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 Simulation Sources and further reading Quadruple integral Difference of two uniform variables ~leemis/chart/UDR/PDFs/ Difference of two uniform variables and solving the double integral Solving the double integral Mean distance polar coordinates change Integral secant cubed Mean distance square Cleve Moler’s blog post 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: 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: Twitter: Facebook: Google : Pinterest: Tumblr: Instagram: Patreon: Newsletter (sent only for big news, like a new book release): 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) “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) “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. “Math Puzzles Volume 2“ is a sequel book with more great problems. (rated 4.3/5 stars on 4 reviews) “Math Puzzles Volume 3“ is the third in the series. (rated 3.8/5 stars on 5 reviews) “40 Paradoxes in Logic, Probability, and Game Theory“ contains thought-provoking and counter-intuitive results. (rated 4.3/5 stars on 12 reviews) “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) “Multiply Numbers By Drawing Lines“ This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 3 reviews)