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Four darts


Best Answer Rob_Gandy, 21 March 2013 - 09:25 PM

Spoiler for My thoughts
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11 replies to this topic

#1 bonanova

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Posted 18 March 2013 - 03:40 PM

This puzzle was discussed at length a few years ago in this forum.
 
It was rich enough that I thought it deserved another airing, with a largely
different set of puzzle solvers. I can't find it using search, to know for sure,
but I'm wondering whether Prime was one of the solvers last time.
 
For those to whom this is a new puzzle, I hope you will enjoy it.
It's simple to state, and challenging to solve.
 
I'm pretty poor at darts, but I managed to put three of them on the board.
Assuming my fourth dart also hits the board [circular], then
 
  • what is the probability it will lie inside the triangle formed by the first three?
  • what is the probability the four darts will not form a convex quadrilateral?
 
Spoiler for clarification

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#2 Prime

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Posted 20 March 2013 - 10:58 PM

I recall vaguely, trying to solve that problem. I seem to recall we did not come to a consensus back then. It seems like a serious math problem. And we were trying to solve only the question 2 there.


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Past prime, actually.


#3 Prime

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Posted 21 March 2013 - 02:22 AM

Since question 1 was not a part of that problem posted in 2008, let me provide a solution to that one real quick.

Spoiler for probability of 4th dart inside

Now, that question 1 is solved, I am inclined to yield the opportunity of solving question 2 to others.


Edited by Prime, 21 March 2013 - 02:23 AM.

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Past prime, actually.


#4 bushindo

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Posted 21 March 2013 - 04:54 PM

Since question 1 was not a part of that problem posted in 2008, let me provide a solution to that one real quick.

Spoiler for probability of 4th dart inside

Now, that question 1 is solved, I am inclined to yield the opportunity of solving question 2 to others.

 

I think the proof is fine as it is, but I believe there are a few typos. See below

Spoiler for

 

Question for bonanova: about question 1- the probability that the 4th point falls within the triangle,

Spoiler for


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#5 Prime

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Posted 21 March 2013 - 07:48 PM

Since question 1 was not a part of that problem posted in 2008, let me provide a solution to that one real quick.

Spoiler for probability of 4th dart inside

Now, that question 1 is solved, I am inclined to yield the opportunity of solving question 2 to others.

 

I think the proof is fine as it is, but I believe there are a few typos. See below

Spoiler for

 

Question for bonanova: about question 1- the probability that the 4th point falls within the triangle,

Spoiler for

Yes, I muddled my explanation. Of course, I meant not forming convex quad, or forming concave quad. But I did mean the point inside a triangle. Perhaps, I should have mentioned re-drawing the lines connecting the points.

At any rate, my post solves the relation between question 1 and question 2.

The straightforward approach with sixtuple integral seems too complex and more a numeric than analytical solution. As far as I remember, Bonanova had done numeric solution back when, by runnig computer simulations.


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Past prime, actually.


#6 Rob_Gandy

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Posted 21 March 2013 - 09:25 PM   Best Answer

Spoiler for My thoughts

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#7 Prime

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Posted 21 March 2013 - 10:11 PM

Spoiler for My thoughts

If we know the average area of a randomly drawn triangle inside a unit circle, then the problem is solved. How do you find the average area of triangle? Where does 35/48Pi come from?


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Past prime, actually.


#8 Rob_Gandy

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Posted 21 March 2013 - 10:30 PM

Spoiler for My thoughts

If we know the average area of a randomly drawn triangle inside a unit circle, then the problem is solved. How do you find the average area of triangle? Where does 35/48Pi come from?

http://mathworld.wol...glePicking.html


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#9 Prime

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Posted 21 March 2013 - 10:32 PM

The history of this problem as I recall it.

I suggested first solving the average distance between two points inside the unit circle. Then solving the average distance from a point to a segment inside the unit circle. Thus finding the average area of triangle and ultimately solving the problem.

I don't recall that anyone coming up with a solid analytical solution. Bonanova ran computer simulations providing the numeric answer and came up with analysis, which I did not quite follow at the time.

There was some confusion about the 4th point inside the triangle formed by the first 3 points, versus concave quadrilateral. I belive I resolved that question this time around in the post #3 with corrections by Bushindo in post #4.


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Past prime, actually.


#10 Prime

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Posted 21 March 2013 - 10:39 PM

 


Spoiler for My thoughts

If we know the average area of a randomly drawn triangle inside a unit circle, then the problem is solved. How do you find the average area of triangle? Where does 35/48Pi come from?

http://mathworld.wol...glePicking.html

But that's not solving the puzzle, that's finding an answer on the internet. I don't understand 5-tuple integrals and would have to study to verify that solution.

It is educational. However, to discover something of our own, I'd look for a simpler more understandable solution.


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Past prime, actually.





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