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.

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

Explanation of Kruskal count

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, automatic object realignment, 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.^{[23][24][25][26][6]} A simplified version using the hands of a clock is as follows.^[27] 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
• Ergodic theory^[16]
• Geometric distribution^[16]
• Overlapping instructions^[24][25][28]
• Pollard's kangaroo algorithm^[4][5][6]
• Self-synchronizing code

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. 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, D.C., USA: 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. 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. Mathematics Department, Plessey Telecommunications Research, Taplow Court, Maidenhead, Berkshire, UK: American Mathematical Society. 32 (143): 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. Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK: International Association for Cryptologic Research. 13 (4): 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. Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK: The Mathematical Association. 84 (500): 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. 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)
10. 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)
11. 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)
12. 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)
13. Durrett, Richard "Rick" Timothy (2005) [1989]. "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 [d]. p. 312. ISBN 0-534-42441-4. ISBN 978-0-534-42441-1. (497 pages)
14. Kovchegov, Yevgeniy V. [at Wikidata] (2007-10-06). "From Markov Chains to Gibbs Fields" (PDF). Corvallis, Oregon, USA: 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 .026. (45 pages)
15. 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)
16. Nishiyama, Yutaka (July 2013) [2012-12-10]. "The Kruskal principle" (PDF). International Journal of Pure and Applied Mathematics [d]. Department of Business Information, Faculty of Information Management, Osaka University of Economics, Osaka, Japan: Academic Publications, Ltd. 85 (6): 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)
17. Farrell, Jeremiah (2010). "Foshee Magically Interpreted". Indianapolis, Indiana, USA. 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)
18. Gardner, Martin (June 1975). "The Kruskal Principle". The Pallbearers Review. Vol. 10, no. 8. L&L Publishing. pp. 967–, 985; Braunmüller, Rudolf, ed. (January 1984). "Das Kruskal-Prinzip" [The Kruskal Principle]. intermagic - Ein Magisches Journal (in German). Vol. 10, no. 3 & 4. Munich, Germany. p. 125.
19. Fulves, Karl (June 1975). "Kruskal Phone Effect". The Pallbearers Review. Vol. 10, no. 8. L&L Publishing. pp. 970–.
20. Kraus, Alexander F. (December 1957). Lyons, P. Howard (ed.). "Sum Total". ibidem. No. 12. Toronto, Ontario, Canada. p. 7 (232). Part 1 (Problem).
21. Kraus, Alexander F. (March 1958). Lyons, P. Howard (ed.). "Sum Total". ibidem. No. 13. Toronto, Ontario, Canada. pp. 13–16 (255–258). Part 2 (Solution).
22. Ransom, Tom; Katz, Max (March 1958). Lyons, P. Howard (ed.). "Sum More". ibidem. No. 13. Toronto, Ontario, Canada. pp. 17–18 (258–259).
23. Lagarias, Jeffrey "Jeff" Clark; Vanderbei, Robert J. (1988). The Kruskal Count. Murray Hill, New Jersey, USA: AT&T Bell Laboratories.
24. 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)
25. 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, USA: 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)
26. 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.
27. 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.
28. 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

• Marlo, Edward "Ed" (1976-12-01). Written at Chicago, Illinois, USA. 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, USA: International Brotherhood of Magicians. pp. 82, 83, 83, 84, 85–87. ISSN 0024-4023.
• Hudson, Charles (1977-10-01). Written at Chicago, Illinois, USA. "The Kruskal Principle". Card Corner. The Linking Ring. Vol. 57, no. 10. Bluffton, Ohio, USA: 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)
• Havil, Julian R. [in German] (2008). Impossible? Surprising Solutions to Counterintuitive Conundrums (1 ed.). Princeton, New Jersey, USA: Princeton University Press. ISBN 978-0-691-13131-3. JSTOR j.ctt7rnph. LCCN 2007051792. Retrieved 2023-08-19. (xii+235 pages) (NB. The English edition contains a significant number of typographical errors.); Havil, Julian R. [in German] (2009). Das gibts doch nicht – mathematische Rätsel [Impossible? Surprising Solutions to Counterintuitive Conundrums] (in German) (1 ed.). Spektrum Akademischer Verlag. ISBN 3-8274-2306-6. ISBN 978-3-8274-2306-1.
• 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, USA. "Cards, Codes, and Kangaroos" (PDF). The UMAP Journal. Modules and Monographs in Undergraduate Mathematics and its Applications (UMAP) Project. Bedford, Massachusetts, USA: Consortium For Mathematics & Its Applications, Inc. (COMAP). 32 (3): 199–236. UMAP Unit 808. (PDF) from the original on 2023-08-19. Retrieved 2023-08-19.
• 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)
• 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" (PDF). Entropy. Basel, Switzerland: Multidisciplinary Digital Publishing Institute. 21 (2). arXiv:1812.01195. doi:10.3390/e21020154. ISSN 1099-4300. PMC 7514636. PMID 33266870. S2CID 54444590. Article 154. (PDF) from the original on 2023-09-01. Retrieved 2023-09-01. 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" (PDF). IEEE Transactions on Information Theory. arXiv:2211.10309. (PDF) from the original on 2023-08-30. Retrieved 2023-08-30. (17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)