This is a puzzle from Rodolfo Kurchan.
Can you place 10 L-shaped tetrominoes on a 8x8 grid, such that:
- No two tetrominoes overlap.
- Tetrominoes can be rotated and flipped.
- Every row and column contains the same number of cells covered by a tetromino.
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asked Mar 17, 2022 at 0:30
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$\begingroup$ Can we used the flipped version i.e. J tetrominoes as well? $\endgroup$– A usernameCommented Mar 17, 2022 at 0:46
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$\begingroup$ Yep. Just updated the conditions. $\endgroup$– Dmitry KamenetskyCommented Mar 17, 2022 at 1:05
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$\begingroup$ I made a more general version of this puzzle: puzzling.stackexchange.com/questions/115341/… $\endgroup$– Dmitry KamenetskyCommented Mar 17, 2022 at 14:07
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2 Answers
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First, we need to work out the number of squares in each row.
Each tetromino takes up 4 squares, and each contributes twice to the total squares in rows and columns.
So each tetromino contributes 8 squares.
There are 10 * 8 = 80 squares and 16 rows/columns, so there are 80 / 16 = 5 squares per row/column.
Then, some trial-and-error does the trick:
I started with this configuration, which looked promising:
Then messed around until I found an inner configuration that worked.
An alternative, asymmetric solution:
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$\begingroup$ Awesome work! That is even more beautiful than the solution I had in mind. $\endgroup$ Commented Mar 17, 2022 at 1:30
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$\begingroup$ I wonder if we can also get the two main diagonals to work, making it fully magic? $\endgroup$ Commented Mar 17, 2022 at 1:33
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$\begingroup$ @DmitryKamenetsky Hm... that wouldn't work with rotational symmetry, I'd have to mess around a bit more. $\endgroup$ Commented Mar 17, 2022 at 1:48
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1$\begingroup$ I think there might be some parity argument, as all the solutions for rows/columns that I've found have even numbers of squares on both diagonals. $\endgroup$ Commented Mar 17, 2022 at 2:21
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1$\begingroup$ @Ausername I've had a quick play with pencil and paper, and arrangements with odd diagonals do exist. Haven't found one with five on both yet. $\endgroup$– fljxCommented Mar 17, 2022 at 9:42
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@Ausername has answered the main question, but an additional question was asked in comments:
I wonder if we can also get the two main diagonals to work, making it fully magic?
And the answer to this is:
Yes
The following is one example:
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$\begingroup$ I love this! Well done. $\endgroup$ Commented Mar 17, 2022 at 14:00