Kruskal count
{{Redir|Kruskal's principle|Kruskal's method|Kruskal's algorithm}}
{{Redir|Sum Total|the statistical quantity|Total sum of squares}}
{{Redir|Dynkin's card trick|optimal stopping problems|Dynkin games}}
{{Use dmy dates|date=August 2023|cs1-dates=y}}
{{Use list-defined references|date=August 2023}}
The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s{{when|reason=We have several citations (Durrett (1989/1991), Kovchegov (2007), Jiang (2010), Weinhold (2011)) referring directly or indirectily to Dynkin demonstrating this trick to his students in his probability class, but it remains unclear if they refer to the Russian or the American classes and we still have to find Dynkin's original publication where he described it. There are vague indications that it might be in "Markov processes" (1965/1963/1962).|date=September 2023}} discussing coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. {{anchor|Kraus}}This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total and later called Kraus principle.
Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navigation, object alignment, and others.
Card trick
File:Kruskal_count_principle.svg
The trick is performed with cards, but is more a magical-looking effect than a conventional magic trick. The magician has no access to the cards, which are manipulated by members of the audience. Thus sleight of hand is not possible. Rather the effect is based on the mathematical fact that the output of a Markov chain, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock performed by David Copperfield is as follows.http://popularmechanics.com/science/a31136091/math-magic-trick/ 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
Notes
{{reflist|group="nb"|refs=
According to {{citeref|Diaconis|Graham|2016|Diaconis & Graham (2012)|style=plain}}, 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 {{citeref|Gardner|1975|Gardner (1975)|style=plain}}.
}}
References
{{reflist|refs=
{{cite magazine |title=How to Do the Math Magic Trick That Will Impress Everyone You Know - Here's the secret. |author-first=Caroline |author-last=Delbert |date=2020-02-27 |department=Science |magazine=Popular Mechanics |issn=0032-4558 |publisher=Hearst Magazine Media, Inc. |url=http://popularmechanics.com/science/a31136091/math-magic-trick |access-date=2021-12-25 |url-status=live |archive-url=https://web.archive.org/web/20211019081906/https://www.popularmechanics.com/science/a31136091/math-magic-trick/ |archive-date=2021-10-19}}
{{cite book |title=The Kruskal Count |author-first1=Jeffrey "Jeff" Clark |author-last1=Lagarias |author-link1=Jeffrey Clark Lagarias |author-first2=Robert J. |author-last2=Vanderbei |author-link2=Robert J. Vanderbei |date=1988 |publisher=AT&T Bell Laboratories |location=Murray Hill, New Jersey, US}}
{{cite book |title=The Mathematics of Preference, Choice and Order. Essays in Honor of Peter J. Fishburn |author-first1=Jeffrey "Jeff" Clark |author-last1=Lagarias |author-link1=Jeffrey Clark Lagarias |author-first2=Eric Michael |author-last2=Rains |author-link2=Eric Michael Rains |author-first3=Robert J. |author-last3=Vanderbei |chapter=The Kruskal Count |series=Studies in Choice and Welfare |author-link3=Robert J. Vanderbei |date=2009 |orig-date=2001-10-13 |arxiv=math/0110143 |editor-first1=Stephen |editor-last1=Brams |editor-first2=William V. |editor-last2=Gehrlein |editor-first3=Fred S. |editor-last3=Roberts |publisher=Springer-Verlag |publication-place=Berlin / Heidelberg, Germany |isbn=978-3-540-79127-0 |doi=10.1007/978-3-540-79128-7_23 |s2cid=18273053 |pages=371–391}} (22 pages)
{{cite conference |title=Towards Integral Binary Execution: Implementing Oblivious Hashing Using Overlapped Instruction Encodings |author-first1=Matthias |author-last1=Jacob |author-first2=Mariusz H. |author-last2=Jakubowski |author-first3=Ramarathnam |author-last3=Venkatesan |author-link3=:d:Q102402462 |conference=Proceedings of the 9th workshop on Multimedia & Security (MM&Sec '07) |location=Dallas, Texas, US |date=20–21 September 2007 |publisher=Association for Computing Machinery |isbn=978-1-59593-857-2 |citeseerx=10.1.1.69.5258 |doi=10.1145/1288869.1288887 |s2cid=14174680 |pages=129–140 |url=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/jacob07overlap.pdf |access-date=2021-12-25 |url-status=live |archive-url=https://web.archive.org/web/20180904062911/https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/jacob07overlap.pdf |archive-date=2018-09-04}} (12 pages)
{{cite web |title=Graph Based Model for Software Tamper Protection |author-first=Mariusz H. |author-last=Jakubowski |date=February 2016 |publisher=Microsoft |url=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/mariuszj-jacob07overlap.ppt |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20191031000757/https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/mariuszj-jacob07overlap.ppt |archive-date=2019-10-31}}
{{cite conference |chapter=On Kruskal's Principle |author-first1=Wayne |author-last1=Haga |author-first2=Sinai |author-last2=Robins |author-link2=:d:Q102195608 |editor-first1=Jonathan |editor-last1=Borwein |editor-link1=Jonathan Borwein |editor-first2=Peter |editor-last2=Borwein |editor-link2=Peter Borwein |editor-first3=Loki |editor-last3=Jörgenson |editor-first4=Robert "Rob" M. |editor-last4=Corless |title=Organic Mathematics |location=Simon Fraser University, Burnaby, British Columbia, Canada |conference=Canadian Mathematical Society Conference Proceedings |issn=0731-1036 |volume=20 |publisher=American Mathematical Society |publication-place=Providence, Rhode Island, US |date=June 1997 |orig-date=1995-12-12 |isbn=978-0-8218-0668-5 |id={{ISBN|0-8218-0668-8}} |lccn=97-179 |pages=407–411 |chapter-url=https://books.google.com/books?id=bEgtaaU0QRoC&pg=PA407 |url-access=limited |url=https://books.google.com/books?id=bEgtaaU0QRoC |access-date=2023-08-19}} (5 pages)
{{cite magazine |title=Sum Total |author-first=Alexander F. |author-last=Kraus |magazine=ibidem |editor-first=Philip Howard |editor-last=Lyons |publication-place=Toronto, Ontario, Canada |date=December 1957 |number=12 |id=Part 1 (Problem) |page=7}} (1 page) (NB. The second part can be found in {{citeref|Kraus|1958|Kraus (1958)|style=plain}}.); {{cite book |chapter=Sum Total (Problem) |author-first=Alexander F. |author-last=Kraus |title=ibidem - P. Howard Lyons |editor-first1=Tom |editor-last1=Ransom |editor-first2=Matthew |editor-last2=Field |editor-first3=Mark |editor-last3=Phillips |others=Lyons, Pat Patterson (illustrations) |volume=1 |page=232 |publisher=Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.) |publication-place=Washington DC, US |date=1993 |isbn= |edition=1 |url=https://www.martinsmagic.com/allmagic/books/ibidem-vol-1-by-p-howard-lyons/}} (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
{{cite magazine |title=Sum Total |author-first=Alexander F. |author-last=Kraus |magazine=ibidem |editor-first=Philip Howard |editor-last=Lyons |publication-place=Toronto, Ontario, Canada |number=13 |date=March 1958 |id=Part 2 (Solution) |pages=13–16}} (4 pages) (NB. The first part can be found in {{citeref|Kraus|1957|Kraus (1957)|style=plain}}.); {{cite book |chapter=Sum Total (Solution) |author-first=Alexander F. |author-last=Kraus |title=ibidem - P. Howard Lyons |editor-first1=Tom |editor-last1=Ransom |editor-first2=Matthew |editor-last2=Field |editor-first3=Mark |editor-last3=Phillips |others=Lyons, Pat Patterson (illustrations) |volume=1 |pages=255–258 |publisher=Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.) |publication-place=Washington DC, US |date=1993 |isbn= |edition=1 |url=https://www.martinsmagic.com/allmagic/books/ibidem-vol-1-by-p-howard-lyons/}} (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
{{cite magazine |title=Sum More |author-first1=Tom |author-last1=Ransom |author-first2=Max |author-last2=Katz |magazine=ibidem |editor-first=Philip Howard |editor-last=Lyons |publication-place=Toronto, Ontario, Canada |number=13 |date=March 1958 |pages=17–18}} (2 pages); {{cite book |chapter=Sum More |author-first1=Tom |author-last1=Ransom |author-first2=Max |author-last2=Katz |title=ibidem - P. Howard Lyons |editor-first1=Tom |editor-last1=Ransom |editor-first2=Matthew |editor-last2=Field |editor-first3=Mark |editor-last3=Phillips |others=Lyons, Pat Patterson (illustrations) |volume=1 |pages=258–259 |publisher=Richard Kaufman & Alan Greenberg (Kaufman and Greenberg); Hermetic Press, Inc. (Jogestja, Ltd.) |publication-place=Washington DC, US |date=1993 |isbn= |edition=1 |url=https://www.martinsmagic.com/allmagic/books/ibidem-vol-1-by-p-howard-lyons/}} (319 pages) (NB. Volume 1 of a three-volume hardcover reprint of ibidem magazine numbers 1 (June 1955) – 15 (December 1958).)
{{cite magazine |title=The Kruskal Principle |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |magazine=The Pallbearers Review |issn= |publisher=L & L Publishing |publication-place=Teaneck, New Jersey, US |volume=10 |number=8 |date=June 1975 |pages=967–970 |postscript=none}} (4 pages); {{cite magazine |title=Cross-Cut Force |editor-first=Karl |editor-last=Fulves |editor-link=Karl Fulves |magazine=The Pallbearers Review |issn= |publisher=L & L Publishing |publication-place=Teaneck, New Jersey, US |volume=10 |number=9 |date=July 1975 |page=985 |postscript=none}} (1 page); {{cite book |title=The Pallbearers Review: Volumes 9–10 |chapter=The Kruskal Principle |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |editor-first=Karl |editor-last=Fulves |editor-link=Karl Fulves |publisher=L & L Publishing - Quality Magical Literature |publication-place=Tahoma, California, US |date=1993 |orig-date=June 1975 |volume=3 |issue=9–10 |pages=967–970, 985 |url=https://www.martinsmagic.com/allmagic/books/pallbearers-review-vols-9-10-by-karl-fulves/ |access-date=2023-09-10 |url-status=live |archive-url=https://web.archive.org/web/20230910074250/https://www.martinsmagic.com/allmagic/books/pallbearers-review-vols-9-10-by-karl-fulves/ |archive-date=2023-09-10 |postscript=none}} [https://web.archive.org/web/20230910083453/https://www.conjuringarchive.com/list/book/1218?highlight=56243] (381 pages) (NB. Volume 3 of a three-volume hardcover reprint of The Pallbearers Review magazine volumes 9 (November 1973) – 10 (1977).); {{cite magazine |title=Das Kruskal-Prinzip |language=de |trans-title=The Kruskal Principle |magazine=intermagic - Ein Magisches Journal |issn= |editor-first=Rudolf |editor-last=Braunmüller |publication-place=Munich, Germany |volume=10 |number=3 & 4 |date=January 1984 |pages=125–}}
{{cite magazine |title=Kruskal Phone Effect |author-first=Karl |author-last=Fulves |author-link=Karl Fulves |magazine=The Pallbearers Review |issn= |publisher=L & L Publishing |publication-place=Teaneck, New Jersey, US |volume=10 |number=8 |date=June 1975 |pages=970– |postscript=none}}; {{cite book |title=The Pallbearers Review: Volumes 9–10 |chapter=Kruskal Phone Effect |author-first=Karl |author-last=Fulves |author-link=Karl Fulves |publisher=L & L Publishing - Quality Magical Literature |publication-place=Tahoma, California, US |date=1993 |orig-date=June 1975 |volume=3 |issue=9–10 |pages=970– |url=https://www.martinsmagic.com/allmagic/books/pallbearers-review-vols-9-10-by-karl-fulves/ |access-date=2023-09-10 |url-status=live |archive-url=https://web.archive.org/web/20230910074250/https://www.martinsmagic.com/allmagic/books/pallbearers-review-vols-9-10-by-karl-fulves/ |archive-date=2023-09-10}} [https://web.archive.org/web/20230910083453/https://www.conjuringarchive.com/list/book/1218?highlight=56243] (381 pages) (NB. Volume 3 of a three-volume hardcover reprint of The Pallbearers Review magazine volumes 9 (November 1973) – 10 (1977).)
{{cite magazine |title=On checker jumping, the Amazon game, weird dice, card tricks and other playful pastimes |series=Mathematical Games |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |magazine=Scientific American |issn=0036-8733 |publisher=Scientific American, Inc. |date=February 1978 |volume=238 |number=2 |jstor=24955629 |pages=19–32}}
{{cite book |title=Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix |chapter=Chapter 19 |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |publisher=W. H. Freeman |date=1989 |orig-date=1988 |edition=1 |isbn= |page=274 |url= |access-date= |postscript=none}}; {{cite book |title=Penrose Tiles to Trapdoor Ciphers ... and the return of Mr. Matrix |chapter=Chapter 19. Sicherman Dice, the Kruskal Count and Other Curiosities |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |publisher=Mathematical Association of America |publication-place=Washington DC, US |lccn=97-70505 |isbn=0-88385-521-6 |series=Spectrum Series |date=1997 |edition=Revised |pages=265–280 [280] |url=https://bobson.ludost.net/copycrime/mgardner/gardner13.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819002849/https://bobson.ludost.net/copycrime/mgardner/gardner13.pdf |archive-date=2023-08-19}} (1+ix+319 pages)
{{cite book |chapter=An old card trick and new Biblical hoax |title=Once Upon a Number - The Hidden Mathematical Logic of Stories |author-first=John Allen |author-last=Paulos |author-link=John Allen Paulos |date=November 1998 |publisher=Basic Books |edition=1 |isbn=978-0-46505159-5 |page=64 |chapter-url=https://math.temple.edu/~paulos/bibhoax.html |url-status=dead |archive-url=https://web.archive.org/web/20150401170957/https://math.temple.edu/~paulos/bibhoax.html |archive-date=2015-04-01}}
{{cite journal |title=Monte Carlo Methods for Index Computation (mod p) |author-first=John M. |author-last=Pollard |author-link=John M. Pollard (mathematician) |location=Mathematics Department, Plessey Telecommunications Research, Taplow Court, Maidenhead, Berkshire, UK |journal=Mathematics of Computation |issn=0025-5718 |volume=32 |number=143 |publisher=American Mathematical Society |date=July 1978 |orig-date=1977-05-01, 1977-11-18 |pages=918–924 |url=https://www.ams.org/journals/mcom/1978-32-143/S0025-5718-1978-0491431-9/S0025-5718-1978-0491431-9.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20130503020626/http://www.ams.org/journals/mcom/1978-32-143/S0025-5718-1978-0491431-9/S0025-5718-1978-0491431-9.pdf |archive-date=2013-05-03}} (7 pages)
{{cite journal |title=Kangaroos, Monopoly and Discrete Logarithms |author-first=John M. |author-last=Pollard |author-link=John M. Pollard (mathematician) |location=Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK |date=2000-08-10 |orig-date=1998-01-23, 1999-09-27 |journal=Journal of Cryptology |issn=0933-2790 |publisher=International Association for Cryptologic Research |volume=13 |number=4 |doi=10.1007/s001450010010 |s2cid=5279098 |pages=437–447 |url=https://link.springer.com/content/pdf/10.1007/s001450010010.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230818234706/https://link.springer.com/content/pdf/10.1007/s001450010010.pdf |archive-date=2023-08-18}} (11 pages)
{{cite journal |title=Kruskal's Card Trick |author-first=John M. |author-last=Pollard |author-link=John M. Pollard (mathematician) |location=Tidmarsh Cottage, Manor Farm Lane, Tidmarsh, Reading, UK |journal=The Mathematical Gazette |issn=0025-5572 |volume=84 |number=500 |date=July 2000 |publisher=The Mathematical Association |jstor=3621657 |id=84.29 |doi=10.2307/3621657 |s2cid=125115379 |pages=265–267 |url=https://web.northeastern.edu/seigen/11Magic/KruskalsCount/Pollard.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230818222322/https://web.northeastern.edu/seigen/11Magic/KruskalsCount/Pollard.pdf |archive-date=2023-08-18}} (1+3 pages)
{{cite web |title=Foshee Magically Interpreted |author-first=Jeremiah |author-last=Farrell |author-link=Jeremiah Farrell |date=2010 |location=Indianapolis, Indiana, US |page=316 |url=https://digitalcommons.butler.edu/cgi/viewcontent.cgi?article=1134&context=wordways |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819172450/https://digitalcommons.butler.edu/cgi/viewcontent.cgi?article=1134&context=wordways |archive-date=2023-08-19 |quote-page=316 |quote=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)
{{cite web |title=Probabilistic couplings for cryptography and privacy |author-first=Gilles |author-last=Barthe |author-link=:d:Q102361529 |publication-place=Madrid, Spain |date=2016 |publisher=IMDEA Software Institute |url=https://software.imdea.org/~gbarthe/mod17/mod-lec1+2.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819090401/https://software.imdea.org/~gbarthe/mod17/mod-lec1+2.pdf |archive-date=2023-08-19}} (66 pages); {{cite web |title=Probabilistic couplings for cryptography and privacy |author-first=Gilles |author-last=Barthe |author-link=:d:Q102361529 |publication-place=Madrid, Spain |publisher=IMDEA Software Institute |date=2016-09-13 |url=https://www.uni-muenster.de/IFIP-WG22/Web/meeting/Singapore16/barthe-slides.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819085622/https://www.uni-muenster.de/IFIP-WG22/Web/meeting/Singapore16/barthe-slides.pdf |archive-date=2023-08-19}} (49 pages)
{{cite book |title=Logic for Programming, Artificial Intelligence, and Reasoning |author-first1=Gilles |author-last1=Barthe |author-link=:d:Q102361529 |author-first2=Thomas |author-last2=Espitau |author-first3=Benjamin |author-last3=Grégoire |author-link3=:d:Q102354334 |author-first4=Justin |author-last4=Hsu |author-first5=Léo |author-last5=Stefanesco |author-first6=Pierre-Yves |author-last6=Strub |chapter=Relational Reasoning via Probabilistic Coupling |series=Lecture Notes in Computer Science |date=2017-07-12 |orig-date=2015 |volume=9450 |publisher=LPAR |publication-place=Suva, France |doi=10.1007/978-3-662-48899-7_27 |id=hal-01246719v2 |arxiv=1509.03476 |isbn=978-3-662-48898-0 |s2cid=3518579 |pages=387–401 |url=https://hal.science/hal-01561433/document |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819085952/https://hal.science/hal-01561433/document |archive-date=2023-08-19}} (17 pages)
{{cite web |title=Codes for PHYD57 Advanced Computing in Physics, UTSC: Dynkin–Kruskal count - convergent Markov chains |author-first=Pawel |author-last=Artymowicz |author-link=:pl:Paweł Artymowicz |date=2020-01-29 |orig-date=2020-01-26 |url=https://planets.utsc.utoronto.ca/~pawel/progD57/ |access-date=2023-08-20 |url-status=live |archive-url=https://web.archive.org/web/20230820134128/https://planets.utsc.utoronto.ca/~pawel/progD57/ |archive-date=2023-08-20 |quote=[...] 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. [...]}} [https://web.archive.org/web/20230820135156/https://planets.utsc.utoronto.ca/~pawel/progD57/game/kruskal-deck-5.py][https://web.archive.org/web/20230820145233/https://planets.utsc.utoronto.ca/~pawel/progD57/game/][https://web.archive.org/web/20230820141044/https://planets.utsc.utoronto.ca/~pawel/progD57/game/jpgs/]
{{cite book |title=Probability: Theory and Examples |author-first=Richard "Rick" Timothy |author-last=Durrett |author-link=Richard Timothy Durrett |date=1991 |orig-date=1989 |edition=1 |publisher=Wadsworth & Brooks/Cole Advanced Books & Software |series=The Wadsworth & Brooks/Cole Statistics/Probability Series |isbn=0-534-13206-5 |publication-place=Pacific Grove, California, US |page=275|mr=1068527 }} (x+453 pages) (NB. This can be found quoted in {{citeref|Jiang|2010|Jiang (2010)|style=plain}}.); {{cite book |title=Probability: Theory and Examples |chapter=Example 5.2. A coupling card trick. |author-first=Richard "Rick" Timothy |author-last=Durrett |author-link=Richard Timothy Durrett |date=2005 |edition=3 |series=The Duxbury Advanced Series in Statistics and Decision Sciences |publisher=Thomson Brooks/Cole Publishing |isbn=0-534-42441-4 |id={{ISBN|978-0-534-42441-1}} |page=312 |url=https://books.google.com/books?id=NPYYAQAAIAAJ&q=Dynkin}} (497 pages) (NB. This can be found quoted in {{citeref|Kovchegov|2007|Kovchegov (2007)|style=plain}}.)
{{cite web |title=From Markov Chains to Gibbs Fields |author-first=Yevgeniy V. |author-last=Kovchegov |author-link=:d:Q56946083 |date=2007-10-06 |publisher=Department of Mathematics, Oregon State University |publication-place=Corvallis, Oregon, US |page=22 |url=http://www.yaroslavvb.com/papers/kovchegov-from.pdf |access-date=2023-09-01 |url-status=live |archive-url=https://web.archive.org/web/20230901195947/http://www.yaroslavvb.com/papers/kovchegov-from.pdf |archive-date=2023-09-01 |quote-page=22 |quote=Here we will quote {{bracket|R. Durrett, "{{citeref|Durrett|2005|Probability: Theory and Examples.|style=plain}}"}}: "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 {{citeref|Gardner|1978|appeared in|style=plain}} 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 {{underline|7}} 8 2 3 7 5 6 {{underline|1}} {{underline|6}} 4 6 5 7 8 {{underline|3}} 1 5 {{underline|3}} 0 7 {{underline|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 {{citeref|Weinhold|2011|Weinhold (2011)|style=plain}}.)
{{cite book |title=Large Sample Techniques for Statistics |author-first=Jiming |author-last=Jiang |author-link=:d:Q100571561 |chapter=Chapter 10 Stochastic Processes; 10.1 Introduction |location=University of California, Davis, California, US |edition=1 |date=2010 |publisher=Springer Science+Business Media, LLC |series=Springer Texts in Statistics |publication-place=New York, US |isbn=978-1-4419-6826-5 |doi=10.1007/978-1-4419-6827-2 |issn=1431-875X |lccn=2010930134 |pages=317–319 |s2cid=118271573 |chapter-url=https://books.google.com/books?id=zUBDAAAAQBAJ&pg=PA317 |url=https://books.google.com/books?id=zUBDAAAAQBAJ |access-date=2023-09-02}} (xvii+610 pages); {{cite book |title=Large Sample Techniques for Statistics |author-first=Jiming |author-last=Jiang |author-link=:d:Q100571561 |chapter=Chapter 10 Stochastic Processes; 10.1 Introduction |location=University of California, Davis, California, US |edition=2 |date=2022 |orig-date=2010 |publisher=Springer Nature Switzerland AG |series=Springer Texts in Statistics |publication-place=Cham, Switzerland |isbn=978-3-030-91694-7 |doi=10.1007/978-3-030-91695-4 |issn=1431-875X |eissn=2197-4136 |pages=339–341 |chapter-url=https://books.google.com/books?id=zRBoEAAAQBAJ&pg=PA339 |url=https://books.google.com/books?id=zRBoEAAAQBAJ |access-date=2023-09-02 |quote-page=339 |quote=[...] 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 ({{citeref|Durrett|1991|1991, p. 275|style=plain}}). 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)
{{cite book |title=Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks |author-first1=Persi Warren |author-last1=Diaconis |author-link1=Persi Warren Diaconis |author-first2=Ronald "Ron" Lewis |author-link2=Ronald Lewis Graham |author-last2=Graham |chapter=Chapter 10. Stars Of Mathematical Magic (And Some Of The Best Tricks In The Book): Martin Gardner |date=2016 |orig-date=2012 |edition=4th printing of 1st |publisher=Princeton University Press |publication-place=Princeton, New Jersey, US & Woodstock, Oxfordshire, UK |isbn=978-0-691-16977-4 |lccn=2011014755 |id={{ISBN|978-0-691-15164-9}} |pages=211–219 [211–212] |chapter-url=https://books.google.com/books?id=W26YDwAAQBAJ&pg=PA211 |url=https://books.google.com/books?id=W26YDwAAQBAJ |access-date=2023-09-06 |quote-pages=211–212 |quote=[...] 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)
{{cite book |title=Impossible? Surprising Solutions to Counterintuitive Conundrums |author-first=Julian R. |author-last=Havil |author-link=:de:Julian R. Havil |chapter=Chapter 12: Two Card Tricks |publisher=Princeton University Press |publication-place=Princeton, New Jersey, US |date=2008 |edition=1 |isbn=978-0-691-13131-3 |lccn=2007051792 |jstor=j.ctt7rnph |pages=131–140 |chapter-url=https://books.google.com/books?id=Ux5QcdVXo6oC&pg=PA131 |url=https://books.google.com/books?id=Ux5QcdVXo6oC |access-date=2023-08-19 |url-access=limited}} [https://archive.org/details/impossiblesurpri0000havi] (xii+235 pages) (NB. The book contains a significant number of typographical errors: {{ASIN|0691150028}}); {{cite book |title=Das gibts doch nicht – Mathematische Rätsel |language=de |trans-title=Impossible? Surprising Solutions to Counterintuitive Conundrums |author-first=Julian R. |author-last=Havil |author-link=:de:Julian R. Havil |translator-first=Michael |translator-last=Zillgitt |chapter=Kapitel 12 - Numerologie und Kartentricks: Das Kruskal-Prinzip |publisher=Spektrum Akademischer Verlag Heidelberg / Springer Science+Business Media |edition=1 |date=2009 |orig-date=2008 |isbn=978-3-8274-2306-1 |id={{ISBN|978-3-8274-2306-1}} |pages=128–135}} (xiv+234 pages)
}}
Further reading
- {{cite book |title=Random Walks (Mathematical Conversations Part 3) |language=en |author-first1=Evgenii Borisovich [Евге́ний Бори́сович] |author-last1=Dynkin [Ды́нкин] |author-link1=Evgenii Borisovich Dynkin |author-first2=Vladimir Andreyevich [Влади́мир Андре́евич] |author-last2=Uspenskii [Успе́нский] |author-link2=Vladimir Andreyevich Uspenskii |editor-first1=Alfred L. |editor-last1=Putnam |editor-first2=Izaak |editor-last2=Wirszup |translator-first1=Norman D. |translator-last1=Whaland, Jr. |translator-first2=Olga A. |translator-last2=Titelbaum |location=University of Moscow, Moscow, Russia |publisher=The University of Chicago / D. C. Heath and Company |series=Survey of Recent East European Mathematical Literature |volume=3 |publication-place=Boston, Massachusetts, US |lccn=63-19838 |date=1963 |edition=1 |url=https://archive.org/details/dynkin-uspenskii-random-walks-mathematical-conversations-part-3 |access-date=2023-09-03}} (1+9+80+9+1 pages) [https://web.archive.org/web/20230902134316/https://mathematikalpha.de/wp-content/uploads/2021/02/Dynkin_Unterhaltungen_3.pdf] (NB. This is a translation of the first Russian edition published as "{{lang|ru|Математические беседы: Задачи о многоцветной раскраске / Задачи из теории чисел / Случайные блуждания}}"[https://web.archive.org/web/20200629114941/http://ilib.mccme.ru:80/djvu/bib-mat-kr/besedy.htm] by GTTI ({{lang|ru|ГТТИ}}) in March 1952 as Number 6 in Library of the Mathematics Circle ({{lang|ru|Библиотека математического кружка}}). It is based on seminars held at the School Mathematics Circle in 1945/1946 and 1946/1947 at Moscow State University.)
- {{cite book |title=Markov Processes-I |language=en |author-first=Evgenii Borisovich [Евге́ний Бори́сович] |author-last=Dynkin [Ды́нкин] |author-link=Evgenii Borisovich Dynkin |location=University of Moscow, Moscow, Russia |translator-first1=Jaap |translator-last1=Fabius |translator-link1=:d:Q77083876 |translator-first2=Vida Lazarus |translator-last2=Greenberg |translator-link2=:d:Q102189194 |translator-first3=Ashok Prasad |translator-last3=Maitra |translator-link3=:d:Q102116681 |translator-first4=Giandomenico |translator-last4=Majone |translator-link4=Giandomenico Majone |series=Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete |issn=0072-7830 |volume=I (121) |date=1965 |orig-date=1963-03-10, 1962-03-31 |edition=1 |lccn=64-24812 |doi=10.1007/978-3-662-00031-1 |isbn=978-3-662-00033-5 |id=Title-No. 5104 |publisher=Springer-Verlag (Academic Press, Inc.) |s2cid=251691119 |publication-place=New York, US / Berlin, Germany |url=https://archive.org/details/markovprocesses0001dynk |access-date=2023-09-02 |url-access=registration}} [https://books.google.com/books?id=vHrpCAAAQBAJ] (xii+365+1 pages); {{cite book |title=Markov Processes-II |language=en |author-first=Evgenii Borisovich |author-last=Dynkin |author-link=Evgenii Borisovich Dynkin |location=University of Moscow, Moscow, Russia |translator-first1=Jaap |translator-last1=Fabius |translator-link1=:d:Q77083876 |translator-first2=Vida Lazarus |translator-last2=Greenberg |translator-link2=:d:Q102189194 |translator-first3=Ashok Prasad |translator-last3=Maitra |translator-link3=:d:Q102116681 |translator-first4=Giandomenico |translator-last4=Majone |translator-link4=Giandomenico Majone |series=Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete |issn=0072-7830 |volume=II (122) |date=1965 |orig-date=1963-03-10, 1962-03-31 |edition=1 |lccn=64-24812 |doi=10.1007/978-3-662-25360-1 |isbn=978-3-662-23320-7 |id=Title-No. 5105 |publisher=Springer-Verlag |publication-place=New York, US / Berlin, Germany |url=https://archive.org/details/markovprocesses0002dynk |access-date=2023-09-02 |url-access=registration}} (viii+274+2 pages) (NB. This was originally published in Russian as "Markovskie prot︠s︡essy" ({{lang|ru|Марковские процессы}}) by Fizmatgiz ({{lang|ru|Физматгиз}}) in 1963 and translated to English with the assistance of the author.)
- {{cite book |title=Markov Processes: Theorems and Problems |language=en |author-first1=Evgenii Borisovich [Евге́ний Бори́сович] |author-last1=Dynkin [Ды́нкин] |author-link1=Evgenii Borisovich Dynkin |author-first2=Aleksandr Adol'fovich [Александр Адольфович] |author-last2=Yushkevish [Юшкевич] |author-link2=:de:Alexander Adolfowitsch Juschkewitsch |translator-first=James S. |translator-last=Wood |location=University of Moscow, Moscow, Russia |publisher=Plenum Press / Plenum Publishing Corporation |publication-place=New York, US |lccn=69-12529 |date=1969 |orig-date=1966-01-22 |edition=1 |url=https://people.math.harvard.edu/~ctm/home/text/class/harvard/219/21/html/home/sources/dynkin.pdf |access-date=2023-09-03 |url-status=live |archive-url=https://web.archive.org/web/20230906170022/https://people.math.harvard.edu/~ctm/home/text/class/harvard/219/21/html/home/sources/dynkin.pdf |archive-date=2023-09-06}} (x+237 pages) (NB. This is a corrected translation of the first Russian edition published as "{{lang|ru|Теоремы и задачи о процессах Маркова}}" by Nauka Press ({{lang|ru|Наука}}) in 1967 as part of a series on Probability Theory and Mathematical Statistics ({{lang|ru|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.)
- {{cite magazine |title=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" |author-first=Edward "Ed" |author-last=Marlo |author-link=Ed Marlo |location=Chicago, Illinois, US |editor-first=Charles |editor-last=Hudson |department=Card Corner |magazine=The Linking Ring |issn=0024-4023 |publisher=International Brotherhood of Magicians |publication-place=Bluffton, Ohio, US |date=1976-12-01 |volume=56 |number=12 |pages=82, 83, 83, 84, 85–87}}
- {{cite magazine |title=The Kruskal Principle |author-first=Charles |author-last=Hudson |location=Chicago, Illinois, US |department=Card Corner |magazine=The Linking Ring |issn=0024-4023 |publisher=International Brotherhood of Magicians |publication-place=Bluffton, Ohio, US |date=1977-10-01 |volume=57 |number=10 |page=85}}
- {{cite magazine |title=Ten Amazing Mathematical Tricks |author-first=Martin |author-last=Gardner |author-link=Martin Gardner |magazine=Math Horizons |issn=1072-4117 |department=Gardner's Gatherings |publisher=Mathematical Association of America / Taylor & Francis, Ltd. |volume=6 |number=1 |date=September 1998 |jstor=25678174 |pages=13–15, 26}} (4 pages)
- {{cite book |title=Taking Chances: Winning with Probability |author-first=John |author-last=Haigh |chapter=7. Waiting, waiting, waiting: Packs of cards (2) |publication-place=Oxford, UK |publisher=Oxford University Press Inc. |date=1999 |edition=1 |isbn=978-0-19-850291-3 |pages=133–136 |url=https://books.google.com/books?id=SqIsAAAAYAAJ&q=%22packs+of+cards+%282%29%22 |access-date=2023-09-06}} (4 pages); {{cite book |title=Taking Chances: Winning with Probability |author-first=John |author-last=Haigh |chapter=7. Waiting, waiting, waiting: Packs of cards (2) |publication-place=Oxford, UK |publisher=Oxford University Press Inc. |date=2009 |orig-date=2003 |edition=Reprint of 2nd |isbn=978-0-19-852663-6 |pages=139–142 |chapter-url=https://books.google.com/books?id=11NxcwgqdJkC&pg=PA139 |url=https://books.google.com/books?id=11NxcwgqdJkC |access-date=2023-09-03}} (4 of xiv+373+17 pages)
- {{cite book |title=Puzzlers' Tribute: A Feast for the Mind |chapter=A Labyrinth in a Labyrinth |editor-first1=David |editor-last1=Wolfe |editor-link1=David Wolfe (mathematician) |editor-first2=Tom |editor-last2=Rodgers |editor-link2=Thomas Malin Rodgers |author-first=Gordon |author-last=Bean |date=2002 |publisher=CRC Press / Taylor & Francis Group, LLC |edition=1 |isbn=978-1-43986410-4 |pages=103–106 |chapter-url=https://books.google.com/books?id=zUBZDwAAQBAJ&pg=PA103 |url=https://books.google.com/books?id=zUBZDwAAQBAJ}} (xvi+421 pages)
- {{cite journal |title=A Random Walk on a Circular Path |author-first1=Wai-Ki |author-last1=Ching |author-link1=:d:Q102247201 |author-first2=Yiu-Fai |author-last2=Lee |journal={{ill|International Journal of Mathematical Education in Science and Technology|d|Q15765610}} |issn=0020-739X |eissn=1464-5211 |publisher=Taylor & Francis, Ltd. |department=Miscellany |volume=36 |issue=6 |date=September 2005 |orig-date=2004-05-05 |doi=10.1080/00207390500064254 |pages=680–683|s2cid=121692834 }} (4 pages)
- {{cite journal |title=On Convergent Probability of a Random Walk |author-first1=Yiu-Fai |author-last1=Lee |author-first2=Wai-Ki |author-last2=Ching
|author-link2=:d:Q102247201 |date=2006-03-07 |department=Classroom notes |journal={{ill|International Journal of Mathematical Education in Science and Technology|d|Q15765610}} |issn=0020-739X |eissn=1464-5211 |volume=37 |issue=7 |publisher=Taylor & Francis, Ltd. |orig-date=2005-09-29 |doi=10.1080/00207390600712299 |location=Advanced Modeling and Applied Computing Laboratory and Department of Mathematics, The University of Hong Kong, Hong Kong |pages=833–838 |s2cid=121242696 |url=http://www.math.hku.hk/imrwww/IMRPreprintSeries/2006/IMR2006-11.pdf |access-date=2023-09-02 |url-status=live |archive-url=https://web.archive.org/web/20230902190323/http://www.math.hku.hk/imrwww/IMRPreprintSeries/2006/IMR2006-11.pdf |archive-date=2023-09-02}} (6 pages)
- {{cite journal |title=Magic Card Maths |author-first=Steve "Dr. Maths" |author-last=Humble |date=July 2008 |journal=The Montana Mathematics Enthusiast |issn=1551-3440 |volume=5 |number=2 & 3 |publisher=University of Montana |publication-place=Missoula, Montana, US |id=Article 14 |doi=10.54870/1551-3440.1111 |pages=327–336 |s2cid=117632058 |url=https://scholarworks.umt.edu/tme/vol5/iss2/14 |access-date=2023-09-02 |url-status=live |archive-url=https://web.archive.org/web/20230903165030/https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1111&context=tme |archive-date=2023-09-03|doi-access=free }} (1+10 pages)
- {{cite conference |title=How Long Does it Take to Catch a Wild Kangaroo? |author-first1=Ravi |author-last1=Montenegro |author-link1=:d:Q102271025 |author-first2=Prasad V. |author-last2=Tetali |author-link2=Prasad V. Tetali |date=2010-11-07 |orig-date=2009-05-31 |conference=Proceedings of the forty-first annual ACM symposium on Theory of computing (STOC 2009) |doi=10.1145/1536414.1536490 |s2cid=12797847 |arxiv=0812.0789 |pages=553–560 |url=https://faculty.uml.edu/rmontenegro/research/kangaroo-journal.pdf |access-date=2023-08-20 |url-status=live |archive-url=https://web.archive.org/web/20230820105311/https://faculty.uml.edu/rmontenegro/research/kangaroo-journal.pdf |archive-date=2023-08-20}}
- {{cite web |title=Kruskal's Count |author-first=James |author-last=Grime |author-link=:d:Q14949225 |date=2011 |website=singingbanana.com |url=https://www.singingbanana.com/Kruskal.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819090025/https://www.singingbanana.com/Kruskal.pdf |archive-date=2023-08-19}} (8 pages)
- {{cite journal |title=Cards, Codes, and Kangaroos |author-first=Lindsey R. |author-last=Bosko |location=Department of Mathematics, North Carolina State University, Raleigh, North Carolina, US |journal=The UMAP Journal |volume=32 |issue=3 |id=UMAP Unit 808 |series=Modules and Monographs in Undergraduate Mathematics and its Applications (UMAP) Project |pages=199–236 |date=2011 |publisher=Consortium For Mathematics & Its Applications, Inc. (COMAP) |publication-place=Bedford, Massachusetts, US |url=https://stitz.wordpress.ncsu.edu/files/2018/12/kangaroos.pdf |access-date=2023-08-19 |url-status=live |archive-url=https://web.archive.org/web/20230819194127/https://stitz.wordpress.ncsu.edu/files/2018/12/kangaroos.pdf |archive-date=2023-08-19}}
- {{cite web |title=Wikipedia's fixed point |date=2011-05-26 |author-first=Bob |author-last=West |author-link=:d:Q28053587 |work=dlab @ EPFL |publisher=Data Science Lab, École Polytechnique Fédérale de Lausanne |publication-place=Lausanne, Switzerland |url=https://dlab.epfl.ch/2011-05-26-wikipedias-fixed-point/ |access-date=2023-09-04 |url-status=live |archive-url=https://web.archive.org/web/20220523101801/https://dlab.epfl.ch/2011-05-26-wikipedias-fixed-point/ |archive-date=2022-05-23 |quote=[...] it turns out there is a card trick that works exactly the same way. It's called the "Kruskal Count" [...]}}
- {{cite news |title=Mathematics in the Streets of Kraków |editor-first=Ehrhard |editor-last=Behrends |editor-link=:de:Ehrhard Behrends |author-first=Steve "Dr. Maths" |author-last=Humble |location=Kraków, Poland |date=September 2012 |orig-date=2012-07-02 |issue=85 |newspaper=EMS Newsletter |issn=1027-488X |publisher=EMS Publishing House / European Mathematical Society |publication-place=Zürich, Switzerland |pages=20–21 [21] |url=https://www.ems-ph.org/journals/newsletter/pdf/2012-09-85.pdf |access-date=2023-09-02 |url-status=live |archive-url=https://web.archive.org/web/20230902182119/https://www.ems-ph.org/journals/newsletter/pdf/2012-09-85.pdf |archive-date=2023-09-02 |quote-page=21 |quote=[...] The Kruscal count [...]}} [https://web.archive.org/web/20230902172632/http://page.mi.fu-berlin.de/bhrnds/publ_papers/nl2012-09-85_1.pdf] (2 pages)
- {{cite conference |title=Instruction-Level Steganography for Covert Trigger-Based Malware |author-first1=Dennis |author-last1=Andriesse |author-first2=Herbert |author-last2=Bos |author-link2=:d:Q56565972 |date=2014-07-10 |editor-first=Sven |editor-last=Dietrich |conference=11th International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment (DIMVA) |publisher=Springer International Publishing |series=Lecture Notes in Computer Science |publication-place=Egham, UK; Switzerland |location=Vrije Universiteit Amsterdam, Amsterdam, Netherlands |doi=10.1007/978-3-319-08509-8_3 |s2cid=4634611 |id=LNCS 8550 |issn=0302-9743 |eissn=1611-3349 |isbn=978-3-31908508-1 |pages=41–50 [45] |url=https://www.cs.vu.nl/~herbertb/papers/stega_dimva14.pdf |access-date=2023-08-26 |url-status=live |archive-url=https://web.archive.org/web/20230826135254/https://www.cs.vu.nl/~herbertb/papers/stega_dimva14.pdf |archive-date=2023-08-26}} (10 pages)
- {{cite book |title=Kruskal's Principle and Collision Time for Monotone Transitive Walks on the Integers |author-first1=Ravi |author-last1=Montenegro |author-link1=:d:Q102271025 |author-first2=Prasad V. |author-last2=Tetali |author-link2=Prasad V. Tetali |date=2014-09-07 |url=https://tetali.math.gatech.edu/PUBLIS/MT14.pdf |access-date=2023-08-22 |url-status=live |archive-url=https://web.archive.org/web/20230822002748/https://tetali.math.gatech.edu/PUBLIS/MT14.pdf |archive-date=2023-08-22}} (18 pages)
- {{cite conference |title=Collision of Random Walks and a Refined Analysis of Attacks on the Discrete Logarithm Problem |author-first1=Shuji |author-last1=Kijima |author-first2=Ravi |author-last2=Montenegro |author-link2=:d:Q102271025 |editor-first=Jonathan |editor-last=Katz |editor-link=Jonathan Katz (computer scientist) |date=2015-03-15 |orig-date=2015-03-30/2015-04-01 |conference=Proceedings of the 18th IACR International Conference on Practice and Theory in Public-Key Cryptography |location=Gaithersburg, Maryland, US |publisher=International Association for Cryptologic Research / Springer Science+Business Media |publication-place=Berlin & Heidelberg, Germany |series=Lecture Notes in Computer Science |id=LNCS 9020 |isbn=978-3-662-46446-5 |doi=10.1007/978-3-662-46447-2_6 |pages=127–149 |url=https://www.iacr.org/archive/pkc2015/90200136/90200136.pdf |access-date=2023-09-03 |url-status=live |archive-url=https://web.archive.org/web/20230903083941/https://www.iacr.org/archive/pkc2015/90200136/90200136.pdf |archive-date=2023-09-03}} (23 pages)
- {{cite web |title=PDCA and the Roads to Rome: Can a lean purist and a Six Sigma purist reach the same answer to a problem? |author-first=Harish |author-last=Jose |department=Lean |date=2016-06-14 |orig-date=2016-06-02 |url=https://www.linkedin.com/pulse/pdca-roads-rome-harish-jose |access-date=2023-09-07 |url-status=live |archive-url=https://web.archive.org/web/20230907192316/https://www.linkedin.com/pulse/pdca-roads-rome-harish-jose |archive-date=2023-09-07}} [https://web.archive.org/web/20230907192503/https://harishsnotebook.wordpress.com/tag/pdca/][https://web.archive.org/web/20221130232206/https://www.qualitydigest.com/inside/lean-column/pdca-and-roads-rome-061416.html]
- {{cite book |author-first1=Daniel |author-last1=Lamprecht |author-first2=Dimitar |author-last2=Dimitrov |author-first3=Denis |author-last3=Helic |author-first4=Markus |author-last4=Strohmaier |title=Proceedings of the 12th International Symposium on Open Collaboration |chapter=Evaluating and Improving Navigability of Wikipedia: A Comparative Study of Eight Language Editions |publisher=Association for Computing Machinery |date=2016-08-17 |pages=1–10 |location=OpenSym, Berlin, Germany |isbn=978-1-4503-4451-7 |doi=10.1145/2957792.2957813 |s2cid=13244770 |url=http://www.daniellamprecht.com/wp-content/uploads/2016/08/Evaluating-and-Improving-Navigability-of-Wikipedia-a-Comparative-Study-of-eight-Language-Editions.pdf |access-date=2021-03-17 |url-status=live |archive-url=https://web.archive.org/web/20230904213930/https://www.daniellamprecht.com/wp-content/uploads/2016/08/Evaluating-and-Improving-Navigability-of-Wikipedia-a-Comparative-Study-of-eight-Language-Editions.pdf |archive-date=2023-09-04}}
- {{cite book |title=On Offensive and Defensive Methods in Software Security |author-last=Jämthagen |author-first=Christopher |date=November 2016 |issn=1654-790X |number=89 |isbn=978-91-7623-942-1 |type=Thesis |publisher=Department of Electrical and Information Technology, Lund University |publication-place=Lund, Sweden |page=96 |url=https://lucris.lub.lu.se/ws/portalfiles/portal/15764406/dissertation.pdf |access-date=2023-08-26 |url-status=live |archive-url=https://web.archive.org/web/20230826135321/https://lucris.lub.lu.se/ws/portalfiles/portal/15764406/dissertation.pdf |archive-date=2023-08-26}} (1+xvii+1+152 pages)
- {{cite journal |title=Sensorless Pose Determination Using Randomized Action Sequences |author-first1=Pragna |author-last1=Mannam |author-first2=Alexander |author-last2=Volkov, Jr. |author-first3=Robert |author-last3=Paolini |author-first4=Gregory Scott |author-last4=Chirikjian |author-link4=Gregory Scott Chirikjian |author-first5=Matthew Thomas |author-last5=Mason |author-link5=Matthew Thomas Mason |journal=Entropy |issn=1099-4300 |publisher=Multidisciplinary Digital Publishing Institute |publication-place=Basel, Switzerland |date=2019-02-06 |orig-date=2018-12-04 |volume=21 |number=2 |page=154 |id=Article 154 |doi=10.3390/e21020154 |s2cid=54444590 |arxiv=1812.01195 |pmc=7514636 |pmid=33266870 |bibcode=2019Entrp..21..154M |quote-page=2 |quote=[...] 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". [...] |doi-access=free }} (13 pages)
- {{cite journal |title=Constructions and bounds for codes with restricted overlaps |author-first1=Simon Robert |author-last1=Blackburn |author-first2=Navid Nasr |author-last2=Esfahani |author-first3=Donald Lawson |author-last3=Kreher |author-first4=Douglas "Doug" Robert |author-last4=Stinson |author-link4=Douglas Robert Stinson |date=2023-08-22 |orig-date=2022-11-18 |journal=IEEE Transactions on Information Theory |arxiv=2211.10309}} (17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)
External links
- {{cite web |title=Dr. Maths Randomness Show |author-first=Steve "Dr. Maths" |author-last=Humble |website=YouTube |location=Alchemist Cafe, Dublin, Ireland |date=2010 |type=Video |url=http://www.youtube.com/watch?v=goIOWcnagP0 |access-date=2023-09-05}} [23:40]
- {{cite web |title=Mathematical Card Trick Source |date=2015–2017 |work=GeniiForum |department=Close-Up Magic |url=https://forums.geniimagazine.com/viewtopic.php?t=46856 |access-date=2023-09-05 |url-status=live |archive-url=https://web.archive.org/web/20230904221639/https://forums.geniimagazine.com/viewtopic.php?t=46856 |archive-date=2023-09-04}}
- {{cite web |title=Kruskal Principle |editor-first=Denis |editor-last=Behr |date=2023 |work=Conjuring Archive |url=https://www.conjuringarchive.com/list/category/1808 |access-date=2023-09-10 |url-status=live |archive-url=https://web.archive.org/web/20230910083843/https://www.conjuringarchive.com/list/category/1808 |archive-date=2023-09-10}}
Category:Recreational mathematics