Follow-up, though nobody paid any attention the first time: Oded Margalit's "Ponder This" question about the 1-dimensional gun was to be answered at the beginning of June, but he has extended it: http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/Solutions/May2014.html "The answer for the general n bullet problem is zero if n is odd (n=2m+1) and \Pi_{i=1}^m ((2*i-1)/(2*i)) when n is even (n=2m). So for n=20, the answer is (1*3*5*7*9*11*13*15*17*19) / (2*4*6*8*10*12*14*16*18*20) = 46189/262144 = 0.1761970520. However, it is not easy to prove this, and several solvers sent us incorrect proofs." This is the probability that if you shoot n bullets, they will all mutually annihilate. I've seen several incorrect proofs of the correct closed form (including for exactly k of n bullets surviving), but no actual proofs. Anyone? --Michael On Tue, May 6, 2014 at 10:50 AM, Michael Kleber <michael.kleber@gmail.com> wrote:
Hey Funsters,
Oded Margalit's "Ponder This" puzzle this month is a bit about the "1-Dimensional Gun" model that David Wilson introduced here in June 2012. I told Oded about it when we were both speaking at the ITA workship in San Diego in February.
http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/Challenges/May2014.html
The question Oded asked:
"Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1.
The speeds of the bullets are independent uniform random variables. Each bullet keeps the exact same speed and when two bullets collide, they are both annihilated.
After shooting 20 bullets, what is the probability that eventually all the bullets will be annihilated?"
Oded has a closed form for the probability of all bullets being annihilated when you shoot 2n of them, but not, I think, the probability of 2k>0 surviving.
--Michael
-- Forewarned is worth an octopus in the bush.
-- Forewarned is worth an octopus in the bush.