The hardest problem on the hardest test

A difficult Putnam question with an elegant solution.
An equally valuable form of support is to simply share some of the videos.

These videos exist thanks to Patreon:

A different write-up of this solution:

1992 Putnam with this problem:

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!

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Korean: tebaioioo

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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:

Stream the music on Spotify:

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).

Various social media stuffs:

[knownbynone]

The video: "The hardest problem on the hardest test"
Me who barely passed math classes: “Hmm, lets see.”

[isaacmalik]

what i find even more amazing is that someone had to come up with all of these problems and solve all of them

[clone]

About 5 years ago, I watched this video and got inspired to study maths in college and take this test. This Saturday, I took the Putnam and solved 1 question!!!!

[matthewganun]

Imagine taking this test only to see some kid start flipping a coin

[sinan.]

Teacher: "The test won't be that hard."

*The test:*

[matthewgeary]

I remember watching this back in 2017, my freshman year of high school, and then falling in love with this channel and math itself. Here I am in 2022, a freshman math undergraduate, still knowing damn well that I could never solve this by myself lmao

[acurefordeath]

You can further extrapolate the simplification to a one dimensional problem: given the space between two random points on a bounded line, how many solutions will contain the center? Given a first random point, there is always a 1/2 chance that the other random point is on the other side of the midpoint. You can then extrapolate from one to three dimensions by cubing the solution. (1/2)^3 = 1/8

[enjoipls.]

"If you choose 4 random points on a sphere...."
I think I changed my mind can I go home now?

[nujak]

This guy: explain all the math


Me: still trying to figure out what a tetrahedron is

[cheese]

I’m in high school and I’m about average in math, I was so close to seeing this video as a write off and leaving. I’m so glad I decided to stick around, you explained this in a way that even the average can understand, you are an amazing teacher and this video was extremely entertaining!

[caitlinphillips]

Your teaching abilities are phenominal, you have explained this in a way that makes perfect sense to me without simply removing relevant details to avoid simplifying them, which is what most 'teachers' would do and consider it a job done.

[tobe]

Me after 3 mins: I understand
Me after 6 mins: I understandn't

[TheRealGuywithoutaMustache]

Imagine coming home tired from learning math at school, only to find this in your recommended.

[QuantumDivergence]

Wow, the animation, and music, the way you explained all of this. This is a work of art. Well done.

[TheSurprises]

I really love how you break it down ...it makes so much sense and you are super understandable! Thanks for what you're doing to teach these concepts. And breaking it down beyond the answer to this question to what you can apply to all other situations is GOLD! 🥰

[nicholastorres]

teacher: im just gonna skip this question, u guys should have learned this last year

[ApenstaartjeYT]

Girl: I don’t get why he doesn’t get my signals
Her signals:

[lordyt]

I love this way of visualization 3Blue1Brown as well as other Math related YouTubers use. It really helps to understand these kinds of problems on a deeper level. For example, I doubt I would‘ve gained the same key insights from just reading through the listed write-up in comparison to your great video. I wish this method would be used in actual classrooms, however, with the dawning age of smart boards and such we might get tho this at some point ^^.

[Ajbolt]

My brother recommended this channel, and this is a great introductory video to your content. The math is above my head, but the discussion about the process of problem solving is relevant to anyone. Cool!

[Kali-bsoj]

You know it’s getting lit when the smart kid brags about getting 3/120 points