links

• Humble, Steve "Dr. Maths" (2010). "Dr. Maths Randomness Show". YouTube (Video). Alchemist Cafe, Dublin, Ireland. Retrieved 2023-09-05. [23:40]
• "Mathematical Card Trick Source". Close-Up Magic. GeniiForum. 2015–2017. from the original on 2023-09-04. Retrieved 2023-09-05.
• Behr, Denis, ed. (2023). "Kruskal Principle". Conjuring Archive. from the original on 2023-09-10. Retrieved 2023-09-10.