Math

# VERY HARD Puzzle: What Is The Distance Between Two Random Points In A Square?

What is the average distance between two randomly chosen points in a square with a side length 1? The video presents how to estimate the answer and also derives the exact answer using integrals. Per suggestions in the comments, I have modified the title to indicate the difficulty level is VERY HARD (a college level calculus course and probability theory course are necessary background).

Blog post
http://wp.me/p6aMk-4FA

Simulation

http://math.stackexchange.com/questions/1254129/average-distance-between-two-random-points-in-a-square

Difference of two uniform variables
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformStandardtriangular.pdf

Difference of two uniform variables and solving the double integral
https://www.quora.com/How-can-I-calculate-the-mean-distance-of-two-points-placed-at-random-in-a-unit-square

Solving the double integral
http://math.stackexchange.com/questions/1294800/average-distance-between-two-randomly-chosen-points-in-unit-square-without-calc

Mean distance polar coordinates change
http://www.mathpages.com/home/kmath324/kmath324.htm

Integral secant cubed
https://en.wikipedia.org/wiki/Integral_of_secant_cubed

Mean distance square
http://mathworld.wolfram.com/SquareLinePicking.html

Cleve Moler’s blog post

How Far Apart Are Two Random Points in a Square?

Thanks to Patrons! A video like this takes a lot of research and you guys make it possible.
Kyle
Alberto Nishikawa
Brian M. Mooney

If you like my videos, you can support me at Patreon:
https://www.patreon.com/mindyourdecisions

Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.

My Blog: https://mindyourdecisions.com/blog/
Pinterest: https://www.pinterest.com/preshtalwalkar/
Tumblr: https://preshtalwalkar.tumblr.com/
Instagram: https://instagram.com/preshtalwalkar/
Patreon: https://www.patreon.com/mindyourdecisions

Newsletter (sent only for big news, like a new book release): http://eepurl.com/KvS0r

If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.

My Books
“The Joy of Game Theory” shows how you can use math to out-think your competition. (rated 3.9/5 stars on 32 reviews)
http://amzn.to/1uQvA20

“The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias” is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.6/5 stars on 3 reviews)
http://amzn.to/1o3FaAg

“Math Puzzles Volume 1” features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews.
http://amzn.to/1GhUUSH

“Math Puzzles Volume 2” is a sequel book with more great problems. (rated 4.3/5 stars on 4 reviews)
http://amzn.to/1NKbyCs

“Math Puzzles Volume 3” is the third in the series. (rated 3.8/5 stars on 5 reviews)
http://amzn.to/1NKbGlp

“40 Paradoxes in Logic, Probability, and Game Theory” contains thought-provoking and counter-intuitive results. (rated 4.3/5 stars on 12 reviews)
http://amzn.to/1LOCI4U

“The Best Mental Math Tricks” teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 4 reviews)