My favorite math problem

Author: mapehe < matias@three.consulting >

Date: 21.11.2021

Intro (safe to skip)

Before growing disillusioned to an academic career and co-founding three.consulting, I used to study math. Math can only be learned by solving problems. Hence, I've spent several years solving math problems of varying difficulty and beauty. I thought it would be fun to share the one I like the most.

For most people math means stuff like $1+1=2$, i.e. computation. There is little room for creativity in computation and, I believe, that's the reason most people view math as a tedious and purely mechanical activity. When a student progresses to a more advanced level, they encounter the infamous "proof courses", where the problems suddenly transform from mechanical challenges such as "find the derivative of $f$" to something more abstract, such as "prove that there is an infinite number of primes". After the student recovers from the initial shock (assuming they don't abandon math at this point), they may realize that these types of problems, or their solutions to be more precise, can be nice, beautiful or even stunning.

What's a beautiful proof, then? A legendary 20th century mathematician Paul Erdős was an avid collector of beautiful proofs. Quoting Wikipedia: He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF). He accused SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from The Book!" This later inspired a book titled Proofs from the Book.

I've glossed over Proofs from the Book (Erdős was actually involved in the writing process) and I think that at least a few of the proofs have the following features, I personally find beautiful too:

• The problem is simple to state
• The problem is at least moderately difficult
• The solution is surprisingly concise

A good example could be finding the value of $Q$: $$ Q = \frac{1}{q} + \frac{1}{q^2} + \frac{1}{q^3} + \frac{1}{q^4} + \cdots, $$ with $q > 1$. Since multiplication is distributative (the careful reader pauses here and verifies we aren't mishandling an infinite series) $$ Q = \frac{1}{q} \; \underbrace{\left( 1 + \frac{1}{q} + \frac{1}{q^2} + \frac{1}{q^3} + \cdots \right)}_{=1+Q} = \frac{1}{q}\left(1 +Q \right), $$ which is easily rearranged to $$ Q = \frac{1}{q-1}. $$

There is something magical in how an infinite series is tamed with a one-liner. While most examples in Proofs from the Book are significantly more sophisticated, I think this type of "magical" insight is a common theme throughout. In my opinion, the problem I decided to share is similar in this sense. I actually like it even better, since it's so concrete and easily presented.

The problem

Let board $A$ be a standard $8 \times 8$ chessboard. Let board $B$ be a $8 \times 8$ chessboard with the squares $a1$ and $h8$ removed. Board $A$ can trivially be covered with $32$ blocks of size $2 \times 1$. Can board $B$ be covered with $2 \times 1$ and $1 \times 2$ blocks (such that there are $31$ blocks in total)?

Solution