A slightly more involved version of this puzzle asks about covering all but one square of the chess board with 21 trominoes (each of size 3x1 or 1x3), since 64 = 21*3 + 1.
What square remains uncovered (unique up to symmetry) ?
My Favorite Math Problem
51–60 of 86 posts
Re: My Favorite Math Problem
#52It's simultaneously fascinating and frustrating when you run into a problem that suits those criteria.
In this case of course the hint is well disguised: if you had an unmarked board you might not think to checker it, but a checkered board is a common thing, so you might not think anything of it when someone presents it as part of the problem.
Reminds me of how you find the area under exp(-x^2). Stare at it for a bit and it looks like it can't be done. But if you add another dimension to it, you find the solution.
And this is what is both fascinating and frustrating. You could add a lot of things to a problem without making any progress, but there are certain things you can do that make it simple.
Re: My Favorite Math Problem
#53Typeset PDF in case you don't want to LaTeX-by-eye: https://cloudflare-ipfs.com/ipfs/QmcZf4xuHEvvqrBCv2uhnyQXk5u...
What? The mathematics shows correctly on the page for me, and that PDF is missing some chessboard images...
Re: My Favorite Math Problem
#54Typeset PDF in case you don't want to LaTeX-by-eye: https://cloudflare-ipfs.com/ipfs/QmcZf4xuHEvvqrBCv2uhnyQXk5u...
What? The mathematics shows correctly on the page for me, and that PDF is missing some chessboard images...
presumably they are running a script blocker that took out the JS that renders the maths.
Re: My Favorite Math Problem
#55Earlier quoted context omitted.
> there is also a very elegant solution that fits in a tweet. (If you don't want to try solving it, or have tried and have given up, here is the solution I tried to understand the tweet a couple of times, but I couldn't follow the proof from the tweet itself, so I wrote it up again with the basic axiom from the tweet as a basis + work out each step of the process. https://gist.github.com/t3rmin4t0r/a953450ac64b686854…
Modulo 9, a decimal integer is equal to the sum of its digits. More precisely, congruent, notated by ≡: For instance 123 ≡ 1+2+3 ≡ 6 (mod 9). We show congruences using ≡, and always have (mod N) on the far right to indicate the modulus for the congurence. More generally, if ABC is a decimal string, then we know that ABC ≡ A + B + C (mod 9). Moreover ABC + DEF + GHI must be congruent to A+B+C + D+E+F + G+H+I (mod 9).…
That was all good, except this bit was not obvious to me:
> Modulo 9, a decimal integer is equal to the sum of its digits. More precisely, congruent, notated by ≡:
> For instance 123 ≡ 1+2+3 ≡ 6 (mod 9).
This is the part where the parent comment's explanation was helpful.
Re: My Favorite Math Problem
#56Earlier quoted context omitted.
Well, that explanation is a total cop out, so I can see why they would think that.
I disagree. Arguments to symmetry like this are found all over the place, especially in the kind of geometry and physics I'm familiar with. In fact the solution I came up with when I read the problem was very similar.
The original comment (by tasty_freeze) was an argument to symmetry. Just saying "because of symmetry" is not a proper explanation.
Re: My Favorite Math Problem
#57This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
I solved this in a hedge fund interview with "Consider the limiting case of a coin the same size as the table... You can only place it at the centre, and you win... Now make the coin smaller. How does the strategy change?... It can't, because of symmetry." They didn't consider it a valid solution :|
It is not clear to me how to play following your strategy. There are things missing. You need to describe what the second move should look like.
Re: My Favorite Math Problem
#58I came across one of my favorite math problems in high school and it blew my mind. ABC DEF + GHI ----- 123J Each letter represents a distinct digit 0-9. What is J? Of course you can write a short program that iterates through the possibilities. But there is also a very elegant solution that fits in a tweet. (If you don't want to try solving it, or have tried and have given up, here is the solution: https://twitter.co…
A nuance here is that you are assuming the existence of a solution. It may be the case there are no values for A to J that can result in that equation. Iterating through all possibilities shows that the solution exists, the proof in the tweet doesn't.
The proof in the tweet gives steps to getting a solution, and the solution works. In what way is that not showing that the solution exists?
Re: My Favorite Math Problem
#59This problem reminds me of another problem. There is a round table and two players, A and B. The players take turns placing a coin on the table in any location they desire, but coins may not overlap. The first person who is unable to place a coin loses. What is the winning first move? Answer: the winning move is for player A to place the first coin in the center of the table. After that, no matter which location play…
I solved this in a hedge fund interview with "Consider the limiting case of a coin the same size as the table... You can only place it at the centre, and you win... Now make the coin smaller. How does the strategy change?... It can't, because of symmetry." They didn't consider it a valid solution :|
This solution doesn't hold - just because a game is symmetrical, doesn't guarantee that the optimal move in some limiting case works in all cases.
For example, consider if the game was played with N>2 players, or if the win condition was to make the first move modulo N, for some N>2. In those cases everything would still be symmetrical, and playing in the center would still be a winning move for huge coins, but not (necessarily) for smaller coins.
Re: My Favorite Math Problem
#60As a start (and possibly as an equivalent proof?) you can mentally brute force it by cutting the board down to 4x4 and just visualising how the blocks might fit. It quickly becomes obvious that it's not possible.