A puzzling inequality ...

it's harder than it looks

This week, I wanted to share a puzzling problem involving inequalities.

Like most of the problems I share in this newsletter, it’s easy to state but hard to solve!

Problem

For any positive integer m, show that there exist integers a,b satisfying

\(|a| \leq m, \hspace{5pt} |b| \leq m, \hspace{5pt} 0 < a + b \sqrt{2} \leq \dfrac{1 + \sqrt{2}}{m + 2}.\)

If you have a solution to the above challenge problem, submit it here for a chance to be featured in the next issue of this newsletter.

Last week’s problem

Here is the problem from last week:

In how many ways can you tile a 3×n rectangle by 2×1 dominoes?

This was considerably harder than the problems from previous weeks.

It involved a lot of tedious case-by-case analysis, and lots of drawing pictures.

See here for the solution to last week’s problem. It was written by Conrad Warren and Ravi Dayabhai, who found a very nice recursive formula for the number of ways you can tile a 3×n rectangle by 2×1 dominoes.

Special shoutout to Ben Elkins (Evanston), jc (New York), and Jack Dunn (Scotland) for submitting solutions to this challenge problem.

Given the difficulty of the problem, anyone who attempted it deserves a huge pat on the back!

Learning resources for Lie Groups

This week, I wanted to share some learning resources for Lie groups.

Roughly speaking, a Lie group is the “continuous” analog of a group. Lie groups come up everywhere in pure math, physics, and engineering.

Here are my favorite resources to learn about them.

Books

• Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall: This book is aimed at folks with a basic understanding of linear algebra and calculus.

• Lie Algebras in Particle Physics by Howard Georgi. I have not used this book personally but many of my friends in physics swear by it. It gives a great introduction to use Lie Algebras are used in particle physics.

Videos

• Richard Borcherd’s Lectures about Lie Groups. These are pure gold. This a full online lecture series that explains Lie groups and Lie algebras in full detail.

• This introductory video by Mathemaniac. If you’d like a general introduction to what Lie theory is about, this video is a great place to start.

Thanks for reading and happy learning!

See you next time,

Adithya