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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 performed by David Copperfield is as follows.[30][31] 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.
^ abMacTier, 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)
^Artymowicz, Pawel[in Polish] (2020-01-29) [2020-01-26]. "Codes for PHYD57 Advanced Computing in Physics, UTSC: Dynkin–Kruskal count - convergent Markov chains". Archived 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. [...]{{cite web}}: CS1 maint: bot: original URL status unknown (link)[1][2]
^ abJiang, 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. ISBN978-1-4419-6826-5. ISSN1431-875X. LCCN2010930134. S2CID118271573. 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. eISSN2197-4136. ISBN978-3-030-91694-7. ISSN1431-875X. 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)
^ abKovchegov, 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. Archived(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 inScientific 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 16 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).)
^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]. ISBN978-0-691-16977-4. LCCN2011014755. ISBN978-0-691-15164-9. 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 cryptographythink 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)
^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. Archived from the original on 2023-09-10. Retrieved 2023-09-10[4] (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–.
^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).)
^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).)
^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).)
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. ISSN0024-4023.
Grime, James[at Wikidata] (2011). "Kruskal's Count"(PDF). singingbanana.com. Archived(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. Archived(PDF) from the original on 2023-08-19. Retrieved 2023-08-19.
