VERY HARD Puzzle: What Is The Distance Between Two Random Points In A Square?
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when discussing average or expected distances between random points in a square, and also ask follow-up questions about steps in its integral derivation.
Recommended by 3 people across 3 different months, 2016 to 2024.
What it is
MindYourDecisions presents the puzzle of finding the average distance between two random points in a unit square. It first estimates the answer with a simulation, then derives the exact value with integrals, including a switch to polar coordinates. The channel notes it needs college calculus and probability.
What people said
Recommends Toby Mak · 2017-07 · Mathematics Stack Exchange · on “Euclidean distance for points in $\mathbb R^2$”
@Josué Ortega This video by MindYourDecisions on YouTube ( [this video] ) uses integrals to find the average distance between two points in the unit square - it's extremely relevant to your question.
Recommends jimjim · 2021-08 · Mathematics Stack Exchange · on “Distance between two random points vs the average distance on a shape”
@Troposphere : just googling "random distance on circle/square" gives number of video/web pages, e.g. [LINK] is that what you mean? or this one : [this video]
Recommends Nate · 2024-03 · Mathematics Stack Exchange · on “Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square”
I would try to estimate this statistically in Excel. A square with each side 1 unit can have its lower left corner at the origin. Randomly select 3 x and y coordinate pairs where one coordinate in the pair is 0 or 1 and the other can be determined with =rand(), use the distance formula for each pair to find the sides of the triangle they form, then use Heron's Area Formula to find the area of the triangle. Do this 10,000 times and count up the number of times the triangle's area is $\geq$ .25. The first part of this video is helpful: [this video]
Cites learning · 2016-07 · Mathematics Stack Exchange · on “Average distance between two points lying on boundary of a square”
Calculate the average distance between two points lying on the boundary of a unit square. I tried to approach it in the same way that this video [this video] does, but I couldn't really wrap my head around how not to include the points lying inside the square.
Cites FlanM · 2019-02 · Mathematics Stack Exchange · on “Average area of a rectangle inside the unit square”
I recently came across this problem while toying with the problem of the average distance between 2 points in the unit square [this video] . These two points also define a rectangle, so I was wondering: What is the expected value of the area of that rectangle? I took this problem to Python [LINK] and it turns out to be somewhere around one-ninth. However, I have no idea where to start actually calculating the exact value. If someone could share how to get a more exact answer (more efficiently, obviously) via computer science, or how to get the precise answer with calculus, that would help out a bunch.
Cites dirtydivider · 2019-03 · Mathematics Stack Exchange · on “Mean distance between two points in a square polar coordinates conversion”
In this video [this video] , around the 5:20 mark, YouTuber MindYourDecisions converts the double integral from rectangular coordinates to polar coordinates so as to make the evaluation easier. However, I fail to see how he managed to get the limits on the radius as $0$ and $\frac{1}{\cos\theta}$ . Since the length of the adjacent side is always $1$ , shouldn't the limits be $1\leq r \leq \sec\theta$ ? The same result is obtained in this [LINK] article. Thanks for any input. Sorry that I'm not seeing this.
Cites emnha · 2019-11 · Mathematics Stack Exchange · on “why this integral represents the average distance between two point on a unit square”
I'm watching this video [this video] and get confusing about the integral. Can anyone explain why this integral represents the average distance between two point on a unit square? [LINK]