Archive for January, 2011

3×3 block Pentominoes and Hexominoes

January 21st, 2011

There are 8 pentominoes and 8 hexominoes that fit in a 3×3 cell. The combined set seems to cry out to be presented in a 4×4 grid of 3×3 blocks, with the pentominoes and hexominoes in checkered positions:

The best I could think to do was make a figure that is connected, hole-free, and has a rotationally symmetrical pattern of connections between blocks. I had hoped to make them into a geomagic square, but now I’m pessimistic about that working. And the trick from my magic 45-ominoes of making all rows and columns have the same number of cells in polyominoes won’t work here because the total number in each row of 3×3 blocks is 22, which isn’t divisible by 3.

I’ve looked before at problems involving moving a single cell at a time to cycle through a set of polyforms. Because this set has equal numbers of pieces at two consecutive sizes, it invites using adding and removing single cells, rather than moving them, as the action for taking one piece to the next in a path:

Because the X pentomino has only one possible predecessor or successor, it cannot be part of a cycle, but it is still possible to make a path through all of these pentominoes and hexominoes with the X as one of its endpoints.

Magic Squares and Polyominoes

January 21st, 2011

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: