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# Four darts

### #1

Posted 18 March 2013 - 03:40 PM

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?

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

- Bertrand Russell

### #2

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.

Past prime, actually.

### #3

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.

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.**

Past prime, actually.

### #4

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 insideNow, 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

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

### #5

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 insideNow, 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

__a triangle. Perhaps, I should have mentioned re-drawing the lines connecting the points.__

*inside*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.

Past prime, actually.

### #6

Posted 21 March 2013 - 09:25 PM Best Answer

### #7

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?

Past prime, actually.

### #8

Posted 21 March 2013 - 10:30 PM

Spoiler for My thoughtsIf 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

### #9

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.

Past prime, actually.

### #10

Posted 21 March 2013 - 10:39 PM

Spoiler for My thoughtsIf 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?

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.

Past prime, actually.

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