On Mon, Jun 11, 2012 at 4:28 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This must work. Figure out what will happen if no bullets were ever fired again: there would be some number of annihilations, followed by an unending era during which more distant bullets are receding faster. Then, just "mark time" until all the annihilations have happened, and then fire a salvo of bullets at a speed greater than that of the current fastest, with the number of bullets in the salvo just sufficient to zap all the bullets that are left. Then mark time again. Why is this not a proof?
Yes, certainly that works. I was just pointing out that you can't necessarily fire n = # living bullets right away, which is what I thought Warren had proposed. So to bring things back closer to David's original question, these constructions prove that the probability of the current rightmost bullet getting replaced eventually is nonzero. Can anyone prove that that probability is 1? Or might there be some configuration of bullets in which we expect the current rightmost bullet to survive forever? --Michael
On Mon, Jun 11, 2012 at 4:12 PM, Michael Kleber <michael.kleber@gmail.com
wrote:
But I think he is right. I think a simpler proof is the next n shots all use exactly the maximum allowed speed. This kills the previous n bullets assuming none used exactly max speed. Then the next M-n shots all are arranged to annihilate in chronologically-successive pairs.
No, that's not correct: some of the n currently-living bullets may annihilate each other before these speed=1 (or 1-e) bullets reach them. But as I said before, there is certainly *some* number of bullets you can fire which will make this plan work.
--Michael
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