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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aldous, D.,&Diaconis P. (1986). Shuffling cards and stopping times. American Mathematical Monthly, 93, 333–348.
Diaconis, P. (1988). Group representations in probability and statistics, IMS Lecture Notes — Mono graph Series No. 11. Hayward, CA: Institute of Mathematical Statistics.
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.
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.
Fulves, C.,&Gardner M. (1975). The Kruskal principle. The Pallbearer's Review, Vol. 10 No. 8 (June), 967–976.
Gardner, M. (1978). Mathematical games. Scientific American, 238(2), 19–32.
Gardner, M. (1988). From Penrose tiles to Trapdoor ciphers. New York: W. H. Freeman (Chap. 19).
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.
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.
Iscoe, I., Ney, P.,&Nummelin E. (1985) Large devations of uniformly recurrent Markov additive processes. Advances in Applied Mathematics, 6, 373–412.
Mallows, C. L. (1975). On a probability problem suggested by M. D. Kruskal's card trick. Bell Laboratories memorandum, April 18, 1975 (unpublished)
Marshall, A. W.,&Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic.
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.
Ney, P.,&Nummelin, E. (1987a). Markov Additive processes I. Eigenvalue properties and Limit theorems. Annals of Probability, 15, 561–592.
Ney, P.,&Nummelin, E. (1987b). Markov Additive processes II. Large deviations Annals of Prob ability, 15, 593–609.
Pitman, J. (1976). On coupling of Markov Chains. Zeitschrift fur Wahrscheinlichkeitheorie, 35, 315–322.
Varadhan, S. R. S. (1984). Large deviations and applications. In CBMS-NSF Regional Conference No. 46. SIAM.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lagarias, J.C., Rains, E., Vanderbei, R.J. (2009). The Kruskal Count. In: Brams, S.J., Gehrlein, W.V., Roberts, F.S. (eds) The Mathematics of Preference, Choice and Order. Studies in Choice and Welfare. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79128-7_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-79128-7_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79127-0
Online ISBN: 978-3-540-79128-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)