Skip to main content Accessibility help
×
Home

On the N-tower problem and related problems

  • F. Thomas Bruss (a1), Guy Louchard (a1) and John W. Turner (a1)

Abstract

Consider N towers each made up of a number of counters. At each step a tower is chosen at random, a counter removed which is then added to another tower also chosen at random. The probability distribution for the time needed to empty one of the towers is obtained in the case N = 3. Arguments are set forward as to why no simple formulae can be expected for N > 3. An asymptotic expression for the mean time before one of the towers becomes empty is derived in the case of four towers when they all initially contain a comparably large number of counters. We then study related problems, in particular the ruin problem for three players. Here we use simple martingale methodology as well as a solution proposed by T. S. Ferguson for a slightly modified problem. Throughout the paper it is our main objective to shed light on the reasons why the case N > 3 is so substantially different from the case N ≤ 3.

Copyright

Corresponding author

Postal address: Université Libre de Bruxelles, Département de Mathématique, CP 210 Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗∗ Postal address: Université Libre de Bruxelles, Département d'Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: louchard@ulb.ac.be
∗∗∗ Postal address: Université Libre de Bruxelles, Département de Physique, CP 231, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.

References

Hide All
[1] Benedetti, R. and Petronio, C. (1992). Lectures on Hyperbolic Geometry. Springer, Berlin.
[2] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.
[3] Bruss, F. T. (1996). A note on Ferguson's ruin problem. Tech. Rep., ISRO, Université Libre de Bruxelles.
[4] Engel, A. (1993). The computer solves the three tower problem. Amer. Math. Monthly 100, 6264.
[5] Feller, W. (1968). Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.
[6] Ferguson, T. S. (1995). Gambler's ruin in three dimensions. Unpublished manuscript. Available at http://www.math.ucla.edu/simtom/.
[7] Itô, K. and McKean, H. P. Jr. (1974). Diffusion Processes and their Sample Paths, 2nd printing. Springer, Berlin.
[8] Lavrentiev, M. A. and Chabat, B. V. (1972). Méthodes de la théorie des fonctions d'une variable complexe. Éditions Mir, Moscow.
[9] Lévy, P., (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.
[10] Li, S. Y. R. (1980). A martingale approach to the study of the occurrence of patterns in repeated experiments. Ann. Prob. 8, 11711175.
[11] Sansone, G. and Gerretsen, J. (1969). Lectures on the Theory of Functions of a Complex Variable. II: Geometric Theory. Wolters-Noordhoff, Groningen.
[12] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, D., Princeton, NJ.
[13] Stirzaker, D. (1994). Tower problems and martingales. Math. Scientist 19, 5259.
[14] Turner, J. W. (1984). On the quantum particle in a polyhedral box. J. Phys. A 17, 27912797.

Keywords

MSC classification

Related content

Powered by UNSILO

On the N-tower problem and related problems

  • F. Thomas Bruss (a1), Guy Louchard (a1) and John W. Turner (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.