Can you paint 10 cells of a 10x10 grid such that the largest unpainted rectangle has area of 10 cells?
Here is a similar question for the 7x7 grid: Paint 7 cells of a 7x7 grid
Good luck!
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asked Sep 25, 2019 at 6:11
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2$\begingroup$ I will delete my incorrect solution and retry $\endgroup$– JS1Commented Sep 25, 2019 at 7:38
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1$\begingroup$ Ok I came up with a new solution. $\endgroup$– JS1Commented Sep 25, 2019 at 7:46
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3 Answers
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I came up with:
.......... .....x.... ..x....x.. .......... ....x..... .x......x. .....x.... .......x.. ..x...x... ..........
Method:
1. The edges should not be painted because anywhere you would paint an edge, you could do the same or better by painting one in from the edge instead.
2. Basically, I tried to leave areas of 3x3 and 2x5 and only "block" areas when they threatened to become 2x6 or 3x4. It was a lot of trial and error.
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$\begingroup$ I think you've done it this time - great work! $\endgroup$ Commented Sep 25, 2019 at 7:57
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It seems to be possible to do this with nine painted cells.
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2$\begingroup$ Very nice. The solution seems to be unique (up to rotation). $\endgroup$ Commented Sep 25, 2019 at 11:27
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$\begingroup$ Very nice indeed. I didn't know it was possible. $\endgroup$ Commented Sep 25, 2019 at 12:19
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Using a computer I found there are quite a few solutions, so I decided to tighten the conditions a bit. If you also disallow unpainted rectangles of size $2\times5$ and $5\times2$, there seems to be only one solution left.
. . . . . . . . . . . . . . X . . . X . . X . . . . . . . . . . . . . . X . . . . . . X . . . . . . . . . . . . . . X . . X . . . X . . . . . . . . . . . . . . . . . X . . . X . . . . . . . . . . . .
answered Sep 25, 2019 at 11:21
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1$\begingroup$ If you are interested you can try fitting n mines on n by n grids that are larger. I have results for that $\endgroup$ Commented Sep 25, 2019 at 12:20