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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.
807:(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.
163:, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock 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.
145:
1412:. 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
1435:
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
1617:
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
980:
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
2828:(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.
1602:) 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
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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
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155:
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
3672:
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Lamprecht, Daniel; Dimitrov, Dimitar; Helic, Denis; Strohmaier, Markus (2016-08-17). "Evaluating and
Improving Navigability of Knowledge (XXG): A Comparative Study of Eight Language Editions".
1590:
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
1468:
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546:
1570:, 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
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371:
2495:. Survey of Recent East European Mathematical Literature. Vol. 3. Translated by Whaland, Jr., Norman D.; Titelbaum, Olga A. (1 ed.). Boston, Massachusetts, US:
3369:
1896:
1892:
1831:
1827:
1723:
2660:. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. II (122). Translated by
2340:
2570:. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. I (121). Translated by
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3096:
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2520:
1665:(6). Department of Business Information, Faculty of Information Management, Osaka University of Economics, Osaka, Japan: Academic Publications, Ltd.: 983–992.
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has turned dozens of innocent youngsters into math professors and thousands of math professors into innocent youngsters. We are living proof; Martin nurtured
3666:
738:
MacTier, Arthur F. (2000). "Chapter 6: Kruskal
Principle (Extraordinary Coincidence) / Chapter 7: Kraus Principle (The Magic of 52, Magical Coincidence II)".
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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
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statistics professor and keen amateur magician, was on the admissions committee and let the kid into
Harvard. Fred became the kid's
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Bosko, Lindsey R. (2011). Written at
Department of Mathematics, North Carolina State University, Raleigh, North Carolina, US.
3111:(7). Advanced Modeling and Applied Computing Laboratory and Department of Mathematics, The University of Hong Kong, Hong Kong:
796:
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from
Harvard in 1974, so Kruskal's reply must have been between 1971 and 1974 the latest. Gardner published the trick in
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1209:
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2984:
2416:
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1579:
1158:; Hsu, Justin; Stefanesco, Léo; Strub, Pierre-Yves (2017-07-12) . "Relational Reasoning via Probabilistic Coupling".
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when the length of element is used to create a jump to the next element are called Dynkin–Kruskal sequences, after
521:
113:
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2199:
1396:
977:
310:
117:
70:
3781:
1040:
1008:
665:
197:
2033:
1987:
1937:
1277:. The Wadsworth & Brooks/Cole Statistics/Probability Series (1 ed.). Pacific Grove, California, US:
3240:. Proceedings of the forty-first annual ACM symposium on Theory of computing (STOC 2009). pp. 553–560.
2818:
2552:
2500:
1388:
1368:
1307:
1270:
969:
427:
303:(February 1978). "On checker jumping, the Amazon game, weird dice, card tricks and other playful pastimes".
261:
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85:
in the early 1970s as a side-product while working on another problem. It was published by
Kruskal's friend
3730:
3597:. Proceedings of the 18th IACR International Conference on Practice and Theory in Public-Key Cryptography.
1986:
Kraus, Alexander F. (1993). "Sum Total (Solution)". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.).
4011:
3886:
3649:"PDCA and the Roads to Rome: Can a lean purist and a Six Sigma purist reach the same answer to a problem?"
2304:
2216:
2187:
1936:
Kraus, Alexander F. (1993). "Sum Total (Problem)". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.).
1870:
1801:
1784:
803:(1924–2014), a Russian-American mathematician, who mentioned them in his work, and American mathematician
192:
172:
74:
17:
4001:
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Towards
Integral Binary Execution: Implementing Oblivious Hashing Using Overlapped Instruction Encodings
2043:
1997:
1947:
1743:
1739:
1571:
1557:
1497:
657:
586:
249:
2851:
Hudson, Charles (1977-10-01). Written at
Chicago, Illinois, US. "The Kruskal Principle". Card Corner.
759:"Codes for PHYD57 Advanced Computing in Physics, UTSC: Dynkin–Kruskal count - convergent Markov chains"
3461:(2014-07-10). Written at Vrije Universiteit Amsterdam, Amsterdam, Netherlands. Dietrich, Sven (ed.).
3088:
3817:
3785:
3417:
2988:
2591:
2525:Математические беседы: Задачи о многоцветной раскраске / Задачи из теории чисел / Случайные блуждания
2047:
2001:
1951:
1735:
1595:
1591:
804:
241:
125:
82:
24:
3232:
2489:(1963). Written at University of Moscow, Moscow, Russia. Putnam, Alfred L.; Wirszup, Izaak (eds.).
2299:. Proceedings of the 9th workshop on Multimedia & Security (MM&Sec '07). Dallas, Texas, US:
446:. Canadian Mathematical Society Conference Proceedings. Vol. 20. Providence, Rhode Island, US:
3871:
2673:
2583:
2309:
2224:
2191:
1875:
1806:
1789:
1759:
1562:
1506:"Chapter 10. Stars Of Mathematical Magic (And Some Of The Best Tricks In The Book): Martin Gardner"
1501:
1408:
305:
2032:
Ransom, Tom; Katz, Max (1993). "Sum More". In Ransom, Tom; Field, Matthew; Phillips, Mark (eds.).
578:
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702:
694:
615:
322:
265:
177:
3404:
2227:(2009) . "The Kruskal Count". In Brams, Stephen; Gehrlein, William V.; Roberts, Fred S. (eds.).
1850:
Braunmüller, Rudolf, ed. (January 1984). "Das Kruskal-Prinzip" [The Kruskal Principle].
93:
in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as
3320:
2551:). It is based on seminars held at the School Mathematics Circle in 1945/1946 and 1946/1947 at
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2924:
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2839:
2791:(x+237 pages) (NB. This is a corrected translation of the first Russian edition published as "
2768:
2723:(viii+274+2 pages) (NB. This was originally published in Russian as "Markovskie prot︠s︡essy" (
2707:
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2689:
2621:
2613:
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832:
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451:
423:
385:
375:
314:
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3591:
Collision of Random Walks and a Refined Analysis of Attacks on the Discrete Logarithm Problem
3589:
3472:
International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment
362:
3843:
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1638:
1514:(4th printing of 1st ed.). Princeton, New Jersey, US & Woodstock, Oxfordshire, UK:
1437:
1183:
1098:
957:
915:
846:
678:
599:
439:
431:
3330:. Modules and Monographs in Undergraduate Mathematics and its Applications (UMAP) Project.
1296:
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1001:
3739:
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3361:
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2587:
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253:
159:
is not possible. Rather the effect is based on the mathematical fact that the output of a
156:
3940:
3400:
3821:
2407:"How to Do the Math Magic Trick That Will Impress Everyone You Know - Here's the secret"
144:
3848:
3789:
3412:
2874:
2138:
2126:
2073:
2051:
2005:
1955:
1813:
1779:
1755:
1587:
1566:), found time to occasionally help with homework, and, when the time came to apply for
1553:
740:
Card Concepts - An Anthology of Numerical & Sequential Principles Within Card Magic
358:
343:
300:
245:
187:
133:
86:
3964:
2654:
2564:
1361:
3990:
3204:
3144:
3074:
2879:
2633:
2229:
The Mathematics of Preference, Choice and Order. Essays in Honor of Peter J. Fishburn
2128:(xii+235 pages) (NB. The book contains a significant number of typographical errors:
2038:. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US:
1992:. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US:
1942:. Vol. 1. Lyons, Pat Patterson (illustrations) (1 ed.). Washington DC, US:
884:
800:
706:
435:
351:
109:
66:
3865:
3707:
3554:
Kruskal's Principle and Collision Time for Monotone Transitive Walks on the Integers
3263:
2336:
2268:
1436:
the coupling time is larger than 100. Numerical computation done by one of Dynkin's
1146:
1144:
1085:
Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages
3521:
2760:
1611:
1205:
1120:
775:
758:
619:
182:
160:
105:
31:
3024:; Lee, Yiu-Fai (September 2005) . "A Random Walk on a Circular Path". Miscellany.
3552:
2992:
2948:
2645:
2250:
2082:
1642:
1509:
1187:
905:
836:
440:
3878:
called the Kruskal Count so we have dubbed the phenomenon as "Kruskal effect".
3614:
3487:
3168:
2798:
2377:
1882:
1865:
1817:
1796:
1316:. The Duxbury Advanced Series in Statistics and Decision Sciences (3 ed.).
960:, one of the classroom examples that struck him the most was given by Professor
207:
90:
2391:
361:(1997). "Chapter 19. Sicherman Dice, the Kruskal Count and Other Curiosities".
3919:
3889:(2023-08-22) . "Constructions and bounds for codes with restricted overlaps".
3875:
3386:
3120:
3050:
2685:
2599:
1924:
Kraus, Alexander F. (December 1957). Lyons, Philip Howard (ed.). "Sum Total".
1854:(in German). Vol. 10, no. 3 & 4. Munich, Germany. pp. 125–.
1614:
1603:
1575:
1560:, published some of our mathematical findings to give a first publication (in
1400:
976:). A modified (and expanded) version is given below. Example 10.1. Professor
919:
850:
784:
78:
3907:(17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)
3839:
3757:
3738:(Thesis). Lund, Sweden: Department of Electrical and Information Technology,
3513:
3495:
3429:
3196:
3136:
3128:
3066:
3058:
2896:
2866:
2843:
2703:
2617:
2423:
2024:
Ransom, Tom; Katz, Max (March 1958). Lyons, Philip Howard (ed.). "Sum More".
1686:
1678:
1395:"]: "Example. A coupling card trick. The following demonstration used by
945:
927:
868:
690:
611:
542:
465:
318:
3689:
3581:
3544:
3289:
3255:
3224:
3187:
3084:
3021:
2998:
2730:
2669:
2665:
2579:
2575:
2453:
2318:
2287:
1763:
1751:
1159:
1155:
1151:
1102:
1080:
1076:
1072:
1029:
997:
897:
828:
419:
104:
Besides uses as a card trick, the underlying phenomenon has applications in
3857:
2877:(September 1998). "Ten Amazing Mathematical Tricks". Gardner's Gatherings.
2141:(2009) . "Kapitel 12 - Numerologie und Kartentricks: Das Kruskal-Prinzip".
1978:. No. 13. Toronto, Ontario, Canada. pp. 13–16. Part 2 (Solution).
1974:
Kraus, Alexander F. (March 1958). Lyons, Philip Howard (ed.). "Sum Total".
1670:
1511:
Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks
3458:
3357:
3335:
3285:
3101:
3031:
2817:) with the assistance of the authors. It is based on lectures held at the
2661:
2571:
2107:
1655:
1357:
603:
3399:
Humble, Steve "Dr. Maths" (September 2012) . Written at Kraków, Poland.
3097:
International Journal of Mathematical Education in Science and Technology
3027:
International Journal of Mathematical Education in Science and Technology
2825:
792:
742:(1 ed.). London, UK: Lewis Davenport Limited. pp. 34–38, 39–46.
244:
explained the trick, which later became known as Kruskal's principle, to
30:"Dynkin's card trick" redirects here. For optimal stopping problems, see
2913:
Haigh, John (1999). "7. Waiting, waiting, waiting: Packs of cards (2)".
2904:
326:
3924:
2739:) in 1963 and translated to English with the assistance of the author.)
2133:
1599:
698:
3885:
Blackburn, Simon Robert; Esfahani, Navid Nasr; Kreher, Donald Lawson;
3830:
2522:(NB. This is a translation of the first Russian edition published as "
3674:
Proceedings of the 12th International Symposium on Open Collaboration
2241:
1928:. No. 12. Toronto, Ontario, Canada. p. 7. Part 1 (Problem).
1093:
3181:(2 & 3). Missoula, Montana, US: University of Montana: 327–336.
1598:(a major mathematician who was most well-known for his discovery of
682:
3901:
3812:
2281:
2279:
1178:
3246:
2115:
1533:
876:
364:
Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix
348:
Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix
269:
143:
3790:"Sensorless Pose Determination Using Randomized Action Sequences"
2149:] (in German). Translated by Zillgitt, Michael (1 ed.).
2056:(319 pages) (NB. Volume 1 of a three-volume hardcover reprint of
2010:(319 pages) (NB. Volume 1 of a three-volume hardcover reprint of
1960:(319 pages) (NB. Volume 1 of a three-volume hardcover reprint of
1910:(381 pages) (NB. Volume 3 of a three-volume hardcover reprint of
1844:(381 pages) (NB. Volume 3 of a three-volume hardcover reprint of
1067:
1065:
673:(500). Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK:
3464:
Instruction-Level Steganography for Covert Trigger-Based Malware
2772:
2711:
2625:
2532:
2508:
2211:
2209:
910:. Springer Texts in Statistics (2 ed.). Cham, Switzerland:
389:
257:
16:"Kruskal's principle" redirects here. For Kruskal's method, see
2231:. Studies in Choice and Welfare. Berlin / Heidelberg, Germany:
2147:
Impossible? Surprising Solutions to Counterintuitive Conundrums
2084:
Impossible? Surprising Solutions to Counterintuitive Conundrums
904:. Written at University of California, Davis, California, US.
835:. Written at University of California, Davis, California, US.
594:(4). Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK:
473:
23:"Sum Total" redirects here. For the statistical quantity, see
2374:
Once Upon a Number - The Hidden Mathematical Logic of Stories
1161:
Logic for Programming, Artificial Intelligence, and Reasoning
69:
concept originally demonstrated by the Russian mathematician
2759:. Translated by Wood, James S. (1 ed.). New York, US:
2750:(1969) . Written at University of Moscow, Moscow, Russia.
2652:(1965) . Written at University of Moscow, Moscow, Russia.
2562:(1965) . Written at University of Moscow, Moscow, Russia.
841:. Springer Texts in Statistics (1 ed.). New York, US:
3780:
Mannam, Pragna; Volkov, Jr., Alexander; Paolini, Robert;
1552:
A blurb that appears on one of his books says: Warning:
1467:(in German). Ulm, Germany: University of Ulm. p. 7.
1440:
show that the probability of error is approximately .026.
1279:
Wadsworth & Brooks/Cole Advanced Books & Software
370:. Spectrum Series (Revised ed.). Washington DC, US:
248:
in a reply to a letter Gardner had sent him to recommend
2028:. No. 13. Toronto, Ontario, Canada. pp. 17–18.
1578:
of the past ten years. You ought to give him a chance."
3732:
On Offensive and Defensive Methods in Software Security
3584:(2015-03-15) . Written at Gaithersburg, Maryland, US.
1034:"Probabilistic couplings for cryptography and privacy"
1002:"Probabilistic couplings for cryptography and privacy"
2060:
magazine numbers 1 (June 1955) – 15 (December 1958).)
2014:
magazine numbers 1 (June 1955) – 15 (December 1958).)
1964:
magazine numbers 1 (June 1955) – 15 (December 1958).)
1874:. Vol. 10, no. 8. Teaneck, New Jersey, US:
1805:. Vol. 10, no. 9. Teaneck, New Jersey, US:
1788:. Vol. 10, no. 8. Teaneck, New Jersey, US:
1651:
International Journal of Pure and Applied Mathematics
1236:"Probabilistic Couplings for Probabilistic Reasoning"
1367:. Corvallis, Oregon, US: Department of Mathematics,
1081:"Coupling Proofs are Probabilistic Product Programs"
902:"Chapter 10 Stochastic Processes; 10.1 Introduction"
833:"Chapter 10 Stochastic Processes; 10.1 Introduction"
438:; Jörgenson, Loki; Corless, Robert "Rob" M. (eds.).
3603:International Association for Cryptologic Research
2945:"7. Waiting, waiting, waiting: Packs of cards (2)"
2653:
2563:
2450:"Graph Based Model for Software Tamper Protection"
1848:magazine volumes 9 (November 1973) – 10 (1977).);
596:International Association for Cryptologic Research
3336:Consortium For Mathematics & Its Applications
2857:. Vol. 57, no. 10. Bluffton, Ohio, US:
2834:. Vol. 56, no. 12. Bluffton, Ohio, US:
1893:L & L Publishing - Quality Magical Literature
1828:L & L Publishing - Quality Magical Literature
1633:
1631:
1629:
1627:
1352:
1350:
1079:; Hsu, Justin; Strub, Pierre-Yves (2016-11-07) .
309:. Mathematical Games. Vol. 238, no. 2.
2492:Random Walks (Mathematical Conversations Part 3)
1914:magazine volumes 9 (November 1973) – 10 (1977).)
1455:
1453:
648:
646:
644:
569:
567:
510:"Monte Carlo Methods for Index Computation (mod
3234:How Long Does it Take to Catch a Wild Kangaroo?
2814:Теория вероятностей и математическая статистика
1310:(2005). "Example 5.2. A coupling card trick.".
1301:(x+453 pages) (NB. This can be found quoted in
1265:
1263:
3804:Multidisciplinary Digital Publishing Institute
2809:Probability Theory and Mathematical Statistics
2676:(1 ed.). New York, US / Berlin, Germany:
2586:(1 ed.). New York, US / Berlin, Germany:
1980:(4 pages) (NB. The first part can be found in
1930:(1 page) (NB. The second part can be found in
992:
990:
733:
731:
500:
498:
338:
336:
295:
293:
237:
2812:
2802:
2792:
2734:
2724:
2546:
2536:
2523:
2068:
2066:
1606:. A few years later, the kid lectured at the
1399:in his probability class is a variation of a
1340:(497 pages) (NB. This can be found quoted in
579:"Kangaroos, Monopoly and Discrete Logarithms"
8:
3089:"On Convergent Probability of a Random Walk"
1758:who had earlier written about their mother,
1443:(45 pages) (NB. This can be found quoted in
823:
821:
3368:. Lausanne, Switzerland: Data Science Lab,
2143:Das gibts doch nicht – Mathematische Rätsel
1462:"Vorstellung der Kopplung bei Markovketten"
1229:
1227:
533:, Taplow Court, Maidenhead, Berkshire, UK:
3870:The phenomenon, while also reminiscent of
1738:had two mathematically inclined brothers,
1722:. Indianapolis, Indiana, US. p. 316.
3900:
3847:
3829:
3811:
3245:
3186:
2370:"An old card trick and new Biblical hoax"
2308:
2286:Jacob, Matthias; Jakubowski, Mariusz H.;
2240:
2088:(1 ed.). Princeton, New Jersey, US:
1484:
1341:
1177:
1092:
3928:(Video). Alchemist Cafe, Dublin, Ireland
3729:Jämthagen, Christopher (November 2016).
3370:École Polytechnique Fédérale de Lausanne
3338:, Inc. (COMAP): 199–236. UMAP Unit 808.
2950:Taking Chances: Winning with Probability
2916:Taking Chances: Winning with Probability
2448:Jakubowski, Mariusz H. (February 2016).
1444:
430:, Burnaby, British Columbia, Canada. In
3892:IEEE Transactions on Information Theory
3167:Humble, Steve "Dr. Maths" (July 2008).
2994:Puzzlers' Tribute: A Feast for the Mind
2953:(Reprint of 2nd ed.). Oxford, UK:
2753:Markov Processes: Theorems and Problems
2151:Spektrum Akademischer Verlag Heidelberg
1891:. Vol. 3. Tahoma, California, US:
1826:. Vol. 3. Tahoma, California, US:
1404:
1392:
973:
289:
273:
256:. Diaconis graduated in 1971, earned a
229:
3405:"Mathematics in the Streets of Kraków"
2859:International Brotherhood of Magicians
2836:International Brotherhood of Magicians
1799:, ed. (July 1975). "Cross-Cut Force".
1782:(June 1975). "The Kruskal Principle".
907:Large Sample Techniques for Statistics
838:Large Sample Techniques for Statistics
1981:
1931:
1868:(June 1975). "Kruskal Phone Effect".
1816:(1993) . "The Kruskal Principle". In
1302:
218:Knowledge (XXG):Getting to Philosophy
7:
3601:. Berlin & Heidelberg, Germany:
3416:. No. 85. Zürich, Switzerland:
2794:Теоремы и задачи о процессах Маркова
2040:Richard Kaufman & Alan Greenberg
1994:Richard Kaufman & Alan Greenberg
1944:Richard Kaufman & Alan Greenberg
1888:The Pallbearers Review: Volumes 9–10
1823:The Pallbearers Review: Volumes 9–10
1362:"From Markov Chains to Gibbs Fields"
843:Springer Science+Business Media, LLC
3682:Association for Computing Machinery
2885:Mathematical Association of America
2748:Yushkevish , Aleksandr Adol'fovich
2301:Association for Computing Machinery
531:Plessey Telecommunications Research
372:Mathematical Association of America
3918:Humble, Steve "Dr. Maths" (2010).
3393:. It's called the "Kruskal Count"
3174:The Montana Mathematics Enthusiast
2838:. pp. 82, 83, 83, 84, 85–87.
1852:intermagic - Ein Magisches Journal
1483:(1+9 pages) (NB. This work quotes
14:
3599:Lecture Notes in Computer Science
3480:Springer International Publishing
3476:Lecture Notes in Computer Science
3334:(3). Bedford, Massachusetts, US:
2807:) in 1967 as part of a series on
2548:Библиотека математического кружка
2543:Library of the Mathematics Circle
2487:Uspenskii , Vladimir Andreyevich
1885:(1993) . "Kruskal Phone Effect".
1393:Probability: Theory and Examples.
1166:Lecture Notes in Computer Science
73:in the 1950s or 1960s discussing
3941:"Mathematical Card Trick Source"
2405:Delbert, Caroline (2020-02-27).
1313:Probability: Theory and Examples
1275:Probability: Theory and Examples
1168:. Vol. 9450. Suva, France:
136:, object alignment, and others.
130:variable-length instruction sets
3975:from the original on 2023-09-10
3951:from the original on 2023-09-04
3874:, is similar to an interesting
3767:from the original on 2023-08-26
3717:from the original on 2023-09-04
3655:from the original on 2023-09-07
3634:from the original on 2023-09-03
3607:Springer Science+Business Media
3567:from the original on 2023-08-22
3531:from the original on 2023-08-26
3439:from the original on 2023-09-02
3376:from the original on 2022-05-23
3362:"Knowledge (XXG)'s fixed point"
3345:from the original on 2023-08-19
3306:from the original on 2023-08-19
3273:from the original on 2023-08-20
3211:from the original on 2023-09-03
3154:from the original on 2023-09-02
3003:Taylor & Francis Group, LLC
2782:from the original on 2023-09-06
2541:) in March 1952 as Number 6 in
2460:from the original on 2019-10-31
2430:from the original on 2021-10-19
2346:from the original on 2018-09-04
2198:. Murray Hill, New Jersey, US:
2155:Springer Science+Business Media
1899:from the original on 2023-09-10
1834:from the original on 2023-09-10
1726:from the original on 2023-08-19
1696:from the original on 2023-08-19
1474:from the original on 2023-09-01
1460:Weinhold, Leonie (2011-05-13).
1378:from the original on 2023-09-01
1308:Durrett, Richard "Rick" Timothy
1271:Durrett, Richard "Rick" Timothy
1248:from the original on 2023-08-19
1212:from the original on 2023-08-19
1127:from the original on 2023-08-19
1050:from the original on 2023-08-19
1018:from the original on 2023-08-19
765:from the original on 2023-08-20
716:from the original on 2023-08-18
629:from the original on 2023-08-18
552:from the original on 2013-05-03
529:(143). Mathematics Department,
399:from the original on 2023-08-19
3887:Stinson, Douglas "Doug" Robert
2217:Lagarias, Jeffrey "Jeff" Clark
2188:Lagarias, Jeffrey "Jeff" Clark
1720:"Foshee Magically Interpreted"
1608:Institute for Defense Analyses
1318:Thomson Brooks/Cole Publishing
956:During the author's time as a
912:Springer Nature Switzerland AG
122:control-flow resynchronization
81:by the American mathematician
77:effects and rediscovered as a
1:
3422:European Mathematical Society
3321:"Cards, Codes, and Kangaroos"
2765:Plenum Publishing Corporation
2078:"Chapter 12: Two Card Tricks"
968:. The example was taken from
535:American Mathematical Society
448:American Mathematical Society
3680:. OpenSym, Berlin, Germany:
3647:Jose, Harish (2016-06-14) .
2981:"A Labyrinth in a Labyrinth"
2955:Oxford University Press Inc.
2921:Oxford University Press Inc.
2744:Dynkin , Evgenii Borisovich
2560:Dynkin , Evgenii Borisovich
2483:Dynkin , Evgenii Borisovich
2251:10.1007/978-3-540-79128-7_23
1754:. All three were friends of
1188:10.1007/978-3-662-48899-7_27
675:The Mathematical Association
238:Diaconis & Graham (2012)
203:Pollard's kangaroo algorithm
4007:Number theoretic algorithms
3920:"Dr. Maths Randomness Show"
3615:10.1007/978-3-662-46447-2_6
3488:10.1007/978-3-319-08509-8_3
2883:. Vol. 6, no. 1.
2417:Hearst Magazine Media, Inc.
1558:a runaway fourteen-year-old
4028:
3868:. Article 154. p. 2:
3478:. Egham, UK; Switzerland:
3113:Taylor & Francis, Ltd.
3043:Taylor & Francis, Ltd.
2919:(1 ed.). Oxford, UK:
2889:Taylor & Francis, Ltd.
2650:Dynkin, Evgenii Borisovich
2200:AT&T Bell Laboratories
2090:Princeton University Press
1516:Princeton University Press
1502:Graham, Ronald "Ron" Lewis
522:Mathematics of Computation
114:software tamper protection
29:
22:
15:
3963:Behr, Denis, ed. (2023).
3802:(2). Basel, Switzerland:
3782:Chirikjian, Gregory Scott
3121:10.1080/00207390600712299
3051:10.1080/00207390500064254
2813:
2803:
2793:
2735:
2725:
2686:10.1007/978-3-662-25360-1
2600:10.1007/978-3-662-00031-1
2547:
2537:
2524:
2497:The University of Chicago
1830:. pp. 967–970, 985.
1574:invented two of the best
920:10.1007/978-3-030-91695-4
851:10.1007/978-1-4419-6827-2
795:. Random sequences that
311:Scientific American, Inc.
118:code self-synchronization
71:Evgenii Borisovich Dynkin
3997:Recreational mathematics
3385:it turns out there is a
2290:(20–21 September 2007).
2035:ibidem - P. Howard Lyons
1989:ibidem - P. Howard Lyons
1939:ibidem - P. Howard Lyons
1618:insight led him astray!
1387:Here we will quote [
1041:IMDEA Software Institute
1009:IMDEA Software Institute
666:The Mathematical Gazette
424:"On Kruskal's Principle"
198:Overlapping instructions
3690:10.1145/2957792.2957813
3256:10.1145/1536414.1536490
3188:10.54870/1551-3440.1111
2975:(4 of xiv+373+17 pages)
2819:Moscow State University
2666:Greenberg, Vida Lazarus
2576:Greenberg, Vida Lazarus
2553:Moscow State University
2501:D. C. Heath and Company
2319:10.1145/1288869.1288887
2288:Venkatesan, Ramarathnam
1643:"The Kruskal principle"
1369:Oregon State University
1103:10.1145/3009837.3009896
428:Simon Fraser University
346:(1989) . "Chapter 19".
262:mathematical statistics
213:Self-synchronizing code
51:Dynkin's counting trick
1912:The Pallbearers Review
1871:The Pallbearers Review
1846:The Pallbearers Review
1802:The Pallbearers Review
1785:The Pallbearers Review
1671:10.12732/ijpam.v85i6.1
1498:Diaconis, Persi Warren
1358:Kovchegov, Yevgeniy V.
1241:(Thesis). p. 34.
658:"Kruskal's Card Trick"
193:Geometric distribution
173:Coupling (probability)
152:
3786:Mason, Matthew Thomas
2979:Bean, Gordon (2002).
2943:Haigh, John (2009) .
2044:Kaufman and Greenberg
1998:Kaufman and Greenberg
1948:Kaufman and Greenberg
1744:University of Chicago
1621:(2+xii+2+244+4 pages)
1518:. pp. 211–219 .
1234:Hsu, Justin (2018) .
604:10.1007/s001450010010
587:Journal of Cryptology
374:. pp. 265–280 .
147:
126:variable-length codes
3776:(1+xvii+1+152 pages)
3609:. pp. 127–149.
3418:EMS Publishing House
3005:. pp. 103–106.
2891:pp. 13–15, 26.
2674:Majone, Giandomenico
2670:Maitra, Ashok Prasad
2592:Academic Press, Inc.
2584:Majone, Giandomenico
2580:Maitra, Ashok Prasad
2303:. pp. 129–140.
2235:. pp. 371–391.
2225:Vanderbei, Robert J.
2192:Vanderbei, Robert J.
2157:. pp. 128–135.
2092:. pp. 131–140.
2054:). pp. 258–259.
2048:Hermetic Press, Inc.
2008:). pp. 255–258.
2002:Hermetic Press, Inc.
1952:Hermetic Press, Inc.
1876:L & L Publishing
1807:L & L Publishing
1790:L & L Publishing
1550:. pp. 211–212:
1172:. pp. 387–401.
1087:. pp. 161–174.
914:. pp. 339–341.
845:. pp. 317–319.
805:Martin David Kruskal
450:. pp. 407–411.
83:Martin David Kruskal
47:Dynkin–Kruskal count
25:Total sum of squares
3965:"Kruskal Principle"
3872:contraction mapping
3822:2019Entrp..21..154M
3482:. pp. 41–50 .
3457:Andriesse, Dennis;
3424:. pp. 20–21 .
3389:that works exactly
3094:. Classroom notes.
2726:Марковские процессы
2656:Markov Processes-II
2647:(xii+365+1 pages);
2519:(1+9+80+9+1 pages)
2221:Rains, Eric Michael
1760:Lillian Oppenheimer
1563:Scientific American
1409:Scientific American
1154:; Espitau, Thomas;
964:in his lectures on
442:Organic Mathematics
306:Scientific American
59:coupling card trick
55:Dynkin's card trick
43:Kruskal's principle
18:Kruskal's algorithm
3943:. Close-Up Magic.
3448:The Kruscal count
3169:"Magic Card Maths"
2957:pp. 139–142.
2923:pp. 133–136.
2826:Marlo, Edward "Ed"
2566:Markov Processes-I
2366:Paulos, John Allen
1792:. pp. 967–970
1208:. hal-01246719v2.
1156:Grégoire, Benjamin
1077:Grégoire, Benjamin
966:Probability Theory
895:(xvii+610 pages);
789:well-shuffled deck
266:Harvard University
178:Discrete logarithm
153:
3969:Conjuring Archive
3831:10.3390/e21020154
3749:978-91-7623-942-1
3699:978-1-4503-4451-7
3684:. pp. 1–10.
3624:978-3-662-46446-5
3549:Tetali, Prasad V.
3401:Behrends, Ehrhard
3297:singingbanana.com
3290:"Kruskal's Count"
3229:Tetali, Prasad V.
2964:978-0-19-852663-6
2930:978-0-19-850291-3
2695:978-3-662-23320-7
2609:978-3-662-00033-5
2412:Popular Mechanics
2368:(November 1998).
2328:978-1-59593-857-2
2260:978-3-540-79127-0
2196:The Kruskal Count
2173:978-3-8274-2306-1
2164:978-3-8274-2306-1
2099:978-0-691-13131-3
1895:. pp. 970–.
1716:Farrell, Jeremiah
1639:Nishiyama, Yutaka
1542:978-0-691-15164-9
1525:978-0-691-16977-4
1438:graduate students
1336:978-0-534-42441-1
1197:978-3-662-48898-0
1039:. Madrid, Spain:
1007:. Madrid, Spain:
974:1991, p. 275
937:978-3-030-91694-7
860:978-1-4419-6826-5
774:We looked at the
755:Artymowicz, Pawel
457:978-0-8218-0668-5
432:Borwein, Jonathan
250:Persi W. Diaconis
97:and later called
4019:
3983:
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3582:Montenegro, Ravi
3575:
3573:
3572:
3566:
3559:
3545:Montenegro, Ravi
3539:
3537:
3536:
3530:
3505:978-3-31908508-1
3469:
3450:
3445:
3444:
3438:
3409:
3395:
3382:
3381:
3353:
3351:
3350:
3344:
3328:The UMAP Journal
3325:
3314:
3312:
3311:
3305:
3294:
3281:
3279:
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3225:Montenegro, Ravi
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3012:978-1-43986410-4
2974:
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2940:
2938:
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2908:
2870:
2854:The Linking Ring
2847:
2831:The Linking Ring
2816:
2815:
2806:
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2796:
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2790:
2788:
2787:
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2719:
2714:. Title-No. 5105
2659:
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2636:. Title-No. 5104
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2390:. Archived from
2387:978-0-46505159-5
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2139:Havil, Julian R.
2125:
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2074:Havil, Julian R.
2070:
2061:
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2015:
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778:, where a given
771:
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654:Pollard, John M.
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575:Pollard, John M.
571:
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506:Pollard, John M.
502:
493:
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408:(1+ix+319 pages)
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313:pp. 19–32.
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3884:
3788:(2019-02-06) .
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3740:Lund University
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3580:Kijima, Shuji;
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3231:(2010-11-07) .
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3087:(2006-03-07) .
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3017:(xvi+421 pages)
3013:
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2875:Gardner, Martin
2873:
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2678:Springer-Verlag
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2588:Springer-Verlag
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2476:Further reading
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2100:
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1985:
1973:
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1968:
1958:). p. 232.
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1878:. pp. 970–
1864:
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1814:Gardner, Martin
1812:
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1780:Gardner, Martin
1778:
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1773:
1762:, a remarkable
1733:. p. 316:
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1355:
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1328:
1320:. p. 312.
1306:
1289:
1281:. p. 275.
1269:
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1249:
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1215:
1213:
1198:
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1021:
1019:
1015:
1004:
996:
995:
988:
954:. p. 339:
950:
948:
938:
896:
889:
887:
861:
827:
826:
819:
780:random sequence
768:
766:
757:(2020-01-29) .
753:
752:
748:
737:
736:
729:
719:
717:
713:
683:10.2307/3621657
660:
652:
651:
642:
632:
630:
626:
581:
577:(2000-08-10) .
573:
572:
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417:
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367:
359:Gardner, Martin
357:
344:Gardner, Martin
342:
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301:Gardner, Martin
299:
298:
291:
286:
281:
280:
268:in 1972, and a
254:graduate school
235:
231:
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169:
157:sleight of hand
148:Explanation of
142:
99:Kraus principle
41:(also known as
35:
28:
21:
12:
11:
5:
4025:
4023:
4015:
4014:
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4004:
3999:
3989:
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3984:
3960:
3937:
3913:
3912:External links
3910:
3909:
3908:
3882:
3777:
3748:
3742:. p. 96.
3726:
3698:
3668:
3644:
3623:
3586:Katz, Jonathan
3577:
3551:(2014-09-07).
3541:
3504:
3454:
3446:. p. 21:
3413:EMS Newsletter
3396:
3360:(2011-05-26).
3354:
3316:
3282:
3221:
3207:. Article 14.
3164:
3083:Lee, Yiu-Fai;
3080:
3018:
3011:
2997:(1 ed.).
2976:
2963:
2929:
2910:
2871:
2861:. p. 85.
2848:
2822:
2821:in 1962/1963.)
2740:
2694:
2608:
2556:
2477:
2474:
2471:
2470:
2440:
2397:
2394:on 2015-04-01.
2386:
2380:. p. 64.
2376:(1 ed.).
2357:
2327:
2310:10.1.1.69.5258
2275:
2259:
2205:
2179:
2163:
2098:
2062:
2052:Jogestja, Ltd.
2016:
2006:Jogestja, Ltd.
1966:
1956:Jogestja, Ltd.
1916:
1857:
1771:
1756:Martin Gardner
1707:
1641:(July 2013) .
1623:
1592:Martin Kruskal
1588:thesis advisor
1580:Fred Mosteller
1554:Martin Gardner
1524:
1489:
1449:
1385:. p. 22:
1371:. p. 22.
1360:(2007-10-06).
1346:
1326:
1287:
1259:
1223:
1196:
1152:Barthe, Gilles
1140:
1111:
1073:Barthe, Gilles
1061:
1032:(2016-09-13).
1030:Barthe, Gilles
998:Barthe, Gilles
986:
984:(xv+685 pages)
936:
859:
817:
746:
744:(vi+301 pages)
727:
640:
563:
508:(July 1978) .
494:
456:
436:Borwein, Peter
426:. Written at
422:(June 1997) .
410:
380:
350:(1 ed.).
332:
288:
287:
285:
282:
279:
278:
274:Gardner (1975)
246:Martin Gardner
242:Martin Kruskal
228:
227:
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200:
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190:
188:Ergodic theory
185:
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141:
138:
134:web navigation
87:Martin Gardner
63:shift coupling
13:
10:
9:
6:
4:
3:
2:
4024:
4013:
4012:Markov models
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3994:
3992:
3974:
3970:
3966:
3961:
3950:
3947:. 2015–2017.
3946:
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3645:
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3627:. LNCS 9020.
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3592:
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3542:
3527:
3524:. LNCS 8550.
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3085:Ching, Wai-Ki
3081:
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2880:Math Horizons
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2020:
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2013:
2007:
2003:
1999:
1995:
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1983:
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1963:
1957:
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1945:
1941:
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1933:
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1853:
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1843:
1833:
1829:
1825:
1824:
1819:
1815:
1809:. p. 985
1808:
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1798:
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1347:
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1337:
1333:
1329:
1327:0-534-42441-4
1323:
1319:
1315:
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1298:
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1290:
1288:0-534-13206-5
1284:
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947:
943:
939:
933:
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921:
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913:
909:
908:
903:
899:
898:Jiang, Jiming
886:
882:
878:
874:
870:
866:
862:
856:
852:
848:
844:
840:
839:
834:
830:
829:Jiang, Jiming
824:
822:
818:
815:
813:
811:
808:
806:
802:
801:Eugene Dynkin
798:
794:
790:
786:
781:
777:
776:Markov chains
764:
760:
756:
750:
747:
741:
734:
732:
728:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
667:
659:
656:(July 2000).
655:
649:
647:
645:
641:
625:
621:
617:
613:
609:
605:
601:
597:
593:
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536:
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528:
524:
523:
515:
513:
507:
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483:
482:0-8218-0668-8
479:
475:
471:
467:
463:
459:
453:
449:
444:
443:
437:
433:
429:
425:
421:
420:Robins, Sinai
418:Haga, Wayne;
414:
411:
395:
391:
387:
383:
381:0-88385-521-6
377:
373:
366:
365:
360:
354:. p. 274
353:
352:W. H. Freeman
349:
345:
339:
337:
333:
328:
324:
320:
316:
312:
308:
307:
302:
296:
294:
290:
283:
275:
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267:
263:
259:
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251:
247:
243:
239:
236:According to
233:
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219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
170:
166:
164:
162:
158:
151:
150:Kruskal count
146:
139:
137:
135:
131:
127:
123:
119:
115:
111:
110:code breaking
107:
102:
100:
96:
92:
89:and magician
88:
84:
80:
76:
72:
68:
67:probabilistic
64:
60:
56:
52:
48:
44:
40:
39:Kruskal count
33:
26:
19:
4002:Cryptography
3977:. Retrieved
3968:
3953:. Retrieved
3944:
3930:. Retrieved
3923:
3890:
3869:
3799:
3793:
3769:. Retrieved
3731:
3719:. Retrieved
3673:
3657:. Retrieved
3636:. Retrieved
3590:
3569:. Retrieved
3553:
3533:. Retrieved
3463:
3459:Bos, Herbert
3447:
3441:. Retrieved
3411:
3391:the same way
3384:
3378:. Retrieved
3365:
3347:. Retrieved
3331:
3327:
3308:. Retrieved
3296:
3286:Grime, James
3275:. Retrieved
3233:
3220:(1+10 pages)
3213:. Retrieved
3178:
3172:
3156:. Retrieved
3108:
3095:
3038:
3025:
2993:
2989:Rodgers, Tom
2985:Wolfe, David
2968:. Retrieved
2949:
2934:. Retrieved
2915:
2878:
2852:
2829:
2808:
2784:. Retrieved
2761:Plenum Press
2752:
2716:. Retrieved
2662:Fabius, Jaap
2655:
2638:. Retrieved
2572:Fabius, Jaap
2565:
2542:
2512:. Retrieved
2491:
2462:. Retrieved
2443:
2432:. Retrieved
2410:
2400:
2392:the original
2373:
2360:
2348:. Retrieved
2292:
2242:math/0110143
2228:
2195:
2182:
2146:
2142:
2119:. Retrieved
2083:
2057:
2034:
2025:
2019:
2011:
1988:
1982:Kraus (1957)
1975:
1969:
1961:
1938:
1932:Kraus (1958)
1925:
1919:
1911:
1901:. Retrieved
1887:
1883:Fulves, Karl
1869:
1866:Fulves, Karl
1860:
1851:
1845:
1836:. Retrieved
1822:
1818:Fulves, Karl
1800:
1797:Fulves, Karl
1783:
1774:
1734:
1728:. Retrieved
1710:
1698:. Retrieved
1662:
1649:
1612:cryptography
1610:, a kind of
1561:
1551:
1545:. Retrieved
1510:
1492:
1476:. Retrieved
1416:8 2 3 7 5 6
1407:
1397:E. B. Dynkin
1386:
1380:. Retrieved
1312:
1303:Jiang (2010)
1274:
1250:. Retrieved
1214:. Retrieved
1160:
1129:. Retrieved
1094:1607.03455v5
1084:
1052:. Retrieved
1027:(66 pages);
1020:. Retrieved
978:E. B. Dynkin
962:David Aldous
955:
949:. Retrieved
906:
888:. Retrieved
837:
773:
767:. Retrieved
749:
739:
718:. Retrieved
670:
664:
631:. Retrieved
591:
585:
554:. Retrieved
526:
520:
511:
485:. Retrieved
441:
413:
401:. Retrieved
363:
347:
304:
232:
183:Equifinality
161:Markov chain
154:
149:
124:, design of
106:cryptography
103:
98:
94:
62:
58:
54:
50:
46:
42:
38:
36:
32:Dynkin games
3366:dlab @ EPFL
3115:: 833–838.
3100: [
3045:: 680–683.
3030: [
2941:(4 pages);
2799:Nauka Press
2409:. Science.
2378:Basic Books
2030:(2 pages);
1794:(4 pages);
1654: [
1576:card tricks
1405:appeared in
1257:(147 pages)
725:(1+3 pages)
677:: 265–267.
598:: 437–447.
537:: 918–924.
208:Random walk
91:Karl Fulves
3991:Categories
3979:2023-09-10
3955:2023-09-05
3945:GeniiForum
3932:2023-09-05
3902:2211.10309
3881:(13 pages)
3876:card trick
3813:1812.01195
3771:2023-08-26
3721:2021-03-17
3659:2023-09-07
3643:(23 pages)
3638:2023-09-03
3576:(18 pages)
3571:2023-08-22
3540:(10 pages)
3535:2023-08-26
3443:2023-09-02
3387:card trick
3380:2023-09-04
3349:2023-08-19
3310:2023-08-19
3277:2023-08-20
3215:2023-09-02
3158:2023-09-02
2970:2023-09-03
2936:2023-09-06
2786:2023-09-03
2718:2023-09-02
2640:2023-09-02
2514:2023-09-03
2464:2023-08-19
2434:2021-12-25
2355:(12 pages)
2350:2021-12-25
2273:(22 pages)
2134:0691150028
2121:2023-08-19
2116:2007051792
2108:j.ctt7rnph
1903:2023-09-10
1838:2023-09-10
1811:(1 page);
1730:2023-08-19
1705:(10 pages)
1700:2023-08-19
1615:think tank
1604:card magic
1547:2023-09-06
1534:2011014755
1478:2023-09-01
1423:4 6 5 7 8
1401:card trick
1389:R. Durrett
1382:2023-09-01
1252:2023-08-19
1221:(17 pages)
1216:2023-08-19
1179:1509.03476
1138:(14 pages)
1131:2023-08-19
1059:(49 pages)
1054:2023-08-19
1022:2023-08-19
951:2023-09-02
890:2023-09-02
877:2010930134
785:card trick
769:2023-08-20
720:2023-08-19
638:(11 pages)
633:2023-08-19
556:2023-08-19
487:2023-08-19
403:2023-08-19
284:References
140:Card trick
79:card trick
3840:1099-4300
3758:1654-790X
3514:0302-9743
3496:1611-3349
3474:(DIMVA).
3453:(2 pages)
3430:1027-488X
3358:West, Bob
3315:(8 pages)
3247:0812.0789
3205:117632058
3197:1551-3440
3163:(6 pages)
3145:121242696
3137:0020-739X
3129:1464-5211
3079:(4 pages)
3075:121692834
3067:0020-739X
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