[math-fun] David Wilson's 1-D gun problem, cropping up again and again
In case anyone is tracing the instances of it cropping up: the "1-dimensional gun" model, which David Wilson first posted about here in June of 2012, has come up again. I think this is among the best open problems ever discussed here. Recall the model: a gun at x=0 fires a bullet along the +x axis at time t=0,1,2,3,... with velocity chosen uniformly at random from [0,1]. If two bullets meet, then they annihilate each other. David's original question was whether the speed of the rightmost bullet tends to 1 over time, but given this overall model there are lots of interesting questions to ask. IBM Research's "Ponder This" from May 2014 asked about the finite case, specifically the probability that if you fire 20 bullets, they all annihilate: https://www.research.ibm.com/haifa/ponderthis/challenges/May2014.html The "solution" was posted there, claiming a formula for all even n, which appears to be empirically correct but was not presented with a correct proof. On Quora, someone asked, maybe in December 2014 (?), for the infinite-bullets case, for "probability that a bullet would reach infinity", without putting any work into defining what that means: https://www.quora.com/A-gun-shoots-bullets-towards-infinity-at-a-constant-ra... Then on Reddit earlier this month, someone asked for "the probability that at least one bullet escapes to infinity if you fire n bullets", with also a nod at the n->infinity case: https://www.reddit.com/r/mathriddles/comments/3sa4ci/colliding_bullets/ The comments there include an interesting observation or two in among lots of junk. Two days later it hopped to Math StackExchange http://math.stackexchange.com/questions/1526292/colliding-bullets The top comment there offers the same formula as in the IBM soluiton for the finite-n probability of all bullets annihilating, but again without a solid proof. --Michael -- Forewarned is worth an octopus in the bush.
It's scary the things I think of when I'm lying in bed staring into the darkness...
Just to chime in: This was, as usual for David, a terrific question. After we batted it around a bit in 2012, I asked it separately of two highly respected probabilists I know, and they each thought it was a great question but could not solve it. —Dan
On Nov 27, 2015, at 6:27 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
In case anyone is tracing the instances of it cropping up: the "1-dimensional gun" model, which David Wilson first posted about here in June of 2012, has come up again. I think this is among the best open problems ever discussed here.
Recall the model: a gun at x=0 fires a bullet along the +x axis at time t=0,1,2,3,... with velocity chosen uniformly at random from [0,1]. If two bullets meet, then they annihilate each other. David's original question was whether the speed of the rightmost bullet tends to 1 over time, but given this overall model there are lots of interesting questions to ask.
IBM Research's "Ponder This" from May 2014 asked about the finite case, specifically the probability that if you fire 20 bullets, they all annihilate: https://www.research.ibm.com/haifa/ponderthis/challenges/May2014.html The "solution" was posted there, claiming a formula for all even n, which appears to be empirically correct but was not presented with a correct proof.
On Quora, someone asked, maybe in December 2014 (?), for the infinite-bullets case, for "probability that a bullet would reach infinity", without putting any work into defining what that means: https://www.quora.com/A-gun-shoots-bullets-towards-infinity-at-a-constant-ra...
Then on Reddit earlier this month, someone asked for "the probability that at least one bullet escapes to infinity if you fire n bullets", with also a nod at the n->infinity case: https://www.reddit.com/r/mathriddles/comments/3sa4ci/colliding_bullets/ The comments there include an interesting observation or two in among lots of junk.
Two days later it hopped to Math StackExchange http://math.stackexchange.com/questions/1526292/colliding-bullets The top comment there offers the same formula as in the IBM soluiton for the finite-n probability of all bullets annihilating, but again without a solid proof.
--Michael
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Dan Asimov -
David Wilson -
Michael Kleber