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).