# Set of magic polyominoes that can tile a square

Let's first look at this square grid of numbers. The 9 squares in yellow is what we are looking at and the green numbers are the sums for the digits within the rows and columns. The red squares are the sums of the numbers along the diagonals.

All rows columns and both diagonals sum to a square number.

Now let turn these numbers into polyominoes of given size according to the number in the above grid. Note that all of the polyominoes tile into a 9x9 grid.

To make this work the 11 is broken into to polyomino of size 2 and 9 and likewise the 18 is broken into 8 and 10.

Now the question

Is it possible to construct polyominoes of the given size where only one or the polyominoes is broken into 2 pieces? Instead of 2.

The ideal solution would be to only Break the 18 into 2 identical polyominoes of size 9 each.

Polyominoes can be of any shape but must form the squares when they are brought together.

Bonus if you can make the 3 into a L shape.

And a second question

Instead of using squares to make the shapes is it possible to create 9 unique polygon shapes that can be any shape at all that will fit into the grid where every piece is a whole piece and none are split into 2 pieces?

• I think question 2 can be understood a little better by phasing it like this....is it possible to to slice a 7x7 square into three regions with each region having an area size of 5, 18 and 26. The answer is yes. However the tricky part is. Is it possible to make the cuts so that the shape with area size 5 also can fit inside a 3x3 square and the 18 inside a 5x5 square and the 26 must fit inside a 6x6 square.
– Maff
Feb 15 at 11:28