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bonanova added an answer to a question Random bullets
I'm not sure which was fired first but either way you can see that two bullets survive.
.4 .8 .6 .2
Assuming .2 was fired first  that they will travel to the right..
.6 will catch .2 (before .8 catches .6, owing to the greater speed difference between .6 and .2 ) and they annihilate.
That leaves .4 and .8 to survive.
Assuming .4 was fired first  that they will travel to the left
.8 catches .4 and they annihilate.
That leaves .6 and .2 to survive.
Assigning expected speeds allows you to calculate collision times, and thus annihilation condition, once for each speed permutation, instead of a million times using random values. It may, but I have been unsuccessful, lead to a derivation of the formula (at least for small n.)
Maybe it's still not clear. You just enumerate and analyze the speed permutations once, and you're done.

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bonanova added an answer to a question Random bullets
I disclaim authorship, since BMAD posed the "infinite bullet" version here last year, and I read the finite bullet version elsewhere, along with the conjectured solution. I share Captain Ed's amazement at it.
Here is my contribution. Change random speeds into expected speeds. That eliminates need for simulation. There is a neat result that says for n random speed [0, 1] bullets the expected speeds are i/(n+1). The order they occur defines all the representative cases. And then you're done. (In this approach you know values in addition to ordering.) Although for large n you're back to millions of cases. For small n, though, maybe that formula can be derived. I haven't succeeded yet.
Clarifying:
The expected speed of the fastest of n bullets is n/(n+1)
The expected speed of the slowest of n bullets is 1/(n+1)
The expected speed of the ithfastest bullet is i/(n+1)

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bonanova added an answer to a question Random bullets
There is a conjecture for the probability p0(n) where n=2m for all n bullets to be annihilated. I do not have a proof. Now that there are two programs that are giving the answers for n=4 and n=10 that agree with p0(4) and with p0(10) obtained from the conjectured formula, I think I will give the formula. It can be tested against simulations for other values of n, and its simple form may suggest a line of attack to derive it analytically.

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bonanova added an answer to a question Double up!
I followed most of this on first read. Nicely done and clearly commented. Thanks.
I'd like to compile and run it. What do I need for that?

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bonanova added an answer to a question Random bullets
dgreening points out that the same can be said if the last bullet is slowest.

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bonanova added an answer to a question Minimize the perimeter
I just thought that a generalization of "perimeter" to mean the length of the boundary between the interior of a shape and its exterior would give

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bonanova added an answer to a question Double up!
@bubbled, Nice job. Would love to see the code. Actually I'm trying to learn python.
@araver, belatedly your gold star!

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bonanova added an answer to a question Badminton
Yeah, my reasoning was wrong. One can say that matches are more likely to end with one of the scores. Julie's half of those matches are then more numerous than her half of the matches that end in the other score. Nice little puzzle.

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bonanova added an answer to a question Random bullets
That's a good result. Curious, what language did you do your simulation in?

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bonanova added a question in New Logic/Math Puzzles
Bent octagonA convex octagon completely contains all 28 of its diagonals. But if you move the vertices around you can make portions of some, but not all, of them pass through its exterior. What is the largest number of such diagonals? Equivalently, how many diagonals of an octagon must remain totally in its interior?
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bonanova added an answer to a question Minimize the perimeter
Thinking about this ... One might make the case for three different values of perimeter for three different placements of the "hole," yielding a distinct best answer.

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bonanova added an answer to a question Random bullets
Yes, this one has a definitive answer. I'm not sure consensus was reached on the infinite puzzle.

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bonanova added a question in New Logic/Math Puzzles
Minimize the perimeterA child's game uses twentyone unique shapes that comprise from one to five squares. This puzzle asks how tightly can the shapes be packed without overlap so as to achieve a figure with the smallest perimeter? The individual shapes may be rotated and turned over (reflected) as desired.
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bonanova added an answer to a question Double up!
araver has it.

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bonanova added an answer to a question dividing infinity by 17
I was going to ask which direction the exponentials are evaluated.
But obviously 3 is first raised to the 4th power; that result is then raised to the 5th power; and so on.
Only finite power towers can be evaluated right to left.

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