# L tetrominoes forming an 8x8 magic square

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.
• Can we used the flipped version i.e. J tetrominoes as well? Mar 17 at 0:46
• Yep. Just updated the conditions. Mar 17 at 1:05
• I made a more general version of this puzzle: puzzling.stackexchange.com/questions/115341/… Mar 17 at 14:07

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:

• Awesome work! That is even more beautiful than the solution I had in mind. Mar 17 at 1:30
• I wonder if we can also get the two main diagonals to work, making it fully magic? Mar 17 at 1:33
• @DmitryKamenetsky Hm... that wouldn't work with rotational symmetry, I'd have to mess around a bit more. Mar 17 at 1:48
• 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. Mar 17 at 2:21
• @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.
– fljx
Mar 17 at 9:42 