The content was interesting but the cookie consent and in-your-face subscription pop-ups are infuriating and annoying. Thought I’d mention it since you did go to the trouble of popping by this discussion!
Nice problem! I wonder if there is a generic way of testing such a problem with different board arrangements. For example, could you apply knot theory or another concept?
This is a classic of course, but there is a lesser known extension to this problem (to be read only after this problem has been solved), which also has a beautiful "proof without words":
> 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
The problem is slightly more challenging if you don't use a chessboard, but just a grid, because then you must first come up with the idea of coloring it.
I do not care for this problem as it is not a real problem.
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."
From the title, I first imagined what my favorite math problem was, then clicked on the article -- and they had the same one!
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.
What I like most about this math problem is explaining it to people who understand what I'm saying but still insist that it might be possible and they're going to do it. It's a nice lesson for me to think about and carry through life.
One of my favorites is one that you should be able to do in your head: The product of two numbers is 37, their sum is 18. What is the sum of their reciprocals?
(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.)
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[ 2.6 ms ] story [ 32.4 ms ] threadI'm Matias. I run a small business (ByteSauna) with a blog on the site. I try my best to serve well thought out content. Here's this weeks post.
Hope you enjoy it!
> 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
Which unique square (up to symmetry) must be left if you cover the 64 squares of a chess board with 21 3x1 trominoes?
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.)
I recommend looking also at the THOG problem. See https://en.wikipedia.org/wiki/THOG_problem.