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This is a follow up question this one on MSE, which can basically be summarised as Robert Abilock originally posed in American Monthly in 1967:

The Rifle-Problem:

$n$ riflemen are distributed at random points on a plane. At a signal, each one shoots at and kills his nearest neighbor. What is the expected number of riflemen who are left alive?

In my answer, I cited two apparently conflicting references

    Vicious neighbor problem [R.Tao and F.Y.Wu; 1986], where the answer of $\approx 0.284051 n$ remaining riflemen was given as the solution in $2$ dimensions.

and

    Mathematical Constants: Nearest-neighbor graphs [S.R.Finch; 2008], where Finch states that

In [Vicious neighbor problem], the absolute value signs in the definitions of $\varphi$ and $\psi$ were mistakenly omitted.)$\dots$

I have tried to replicate even the partial results in Tao/Wu's paper (despite leaving out the absolute values of $\varphi$ and $\psi,$) leaving me unsure as to whether I am missing something in my "translation" of the problem into Finch's more modern notation. I should be most grateful if someone could illuminate me further in this matter.

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  • $\begingroup$ You write random points on a plane in the first source, and the second source mention a unit cube. Can the discrepancy depend on the shape of the bounded region where the shooters are? $\endgroup$
    – Per Alexandersson
    Commented Oct 22, 2015 at 23:34
  • $\begingroup$ @PerAlexandersson the Tao/Wu paper disregards boundary conditions, but as far as I am aware, in $3$ dimensions, they give $\operatorname{E}[n]/n \approx 0.303,$ which Finch doesn't dispute. $\endgroup$
    – martin
    Commented Oct 22, 2015 at 23:38
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    $\begingroup$ So I did a python simulation, in which there were 20 shooters uniformly distributed in the unit square together with 1 at the centre. In 1000000 trials, the guy at the centre survived 28.1% of the time. This should be within about $\pm 0.1$% of the correct value if there are 20+1 shooters. (by working with a shooter at the centre, we're avoiding edge effects). I might try increasing 20 to see if anything changes... $\endgroup$
    – Anthony Quas
    Commented Oct 23, 2015 at 0:55
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    $\begingroup$ With 40+1 shooters, the central guy survived with probability 28.4%$\pm 0.1$%. $\endgroup$
    – Anthony Quas
    Commented Oct 23, 2015 at 18:06
  • $\begingroup$ good question, an interesting side-glance, how many rounds are fired before a stalemate? $\endgroup$
    – JMP
    Commented Oct 28, 2015 at 15:37

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