I just drew the figure and manually tried the question but I am wondering is there a way to do this problem via permutations and combinations.
PS: I got answer as 7.
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$\begingroup$ Can you elaborate on how you got 7? $\endgroup$– Calvin LinCommented Nov 22, 2020 at 4:54
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$\begingroup$ I drew a 6x6 grid and it filled it with the T shaped figures shown above using rotations and made sure there were no overlaps and got answer as 7. $\endgroup$– Aditya1256Commented Nov 22, 2020 at 4:58
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3 Answers
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(Fill in the gaps as needed. IF you're stuck, explain where you are stuck and show your work.)
Hint: The answer is
8
Find a configuration that has that many T's.
To prove that we can't do 9 T's, color the grid like a chessboard. Notice that there are 18 Black and 18 White squares. Notice that every T either covers 3 Black 1 White or 3 White 1 Black squares. Find a contradiction.
There is no non-negative integer solution to $a+b = 9, 3a+b = 18$, hence we can't place 9 tiles.
answered Nov 22, 2020 at 5:01
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$\begingroup$ Here is how i did it, i think it maybe wrong. $\endgroup$ Commented Nov 22, 2020 at 5:11
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I filled the 6x6 square manually withou any calculation or considering possibility of other cases.
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1$\begingroup$ Hint: Can you fill a 4x4 square with these T-tetrominos? $\endgroup$ Commented Nov 22, 2020 at 5:14
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$\begingroup$ As I mentioned, you can do 8. There's a slight modification to your digram that will allow us to place a T on the right most column. $\endgroup$ Commented Nov 22, 2020 at 5:23
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$\begingroup$ Nice symmetric shape @achillehui ! :) $\endgroup$– cosmo5Commented Nov 22, 2020 at 18:15
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As noted above it's possible to tile the 6x6 square with 8 T-tetrominoes, with 4 squares left over. FWIW, there's a simple tiling of the 4 x 4 square, so there are trivial tilings of any 4n x 4m rectangle.
Here's a symmetrical 6×6 solution I found by hand:
I hacked my code from my answer to Tiling of a $9\times 7$ rectangle to solve this new problem. Due to the nature of the hack, I'm not totally confident that I've found all the solutions, but I managed to find 184, which include rotations and reflections. So the true answer is something higher than 184 / 8 = 23.
Anyway, here's a slow-motion GIF anim of those results. I suppose it would look nicer if the 4 empty squares were all the same colour, and a different colour to the tetrominoes, but I didn't feel up to hacking that part of the program too.
