The coloring is kind of additional structure that is applied on the object you are working with. And I think this idea of "applying structure" is a very generic. You can solve similar combinatorial arrangement problems that way, but it goes beyond that.
I think that a nice, classic (and significantly more advanced) example is showing that plane and punctured plane (a plane with one missing point) are topologically different. The fundamental (homotopy) groups of these spaces are different, and hence the spaces cannot be continuously deformed to each other.
Somehow the spirit is the same, I feel. In this topology proof it's not a grid you are working with, but a topological space. And the structure you apply is not a coloring, but something quite abstract (a homotopy group). The idea in both cases is similar, though: You apply structure and this structure reveals something that's not easy to see directly.
The magic part is figuring out the structure that produces the data you need.
The chessboard in the article is a bipartite graph with different number of vertices in the two groups, so it cannot have a perfect matching.
> an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, [...]. But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are
Proof/spoiler: https://mathoverflow.net/a/17328/111
* web ui: https://openprocessing.org/sketch/126042/
* Numberfile video: https://youtu.be/lFQGSGsXbXE
I believe your comment is actually about the extension (to this problem and solution) that I posted in another comment: https://news.ycombinator.com/item?id=46005842
Which unique square (up to symmetry) must be left if you cover the 64 squares of a chess board with 21 3x1 trominoes?
Proving that the solution is unique may be more subtle.
Kaprekar's constant is interesting. This one is not.
As for explaining complex math to children, I like to start with zero not being a real number. "If you have zero cookies why are we talking about cookies? There are none. You're now thinking of cookies, which means you have zero cookies, and if you want one then you have negative cookies."
For me, the reason this problem is cool is that it exemplifies mathematical thinking: superficially the problem is about placing individual dominos but the solution is about seeing the underlying structural properties. Similar to Euler realizing the bridges in Königsberg were a graph.
A "lattice point" on the plane is a point where both coordinates are integers, like (3, 4) or (-2, -1). Prove that for any five lattice points, there will be two of them that if you connect them with a line segment, there's another lattice point between them on that line.
Cut one corner off a chessboard. Is it possible to tile the remaining board with 3-by-1 dominoes?
(Spoiler/solution: https://www.jeremykun.com/2011/06/26/tiling-a-chessboard/)
In a typical "tournament" -- say 64 teams, how many matches/games are played before declaring the final winner?
Not sure if there's a way to do spoilers here, but there's a very easy one sentence explanation that involves very close to "no math at all."
(I happened to encounter this two times in close succession when I was getting my teaching credential: first in a teaching manual and then a day or two later, a couple teachers at the school where I was doing my student teaching were puzzling over it and thought they’d challenge me with it and I gave them the answer immediately which shocked them since they’d spent a long time on solving this with algebra and I did it in my head in less than a second. To be honest, I probably wouldn’t have been so quick at the solution without having already seen it.)
but the problem does state that you should be able to do it in your head. who exactly should be able to formulate and reduce simultaneous equations in xy then apply the quadratic formula (with some spicy +/- to keep track of) to get an answer with an irrational number, all in their head? usually, when a problem like this is given there is a shortcut that leads to a simple, not only rational but integer, answer.
the statement "you can do it in your head" generally does not entail this much complexity, as the person who said "you can do it in your head" comes out and says after previously spending a fair amount of time working on it.
words matter, people, that's why I didn't throw in the adjective integral even though I could have.
It's funny that you jump to accusing OP of falsely claiming you can do it in your head, without apparently considering the alternative: that the intended solution is a simpler one than you outlined.
Trust me, you can do this in your head if you know basic high school level math, and you don't need to solve quadratic equations or keep a ton of numbers in your head at the same time.
If I ask you if 123456789 is a prime number, do you complain that it's not fair to make you perform division on such a long number?
yeah, i guess it was a mistake to graduate from MIT undergrad and grad school in quant fields, i should have just stuck with high school math
>If I ask you if 123456789 is a prime number, do you complain that it's not fair to make you perform division on such a long number?
you tell me, is 13717421 prime?
dhosek understood the assignment by making an argument that 123456789 is composite without relying on explicit division of a 9-digit number, which most people would find rather difficult to do in their heads.
Similarly, the posted link is about tiling a mutilated chessboard with dominos. Tiling problems in general are NP-hard, so clearly this isn't something you can solve in your head _in general_, but the charm of that specific problem is that you _can_ solve it by making an insightful observation to avoid the brute force computations.
Similarly, for the puzzle you complained about: we are asked to find 1/a + 1/b where a × b = 37 and a + b = 18. The general solution is to solve a system of two linear equations which involves solving a quadratic equation, which is possible, but tedious and difficult to keep in your head, but the entire point of the question is that there is a better way to figure out the result.
If you have a+b and a-b you’ll get 2a when added together.
So knowing just the sum we can say that a is 9 in this setup.
Now we need to figure out b.
Multiplying out those you get
a^2 + ab -ab - b^2
And I get a longing for not having started this a phone.
Cancels and fill in what we know and we get 81 - b^2 = 37
b = sqrt(44) = sqrt(4)*sqrt(11) = 2sqrt(11)
I don't get it. I don't see why / how it would take any longer than a second or two to solve 'with algebra'. What does that even mean? You would just maybe write down the steps rather than doing them in your head. Is there any other way to solve the problem?
I recommend looking also at the THOG problem. See https://en.wikipedia.org/wiki/THOG_problem.
mapehe•2mo ago
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loloquwowndueo•2mo ago