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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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2 Answers 2

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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:

enter image description here

I started with this configuration, which looked promising:

enter image description here

Then messed around until I found an inner configuration that worked.

An alternative, asymmetric solution:

enter image description here

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  • $\begingroup$ Awesome work! That is even more beautiful than the solution I had in mind. $\endgroup$ Mar 17 at 1:30
  • $\begingroup$ I wonder if we can also get the two main diagonals to work, making it fully magic? $\endgroup$ Mar 17 at 1:33
  • $\begingroup$ @DmitryKamenetsky Hm... that wouldn't work with rotational symmetry, I'd have to mess around a bit more. $\endgroup$
    – A username
    Mar 17 at 1:48
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    $\begingroup$ 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. $\endgroup$
    – A username
    Mar 17 at 2:21
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    $\begingroup$ @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. $\endgroup$
    – fljx
    Mar 17 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:
enter image description here

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  • $\begingroup$ I love this! Well done. $\endgroup$ Mar 17 at 14:00

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