Advertisement

The Kruskal Count

  • Jeffrey C. Lagarias
  • Eric RainsEmail author
  • Robert J. Vanderbei
Chapter
Part of the Studies in Choice and Welfare book series (WELFARE)

The Kruskal Count is a card trick invented by Martin D. Kruskal (who is well known for his work on solitons) which is described in Fulves and Gardner (1975) and Gardner (1978, 1988). In this card trick a magician “guesses” one card in a deck of cards which is determined by a subject using a special counting procedure that we call Kruskal's counting procedure. The magician has a strategy which with high probability will identify the correct card, explained below.

Kruskal's counting procedure goes as follows. The subject shuffles a deck of cards as many times as he likes. He mentally chooses a (secret) number between one and ten. The subject turns the cards of the deck face up one at a time, slowly, and places them in a pile. As he turns up each card he decreases his secret number by one and he continues to count this way till he reaches zero. The card just turned up at the point when the count reaches zero is called the first key card and its value is called the first key number. Here the value of an Ace is one, face cards are assigned the value five, and all other cards take their numerical value. The subject now starts the count over, using the first key number to determine where to stop the count at the second key card. He continues in this fashion, obtaining successive key cards until the deck is exhausted. The last key card encountered, which we call the tapped card, is the card to be “guessed” by the magician.

Keywords

Markov Chain Failure Probability Success Probability Coupling Method Geometric Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldous, D.,&Diaconis P. (1986). Shuffling cards and stopping times. American Mathematical Monthly, 93, 333–348.CrossRefGoogle Scholar
  2. Diaconis, P. (1988). Group representations in probability and statistics, IMS Lecture Notes — Mono graph Series No. 11. Hayward, CA: Institute of Mathematical Statistics.Google Scholar
  3. Doeblin, W. (1938). Exposé de la theorie des chaines simple constantes de Markov á un nombre fini d'etats. Revue Mathematique de l'Union Interbalkanique, 2, 77–105.Google Scholar
  4. Donsker, M. D.,&Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time I. Communications on Pure and Applied Mathematics, 28, 1–47.Google Scholar
  5. Fulves, C.,&Gardner M. (1975). The Kruskal principle. The Pallbearer's Review, Vol. 10 No. 8 (June), 967–976.Google Scholar
  6. Gardner, M. (1978). Mathematical games. Scientific American, 238(2), 19–32.CrossRefGoogle Scholar
  7. Gardner, M. (1988). From Penrose tiles to Trapdoor ciphers. New York: W. H. Freeman (Chap. 19).Google Scholar
  8. Griffeath, D. (1978). Coupling methods for Markov processes. In G. C. Rota (Ed.) Studies in probability and ergodic theory (pp. 1–43). New York: Academic.Google Scholar
  9. Haga, W.,&Robins, S. (1997). On Kruskal's principle. In J. Borwein, P. Borwein, L. Jorgenson,&R. Corless (Eds.) Organic mathematics, Canadian Mathematical Society Conference Proceedings (Vol. 20, pp. 407–412). Providence, RI: AMS.Google Scholar
  10. Iscoe, I., Ney, P.,&Nummelin E. (1985) Large devations of uniformly recurrent Markov additive processes. Advances in Applied Mathematics, 6, 373–412.CrossRefGoogle Scholar
  11. Mallows, C. L. (1975). On a probability problem suggested by M. D. Kruskal's card trick. Bell Laboratories memorandum, April 18, 1975 (unpublished)Google Scholar
  12. Marshall, A. W.,&Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic.Google Scholar
  13. Miller, H. D. (1961). A convexity property in the theory of random variables defined on a finite Markov chain. Annals of Mathematical Statistics, 32, 1260–1270.CrossRefGoogle Scholar
  14. Ney, P.,&Nummelin, E. (1987a). Markov Additive processes I. Eigenvalue properties and Limit theorems. Annals of Probability, 15, 561–592.CrossRefGoogle Scholar
  15. Ney, P.,&Nummelin, E. (1987b). Markov Additive processes II. Large deviations Annals of Prob ability, 15, 593–609.Google Scholar
  16. Pitman, J. (1976). On coupling of Markov Chains. Zeitschrift fur Wahrscheinlichkeitheorie, 35, 315–322.CrossRefGoogle Scholar
  17. Varadhan, S. R. S. (1984). Large deviations and applications. In CBMS-NSF Regional Conference No. 46. SIAM.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jeffrey C. Lagarias
    • 1
  • Eric Rains
    • 2
    Email author
  • Robert J. Vanderbei
    • 3
  1. 1.Department of MathematicsUniversity of MichiganAnnArborUSA
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations