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THE OK CORRAL AND THE POWER OF THE LAW (A CURIOUS POISSON-KERNEL FORMULA FOR A PARABOLIC EQUATION)

Published online by Cambridge University Press:  01 March 1998

DAVID WILLIAMS  and
PAUL McILROY
DAVID WILLIAMS
Affiliation:
School of Mathematical Sciences, Bath University, Bath BA2 7AY
PAUL McILROY
Affiliation:
Synectics Group, Systems Research Unit, British Telecom Laboratories, Ipswich IP5 3RE
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Abstract

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ(m, n) be the expected number of survivors. Clearly, we have boundary conditions:

μ(m, 0)=m, μ(0, n)=n. (1.1)

We also have the equation

formula here

This is because the probability that the first successful shot is made by the side with m gunmen is m/(m+n). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which

m=8192+k, n=8192−k, d(m, n) =√(m2−n2 =128√(2k).

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 2 , March 1998 , pp. 166 - 170
Copyright
© The London Mathematical Society 1998

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