# Sets of tetrominoes forming a magic square

Is it possible to place $$n$$ sets of five free tetrominoes on a $$K \times K$$ square grid, such that:

• No two tetrominoes overlap.
• Tetrominoes can be rotated or flipped.
• Every row, column and two main diagonals contain the same number of cells covered by a tetromino.

What are the smallest positive $$n$$ and $$K$$ for which this is possible?

• I have a solution, but it has a large $n$. Hopefully that can be reduced. Mar 17, 2022 at 14:11
• Just to clarify, the square grid can be arbitrary size? (independent of n). Mar 17, 2022 at 14:30
• What is the smallest positive n and K for which this is possible. Mar 18, 2022 at 12:09
• @DmitryKamenetsky just curious: how large was the n in your solution? Mar 21, 2022 at 9:48
• @Oliphaunt like 8, so it is not even worth discussing. Mar 21, 2022 at 10:03

I believe this does the trick:

n=2, k=10
Each row, column and main diagonal has 4 cells covered. Perhaps this can be improved, with the same tetrominos covering 5 cells per row/column/diagonal on an 8x8 square grid.

And with a bit of programming, we find this:

n=2, k=8
Each row, column and main diagonal has 5 cells covered. This is just one of several hundred solutions with the two square tetrominoes together in the top left corner. There are no doubt thousands of other solutions, but I did not optimize the search to enumerate them within any reasonable timeframe.

• That's a great solution! Mar 18, 2022 at 1:48
• What about n=1 and K=5? Can we show that it's impossible? Mar 18, 2022 at 2:08
• A standard parity argument shows that an n=5,k=10 filled grid is impossible. (With checkerboard colouring, all but the T tetromino cover two black and two white squares each. So you need your five T tetrominos to cover the remaining 10 black + 10 white squares, which isn't possible when each covers 1 black + 3 white or 3 black + 1 white)
– fljx
Mar 18, 2022 at 9:01
• I have ruled out the n=1, k=5 case by essentially trying all possibilities for where the holes are. I don't have a neat argument for impossibility, though some are ruled out by a similar parity argument as above. Mar 18, 2022 at 12:54
• @DanielMathias you total legend! Well that saves me from coding it up :) Mar 20, 2022 at 14:10