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Three Points in a Square (Posted on 2008-03-17) Difficulty: 3 of 5
Three points are chosen at random inside a square. Each point is chosen by choosing a random x-coordinate and a random y-coordinate.

A triangle is drawn with the three random points as the vertices. What is the probability that the center of the square is inside the triangle?

No Solution Yet Submitted by Brian Smith    
Rating: 3.2000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
I give up! (spoiler) | Comment 2 of 18 |
I've been before, and suspected right away that the math is too difficult for me (and probably any of us) to do analytically.

Here's a url that investigates the whole problem:
http://mathworld.wolfram.com/SquareTrianglePicking.html

The general solution for any (x,y) in the unit square (not just the center) is an ugly expression that involves the natural log, but the logarithmic terms drop out  when x = y = 1/2, so it could conceivably evaluate to 1/4 (Charlie's number).

I'll be very interested if anybody finds an analytic solution for the special case where the point of interest is the center.

  Posted by Steve Herman on 2008-03-17 13:45:20
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