Problem of the Month (February 2010)

Tiling regions with polyominoes has been well studied. For example, some variations on this theme can be found at the February 1999, January 2001, October 2002, December 2004, July 2005, May 2006, March 2007, January 2008, and June 2009 Math Magic.

This month we consider another variation. Identify each polyomino with the points at the centers of its component squares. As long as these center points have the shape of the polyomino, the magnification of the polyomino does not matter. For a given spaced-out polyomino, what is the smallest rectangle it can tile without rotations? What if rotations are allowed? What about tiling other shapes, such as triangles, pyramids, and diamonds, with spaced out polyominoes?

Rectangles

 1-4

 5 (George Sicherman)
 (George Sicherman) (George Sicherman)
 (George Sicherman)

 6
 (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman)

 7 (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman)

 8
 (George Sicherman) (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)

 9
 (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman) (George Sicherman)

 10
 (George Sicherman) (George Sicherman) (George Sicherman)

 11
 (George Sicherman) (George Sicherman) (George Sicherman)

Rectangles With Rotations

 1-4

 5 (George Sicherman) (George Sicherman) (George Sicherman)

 6 (George Sicherman) (George Sicherman) (George Sicherman)

 7 (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)

 8
 (George Sicherman) (George Sicherman)
 (George Sicherman) (George Sicherman) (George Sicherman)

 9

 10

 11

George Sicherman sent these solutions for triangles, pyramids, and diamonds:

Triangles

 1-4

 5

Triangles With Rotations

 1-4

 5

Pyramids

 1-4

 5

Pyramids With Rotations

 5

Diamonds

 1-4

 5

Diamonds With Rotations

 5

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/14/10.