I Exploring Interesting Mathematical Puzzles

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

Edit:

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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.
 
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Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

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