# Magic Squares and Polyominoes

January 21st, 2011 by munizao

Lee Sallows recently created a new site, geomagicsquares.com, about geometric magic squares. These differ from standard magic squares in that the numbers are replaced with shapes, and instead of having a magic sum which all of the rows, columns, and main diagonals must add up to, they have a target shape that the shapes in each row, column, and main diagonal must tile. (As in standard magic squares, each entry in the square must differ from all of the others. I really recommend the site highly; the presentation of the geometric magic squares is nearly as beautiful as the underlying mathematics. Many (but not all) of the geometric magic squares there use polyominoes or other polyforms.

Several years ago, I came up with a different way of combining polyominoes and magic squares. My magic 45-ominoes are polyominoes contained in a 3×3 configuration of 3×3 blocks, such that each row, column, and main diagonal has 5 cells within the polyomino, and each 3×3 block has a number of cells corresponding to a number in a magic square.

After reading Sallows’ site, I wanted to try my own hand at a geomagic square, and I came up with a variation that incorporates ideas from my magic 45-ominoes:

The rows and columns in the diagram all contain 5 cells. I wasn’t able to make the main diagonals work out. Maybe you can?

#20: Find a geomagic square of polyominoes that can be presented in a 3×3 grid of 3×3 blocks as above, where all rows, columns, and main diagonals have an equal number of cells that are contained within polyominoes.

By the way, I’m still looking for what I call a Magic Magic 45-omino; that is, a Magic 45-omino where each cell contains a different number between 1 and 45, and each row and column adds up to 115. (Make that problem #21.) Here’s a near solution:

1. Bryce Herdt says:

For #21, you have a *really* near-solution. I found a solution by moving around fourteen numbers in R4-9, C4-7. Here’s the layout (if I don’t bork the formatting too badly):

```31 21 2              32 29
23 24 26             9  33
10    1  43       25 36
7        30 42 22 14
44 5              4  35 27
19 40 38 3  15
28    17 20 39       11
37 41 13 16 8
45 12 18 6  34```

Actually, this might be more accurately called a Semimagic Magic 45-omino, since the diagonals work for the 45-omino but their entries don’t have the right sums. I thank that’s closer to the spirit of your problem, wouldn’t you say? (I noticed that swapping C56 of my solution makes one diagonal fit, but puts the other diagonal at 142.)

2. Bryce Herdt says:

Ggaaaggh. Should I be using brackets around my tags?

3. munizao says:

Nice! I fixed the formatting by putting <pre> tags around it.

4. Bryce Herdt says:

I… I thought I’d done that. Maybe I can’t do it.

5. munizao says:

Oops. You’re right; WordPress strips <pre> tags by default. That should be fixed now. I’ve also added a display of the allowed html tags below the comment box.