Undergrad Exploring Interesting Mathematical Puzzles

[wrong thunder]

TL;DR

Let's discuss and solve some intriguing mathematical puzzles. Share your favorite puzzles and solutions!

I thought it would be fun to start a thread where we can share and discuss interesting mathematical puzzles. Whether you have a puzzle you've been pondering or a favorite one you'd like to challenge others with, feel free to post it here. Let's see how many we can solve together!

Here's one to get us started:Puzzle:
A farmer has a rectangular field and wants to split it into two equal parts. What is the simplest way to do this, and how can we prove that both parts are indeed equal?

 

[fresh_42]

Draw the diagonal, done. Equality follows from congruence. It can also be done with a compass and a straightedge.

Edit:

 

[fresh_42]

How about that one?

What is the least odd, positive integer $m$ such that
$$
n^2=(m+164)(m^2+164^2)
$$
for some integer $n$?

Elegant solution: 1 page.
Brute force: 2 pages.

But maybe there is even a shorter clue than the one I have found.