The most difficult mathematical question on the most difficult mathematical test

A difficult geometry puzzle with an elegant solution.
Home page: https://www.3blue1brown.com/
Brought to you by you: http://3b1b.co/putnam-thanks
And by Brilliant: https://brilliant.org/3b1b
Solution to the puzzle mentioned at the end: https://brilliant.org/3b1bindicator/
These videos exist thanks to Patreon:
https://www.patreon.com/3blue1brown
A different write-up of this solution:
http://lsusmath.rickmabry.org/psisson/putnam/putnam-web.htm
1992 Putnam with this problem:
http://kskedlaya.org/putnam-archive/1992.pdf
A problem with a similar flavor came up on the 2005 Putnam A6. Give it a try! The solution for that problem, by the way, was written by Calvin Lin, a friend of mine who works at Brilliant. Small world!
http://kskedlaya.org/putnam-archive/2005.pdf
http://kskedlaya.org/putnam-archive/2005s.pdf
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Animations largely made using manim, a scrappy open source python library. https://github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various social media stuffs:
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The horses name is friday

::*knows nothing about math :(Still watches video)

:WOW That was soooo Easy im only 3 weeks old and i solved it no problem in 2 mins

:To solve this, you need. 1. A full stack of paper. 2. 50 pencils 3. 200 IQ 4. Probably some more paper

:*In class:* 2 x 2 = 4 4 ÷ 2 = 2 *In test:* The circumference of a circle is 92cm, the radius of a circle is 83cm. What is in my back garden?

:*Starts Exam and turns first page over* Formula Sheet: OMAE WA MOU SHINDERU! Me: NANI!?

:There YouTube. I watched it. Happy? btw, nice vid bro

:Why am I watching a math video at 12:02 am on a Saturday

:Welcome to part 598 of "why is this in my recommended?"

:Question: Probability that this tetrahedron contains the sphere's center? Me in the test: Yes

:"how you might have stumbled across the solution yourself" Didn't know this was a comedy channel

:Somewhere between 0-100%. Looks like I got it right

:Probability on circle is 1/2², probability of a sphere is 1/2³. I find the simplicity of the outcome rather beautiful

:Ppffft...hard? I could do these questions in my sleep. Could you explain this one more time tho?

:Idk why I'm watching this, I don't understand a single term lmao

:In an online test: 43/7 = Your incorrect answer: 6.14 Test answer: 6.142857142857143

:meanwhile i have to do basic division in my calculator almost every time

:Math classes in high school: Find X / 3x=12 Math exams in high school: F I N D T H E P R O B A B I L I T Y of the C E N T E R of the S P H E R E being inside this pointy B O I

:Fastest 3 things in the universe: 1- light 2- universe expansion 3-me clicking on a 3blue1brown notification

:The discomfort that this three-dimensional thinking can give is far more than the happiness it will give after solving it.

:11:15 The only part I understood

:Homework: 8x4 Test: John has 3 apples, the train leaves in two hours, calculate the mass of the sun.

:See you lost me at "math"

:I never thought I would be able to understand how to solve a putnam problem... i am completely overwhelmed with joy... your style of explaining the problem just blew my mind.... i have been a subscriber since last year or so, thank you for making me realise the elegance in math

:Me: Not as probable as me failing this test... *walks outside and chain smokes an entire pack of cigarettes*

:The answer is 2

:The real hardest question. Explain the entire storyline of Kingdom Hearts to some who doesn’t know the series

:easy 2+2 is 4 minus 1 that's 3 quick mafhs

:This guy is amazing at explanations. If he can teach the hardest problem this well, imagine what he would be like as a high school teacher.

:The probability of me understanding this explanation is 0.000002

:Me: uses calculator to find out what is 7+7 Also me: knows how to solve hard math problems easily

:In class: 1 + 1 = 2 2 + 2 = 4 In test: Given the implicit equation of a sphere, calculate the gradient vector and explain why the gradient vector points in the direction of maximum up slope.

:In class: 1-3= -2 -2+3=1 During the test: If (3,-2) is the center of a triangle why would the communist revolution in Guatemala have an impact on the American continent?

:yo what language is this

:Well this is why I gave up on math lol

:Surface integral *traumatic flashbacks to Calculus III* *shivers*

:Question: Probability that the tetrahedron contains the sphere's center? Me: 50% Either it does or doesn't.

:So seriously, solidworks would've helped me with the most difficult math EVER? Holy shit

:I just saw the problem, nothing else, and... Can I go home?

:Fake, everybody know that Kobayashi-Maru Test is the hardest one.

:the hardest problems require the strongest wills. (Thanos from avengers)

:Brain.exe has stopped working

:Wait, they put numbers on the test?!?! Screw that

:I guessed 12% at the beginning lol I was so close with a random guess

:Alright smarty pants What falls faster a pound of bricks or a pound of dogs?

:Monte Carlo Simulation works very best.

:Why am I watching this at 1 in the morning

:Your voice is a mix between Mordecai from Regular Show and Fred from Scooby Doo

:Math : solve this tehtraderon Me : nope

: