Everything about Recreational mathematics
Today's topic is Recreational mathematics.
This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week.
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Next week's topic will be Exceptional objects
The question that got me into mathematics in the first place was the mutilated chessboard problem.
For those of us who haven't heard it before, it goes like this. Given an 8x8 chessboard with opposite corners cut out, and thirty-one 1x2 dominoes, can you completely cover the chessboard with the dominoes?
Go on, try it right now. Try every single possible covering of dominoes you can think of.
>! It's impossible. And you don't need to check any coverings of dominoes.
>! The mutilated chessboard had two corners of the same colour cut out; either both opposite corners are white, or both opposite corners are black. That means that there are either 32 black squares and 30 white squares, or 30 black squares and 32 white squares.
>! Now, each domino covers two adjacent squares; a black square and a white square. So 31 dominoes must cover the same number of black squares as white. But in neither case above does this happen! So it's impossible.
>! When I heard this problem and the solution when I was 5, I was amazed. Such a seemingly-impenetrable problem, felled by the observation that a chessboard is black and white!
That is what good, elegant mathematics is like, to me. A key, obvious insight, that whittles down something so seemingly-impossible into something so trivial.
This is the proof I like to give to people to explain what doing "real" mathematics is like.
Assuming you meant "i-shaped", isn't the answer no? You'd need to be able to place 9 tetrominoes and no matter how you arrange them you can get at most 8?
I meant "J" or "L", a row of three blocks with one added above or below one on the end. I see that lower-case "L" is confusing here.
"i-shaped" is a separate question. You're right that it's not possible, the challenge is to prove it. (Not sure if you were deliberately trying to avoid spoilers.
Color the 6x6 grid with alternating 2x2 squares of black and white
I was having trouble getting spoiler tags to work, but I essentially made a combinatorial argument about states the I-blocks could occupy and showed you could only place 8 in a 6x6 grid. The one thing I wasn't happy with is that it did not generalize in an obvious way to larger grids or lend insight into the J or L cases.
The trick you put in spoilers is also very clever and I'll probably play with that more after class. I'll have to play with the other two you posed.
Gosh, broad topic. And next week it's "counterexamples"!
In Conway's Game of Life there's a line of research into "glider constructions". Basically the question of which patterns you can make by colliding a bunch of [gliders](https://en.wikipedia.org/wiki/Glider_(Conway%27s_Life)). An interesting result that was proven recently is that if you can make a pattern by colliding gliders then you only ever need 35 gliders to do so.
Also, to give a bit of context into just how active the Conway GoL research community is, there is an annual "pattern of the year" contest, and the construction mentioned by this comment only got 3rd place (in 2018, when it was discovered). There has been an absolutely insane amount of cool things done in the Game of Life lately.
I'm running out of ideas and just noticed that we already done pathological examples so I might as well change next week's
Heres a few pitches:
Linear Algebra
Nonstandard Analysis
Statistical Learning theory
Tensor Analysis
Bayesian Statistics
Algorithmic complexity theory
ODEs
DAEs (Differential Algebraic Equations)
SDE (Stochastic Differential Equations)
Numerical Analysis
Functional Equations (Schröders equation, fixed point method,...)
Optimization (Convex/Non-convex/...)
Other good ideas:
Recursion theory
Fredholm Theory
Cardinal Characteristics of the Continuum
Matroids
Large Cardinals
Stability theory
Algebraic Statistics
Applied category theory
Topological combinatorics
Index theory
Mathematical gastronomy
Prove that a magic square interpreted as a matrix has its magic constant as an eigenvalue.
The solution itself is not very difficult or interesting but I love magic squares and I was pleased to find this connection while doing homework for my linear algebra class.
Not sure if a stronger statement about magic squares as matrices can be made, but if so that would probably be cool.
At first this seemed difficult to me, but then I realized this is a weird scenario where finding the eigenvector is the easiest thing to do.
This exact question seems to come up a lot on GRE subject test stuff, for some reason.
It does in fact work for the Parker square, along with other semimagic squares.
What your favorite cute puzzles or questions to mull over and solve when you’re bored or want some idle distraction?
Here's something I've never fully understood: what makes some math recreational and other math "serious"? Certainly a lot (most?) of modern math is quite far from real-life applications so applicability can't be the criterion. It seems to me that answering this question is tantamount to explaining why certain areas of math are studied more than others. The reasons almost certainly have to do with history and taste.
If some problem can be explained in simple enough terms for non-mathematicians to understand, or made into a game/puzzle, then it's recreational.
This isn't quite true though. Lot's of problems in number theory like Fermat's last theorem can be explained for non-mathematicians to understand and plenty of ideas in math can be explained via games, but that doesn't make them recreational.
I was on the math team for a few years in college. I never got more than a zero on the Putnam but I won a silver medal in the 2016 University Physics Competition (if you've done the Mathematical Contest in Modeling, it's like that, but with physics) and a bronze in 2018 (probably could have scored way higher if I hadn't fucked up and accidentally submitted it 4 minutes after the soft deadline), so that kind of makes me feel a little better about myself.
If you're into math contests and you have reasonable programming skills and some knowledge of physics but you're looking for something a little gentler than the Putnam or the MCM, then I would definitely, definitely recommend the UPC, which is held every year in November.
Here are three recreational math questions that I've asked on Stack Exchange:
kind of off-topic, but related to recreational mathematics:
is there any good (non-pop-sci, but also not for experts) mathematical review or book about strange attractors? I know only paper by Étienne Ghys, "The Lorenz Attractor, a Paradigm for Chaos".
E. g., I have questions, like: when we look at Lorenz system attractor
- we see some 2-dimensional shape, it is also said not to be a manifold, it's called a fractal, "we finally conclude that there is an infinite complex of surfaces, each extremely close to one or the other of the two merging surfaces". Is there any concise way to describe the shape of attractor? Maybe if we take some 2d projection on some plane - then it will have some nice shape with easy formulas describing it? Maybe there's some nice animation of this infinite complex of surfaces?
- it's called butterfly wings - as if we have 2 planes, on which these wings are located (should be false) or nicely approximated (didn't find any reference for this or calculations of these approximations). Rössler attractor also looks like half of it lies on plane z=0, but probably it isn't.
- it's always drawn with holes. Does the attractor really has holes? Both near the repelling points and between the curves inside the attractor? If they have holes or 1-d curves as part of attractor - in what sense are they 2-dimensional? And is it possible to fill the holes to make the attractor look like a real 2-dimensional surface without boundary?
- do these objects appear in other subfields of mathematics, not only in dynamical systems?
etc.
Also, is there any systematic study of these objects (e. g. like 3-dimensional manifolds) (like some kind of classification)?
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