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

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

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Kyle
Alberto Nishikawa
Brian M. Mooney

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[ImranZakhaev]

"Did you figure this out?"
:l

[carryon]

"Did you figure this out?"
-
-
I find this question to be mildly humiliating.

[ФилиппЛыков-де]

The quadrupe integral _is_ tractable but it took me five A4 pages. And I have miscalculated the coefficients somewhere. So yes, your solution is better than what I could figure out.

[CollinInGame]

I'll never solve this. it took me 3 and a half minutes before I realised I could just close the video and not have to worry about it.

[unvergebeneid]

"This doesn't look too hard. I'll just have to ... wait ... ok, now this is a bit more complicated than I .... uh. Holy ..."

[wurm]

I immediately thought “about 0.5“. Turns out I was off only by <5%. I am an engineer. So, good enough :)

[geovaniraffaelli]

I lost you at ''Consider a square...''

[sigitrendang]

No one:

Automatic translator: *"Hey this is pressure locker"*

[HakanTheUltimateHoca]

Just guessing before the video, is it
(2+√2+5ln(√2+1))/15 ?

[FilipeSantos-wgpd]

You know the maths are hard when Preshtal Walker stutters while explaining xD

[flamingpaper]

I mean I dont get what happened but it looked cool and seems real

[arrogantuser]

This exact problem was a question on a homework set in my probability theory course. Our textbook covered a similar problem that I referenced heavily while working on it and it still took a good part of an hour to finish. This video makes it seem so simple, well done.

[whozz]

What is the average distance between two random points in a unit circle?

[dummdmdyummmmmmmmmme]

"Did you figure it out?" Really... you actually asked that question?

[MathAdam]

4:21 Simplifies into a double integral. This is going to be a long haul.

[BixbyConsequence]

"Did you figure it out"? Yeah, I did it in my head.

[AnstonMusic]

I thought that there have been too many easy questions in the past weeks, but this was really top notch!
Loved every moment of it.
I'm gonna miss maths a little going to Med School, but hey, I'll find my own problems that need mathematical modeling. Thanks for quenching my thirst and keeping up the drive!

[majorinconvenience]

LMAO when the question was posed I said to myself: “About 0.5?” Then after complex mathematical disambiguation it turned out to be just ABOUT that

[Kenlimepie]

"Did you figure it out?" What am I? Einstein?

[onenameleft]

Wow.. Even when I don't come up with the answer myself, I can usually follow your explanations. this one was just beyond. I took Calculus in college 20 years ago, but I'm clearly rusty. I understood the estimate process, but the double integrals and how you got the numbers you did when you switched to absolute values of delta instead of calculating distances between all points... that was beyond me. I guess that happens when you take something that could be a class long lecture and instead of having time to teach it, you have to condense it down to a 10 minute follow me while I do the problem. It's still enjoyable to watch. And it make sense to me that the average would be lose to .5. I'm not sure i can explain why, but i just expected the average to be close to half the distance of the lengths of the legs. since the hypotenuse of a right triangle with sides 1 is only 1.4 does that help the average stay closer to half the legs. Would the average distance of a square with sides 3 be way larger than 1.5? and would it get further away from the middle as the sides got bigger because distances can go diagonally?