After studying mathematics a lot, I have solved a lot of problems. And when you do anything enough, you develop a taste for things. I like good problems. I also enjoy sharing them with people. There’s one I particularly like – in fact, in many ways, it’s the best one I know.
The statement is quite simple: consider a normal chessboard. produce one warped chessboard By removing two opposite diagonal corner squares (illustrated below).
Now, it is clear that a normal chessboard can be covered by 2×1 blocks. So the question is as follows: Can a deformable chessboard be covered using (exactly 31) 2×1 blocks?
Just to clarify: This is a yes or no question. the question is not How The cut-up chessboard can be covered; the question is If it can be done at allObviously, one way to positively answer the question is to actually produce a wrapper: if a wrapper was successfully produced, then yes, of course, a wrapper exists, However, the point is that an argument that proves either Existence of a cover or a non-existence Any cover is enough,
Solution prescribed below:
Now, why do I like this problem so much? Well, for one, it’s simple. You could explain it to a 7-year-old, and they’d probably get the gist of it. And although it is simple, it is still very difficult. Of course, this is subjective, but I think this is common in these types of combination problems. It’s easy to tell them, the solution is easy to understand – and yet, it is often extremely difficult to figure out the solution on your own.
However, there is a broader context where this problem largely fits. I think this problem is a way to deliver a pinch of “higher mathematics” in an accessible way. Although this is an initial setting, it shows how questions are asked Existence come into play. This is the core: advanced mathematics is often not about creating something directly through a calculation or algorithm, but about showing it. Existence Of something. And when you deal with existence, you start dealing with definitions And evidence,
Abstract definitions and proofs about the objects governed by those definitions are interesting in many ways. They can be extremely creative, and this is where, I think, mathematics can be compared to art. In particular, human intelligence is required to understand and formulate definitions and proofs. However, surprisingly, we are now living in a moment where computers are starting to take over in this field.
Modern mathematics is extremely abstract. I would say that the change began in the late 19th century when Cantor presented important results in set theory (such as it Famous one). Now that this has been going on for a century or two, we are so far down the path of abstraction that at first glance it seems hopeless for computers to deal with it. an original However, the fact is that the evidence can be seen as TypeOn the other hand, types are ubiquitous in programming languages, It becomes a direction that enables mathematical theory to be formulated in a form that computers understand, Microsoft has a project related to type systems of programming languages, In the hands of mathematicians, it has evolved into a project with the lofty goal of formalizing our mathematical knowledge in computer-readable form, Although this system is still highly experimental Proved itself in serious research.
Then there is AI.
Since LLMs can produce any type of text or code, they can equally well attempt to produce type-theoretic formulations of mathematical statements. Terence Tao, one of the leading mathematicians of our time, wrote About recent developments in this field.
It seems to me that something big is happening right now, and it won’t be long before mathematical research starts to look very different from what it is today.