Saturday, June 18, 2011

Square Dartboard Probability

Found a nice little puzzle over at Infinigons:
You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?
Go on. Try it now.

Here are some of my sketches as I attempted solutions.
Here's a different formulation as I thought more about the question.
Here's a a quick check to see if I'm on the right path.

Extension: Given an integer n>2, what is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge of a regular polygon with n sides? What is the probability as n approaches infinity?

I had a lot of fun working this out. Be sure to read Allison's original post on problem solving, persistence, and fearlessness.

If you just want to see the solution, you can find it at the bottom of the next post. In that post, I first find the solution to the Ngon Dartboard Probability (the extension offered above) followed by finding the solution to the instance when n=4 or square.

Square Dartboard Probability
Ngon Dartboard Probability
Ngon Dartboard Probability Simulation
Ngon Dartboard Probability Limit

UPDATE: Added conditions and changed wording.
UPDATE: Added links to the other posts.

3 comments:

  1. for fun I tried the brute force method in Mathematica. I "threw" 30000 darts at the board. If it hit the board AND was closer to the center, I counted it and drew it in the pic. I did it for n=3-10. The percentage is listed at the top of each image: http://yfrog.com/z/kh6xdp -Andy (@arundquist)

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  2. by percentage I mean the fraction of those closer to the center to those that hit the board.

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  3. @Andy : Awesome! The results from your simulation closely matches my results. I'm finishing up the typesetting and diagrams, should have it up "soon"

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