Math Resources for students

Here is a roundup of some of our favorite Mathematical resources.

Sites:
The site Art of Problem Solving is the ultimate resource for problem solving, info on competitions at a number of levels and other programs, and discussion forums about problem solving.

We used to post extensive notes after each Math Circle session.  Notes from 2020 and earlier can be found here.

Videos:
The web is full of great learning resources. A few classics are here and here. In particular, these show you, in beautifully animated fashion, how sometimes going to a higher dimension makes complicated things simple, and how to visualize the very strange fact that its possible to turn a sphere “inside out.”

The channel here also regularly posts interesting new content explaining big math ideas in fun and broken-down ways.

Fun math books:
The Man Who Knew Infinity This tells the inspiring story of Srinivasa Ramanujan, who flunked out of college twice and with almost no formal mathematical training re-discovered much of modern mathematics. He was truly a man who loved numbers and formulas, and wrote down so many shocking formulas that leading mathematicians of the day almost could not believe that he was real. In a short period of a few years after his chance “discovery” by a leading mathematician of the day, he went on to revolutionize entire fields of mathematics; today, his formulas have been shown to contain secrets behind the theory of black holes, and they are being implemented into signal processing methods in cell phone computer chips. A truly breathtaking read and proof that mathematical ideas have the power to change the world, even from the humblest of origins.

Also, this book was very recently turned into a Hollywood movie of the same name with several A-list actors like Dev Patel and Stephen Fry, which is also highly recommended!

The Pea and the Sun: a mathematical paradox
This brilliant book is one of the first exposures Dr. Rolen had to deep mathematics, and is a fun ride through various riddles, geometric puzzles, and paradoxes, all leading to a demonstration of the perplexing Banach-Tarski paradox. This paradox states that it is possible to cut, with very very precise scissors, a ball into just a few pieces, then to rearrange these pieces without stretching or adding new material, only using rotations and translations, into two balls of the same volume as the original! The title refers to the resulting fact that you can therefore cut a ball the size of a pea into finitely many pieces, rearrange, and turn it into a ball the size of the sun! In this book, you will be introduced in a very friendly and approachable way to different ideas about “infinity.” For example, you will see why some infinities are bigger than other infinities, learn how to fill a hotel with no vacancies with more guests without kicking any guests out, and see a complete explanation of the paradox as depicted by a big machine with various tubes and hoppers.

The Number Devil was originally written in German, but its Enlgish translation is also a classic.  It follows the story of a boy who hates math as a number devil takes him on an adventure.

Math summer programs:

The Summer Mathematics Programs Consortium consists of US-based summer math programs for middle and hich school students.  Much of their website is aimed at the directors of such programs, but their Current Members page has a list of member programs with outgoing links you can follow to apply to these programs.