See all monthly puzzles: https://momath.org/mindbenders/

Where the loss function for training LLMs comes from. Job opportunities aligned to this audience: https://3b1b.co/talent Early views and other perks for supporters: https://3b1b.co/support Home page: https://www.3blue1brown.com Part 1: https://youtu.be/l6DKRf-fAAM If you want to learn more about Lagrange Multipliers, I made multiple videos and articles for the topic back in my Khan Academy days: https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/lagrange-multipliers-and-constrained-optimization/v/constrained-optimization-introduction Manim animations by Aaron Gostein and Grant Sanderson NanoGPT animation by Clayton Rabideau 3d black-box model by Paul Dancstep Chess game and Lagrange Multiplier scenes by Nishad Deulkar Music by Vince Rubinetti Timestamps 0:00 - Language trees and zipping 3:02 - Recap optimal codes 5:20 - Defining cross-entropy 8:26 - Intuition and examples 12:59 - Application to language trees 14:55 - Pre-training LLMs 20:38 - What makes this loss function best? 26:13 - Distillation 30:12 - 3b1b Talent 31:35 - KL Divergence ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

Part of a series of monthly puzzles done in collaboration with MoMath.

Full video: https://youtu.be/l6DKRf-fAAM

Full video: https://youtu.be/l6DKRf-fAAM

How to study the compressibility of language. Check out our virtual career fair: https://3b1b.co/talent See new projects before they go live: https://3b1b.co/support Animation credit: Manim scenes by Aaron Gostein and Grant Sanderson Shannon’s story, as well as those for various pi creatures, by Mitchell Zemil. Lunar robot and prediction/compression coin by Paul Dancstep NanoGPT animations by Clayton Rabideau The way of visualizing entropy shown here is something I first came across in this excellent post by Chris Olah: https://colah.github.io/posts/2015-09-Visual-Information/ Shannon’s “A Mathematical Theory of Communication” https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf Shannon’s “Prediction and Entropy of Printed English” https://www.princeton.edu/~wbialek/rome/refs/shannon_51.pdf Scientific American article that mentions the story with Von Neumann suggesting the name Entropy: https://www.esalq.usp.br/lepse/imgs/conteudo_thumb/Energy-and-Information.pdf Timestamps: 0:00 - On “Compression is intelligence.” 3:28 - The warmup example 10:46 - What perfect compression looks like 14:47 - Defining information 17:40 - Information of language 24:29 - Defining Entropy 31:14 - 3b1b Talent ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

Recent puzzle solutions on Patreon: https://members.3blue1brown.com/posts/158885046?pr=true

Made as part of a monthly series of puzzles for the 2026 Year of Math.

On "The Print Gallery", by M.C. Escher Full video: https://youtu.be/ldxFjLJ3rVY

Part of a series of monthly puzzlers. Stay subscribed to see the solution

Escher's Print Gallery, and the tour of complex analysis it invites. Explore our virtual career fair: https://3b1b.co/talent Join channel supporters to see videos early: https://3b1b.co/support An equally valuable form of support is to share the videos. Home page: https://www.3blue1brown.com Original paper by de Smit and Lenstra: https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-lenstra-escher.pdf The book I was showing is "Magic of MC Escher" by J. L. Locher https://amzn.to/4d7zXTT If you want to play with this concept interactively, Jürgen Richter-Gebert put together a nice page: https://mathvisuals.org/PrintGallery/ This piece was co-written by Paul Dancstep, who handled many of the animations in the art section, including the delightful mesh warp scene. Aaron Gostein helped with the manim animations in the section introducing complex functions. Artwork provided by Talia Gershon, Mitchell Zemil, and Anna Fedczuk. https://sites.google.com/view/taliagershon https://mitchellzemil.com/ https://anna-fedczuk.framer.website/about Music by Vincent Rubinetti Timestamps: 0:00 - The print gallery 13:04 - Conformal maps from complex analysis 21:41 - The complex exponential 25:56 - The complex logarithm 32:32 - 3b1b Talent 33:14 - Constructing the key function 40:16 - The deeper math behind Escher ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

Part of a monthly series of puzzlers, in collaboration with MoMath and Peter Winkler

On the volumes of higher-dimensional spheres Explore the 3b1b virtual career fair: See https://3b1b.co/talent Become a supporter for early views of new videos: https://3b1b.co/support An equally valuable form of support is to simply share the videos. Home page: https://www.3blue1brown.com Thanks to UC Santa Cruz for letting me film there, and special thanks to Pedro Morales-Almazan for arranging everything. My video on Numberphile with a fun application of this problem: https://youtu.be/6_yU9eJ0NxA Timestamps: 0:00 - Introduction 1:01 - Random puzzle 6:16 - Outside the box 14:35 - Setting up the volume grid 21:14 - Why 4πr^2 25:21 - Archimedes in higher dimensions 36:17 - The general formula 40:40 - 1/2 factorial 44:58 - Why 5D spheres are the biggest 50:16 - Concentration at the surface 54:27 - A unit-free interpretation 57:50 - 3b1b Talent 59:13 - Explaining the intro animation ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

Part of a series of monthly puzzles, done in collaboration with MoMath. https://momath.org/mindbenders

Solution to last month's probability puzzle.

Unexpected applications and a beautiful proof. Looking for a new career? Check out https://3b1b.co/talent Supporters get early access to new videos: https://3b1b.co/support An equally valuable form of support is to simply share the videos. Home page: https://www.3blue1brown.com Credits: Senia Sheydvasser: Co-writing and sphere deformation animations, made in Blender Paul Dancstep: Those lovely fluffy sphere animations, made in Cinema4D Vince Rubinetti: Music Sphere Eversion clip by Carsten Steger https://commons.wikimedia.org/wiki/File:Thurston_Sphere_Eversion.webm Timestamps: 0:00 - To comb a hairy ball 1:24 - Applications 8:46 - The puzzle of one null point 12:12 - The proof outline 16:41 - Defining orientation 21:44 - Why inside-out is impossible 25:59 - 3b1b Talent 27:44 - Final food for thought ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

This is the first in a set of monthly puzzles, curated by Peter Winkler. This one was originally suggested by Richard Stanley. You can sign up to hear his description of the answer at http://momath.org/mindbenders