# Problem of the Month (April 2010)

A strata tiling of n-ominoes is a tiling of a rectangle with n-ominoes so that congruent tiles form connected groups, and each connected group touches the top and bottom of the rectangle. If you rotated these, they would look like deposited layers of rock. For a given n, we ask what heights of rectangles have strata tilings, and what the smallest one is. Can you find smaller strata tilings? What about larger heights? What about larger polyominoes? Can you find ANY strata tiling of hexominoes?

A ringed tiling of n-ominoes is a tiling of a rectangle with n-ominoes so that congruent tiles form connected concentric rings. Can you find a smaller ringed tiling for tetrominoes? Can you find ANY ringed tiling of pentominoes?

Because ringed tilings are hard to find, and get large quickly, we can also search for sub-ringed tilings, ringed tilings of some subset of n-ominoes. Can you improve any of these results?

This month's solvers include Patrick Hamlyn, Alexandre Muniz, Joe DeVincentis, George Sicherman, Bryce Herdt, Mike Reid, Rodolfo Kurchan, Berend van der Zwaag, Jeremy Galvagni, and Maurizio Morandi.

Here are the best known strata tilings:

Triominoes
 2×6 3×3
Tetrominoes
 4×10 5×16 6×12 7×16 9×12(Mike Reid) 10×12 12×9(Mike Reid) 18×10(Mike Reid)

Mike Reid proved that tetromino strata tilings exist for all heights larger than 3. For large heights 4n, there exist width 9 tilings. For large heights 4n+2, there exist width 10 tilings. And for large odd heights, there exist width 12 tilings.

Pentominoes
 3×30 (Patrick Hamlyn) 4×50 (Alexandre Muniz)
 5×32 (Alexandre Muniz) 7×40 (Mike Reid)
 8×35 (Rudolfo Kurchan) 10×29 (Rudolfo Kurchan)
 11×40 (Rudolfo Kurchan) 13×40 (Rudolfo Kurchan)
 15×26 (Rudolfo Kurchan) 17×40 (Rudolfo Kurchan)
 30×25 (Mike Reid) 40×28 (Rudolfo Kurchan)

Rodolfo Kurchan suggested that we also study quarter-ringed and half-ringed tilings. So here are the best known ringed tilings:

Triominoes
 3×4 3×5 4×6
Tetrominoes
 9×8 (Mike Reid) 14×8 (Mike Reid) 16×12 (Rudolfo Kurchan)
Pentominoes
 25×22 (Rudolfo Kurchan) 45×41 (Mike Reid)

Here are the best known sub-ringed tilings:

Tetrominoes
 1

 2 (GS) (GS)

 3 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)

 4 (GS) (GS) (BZ) (GS) (MR)

Pentominoes
 1 (GS)

 2 (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (BH) (GS) (GS) (MR) (GS) (GS) (GS) (GS)

 3 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (BH) (BH) (MR) (GS)
 (GS) (MM) (MM) (MM) (GS) (GS) (MM) (MM) (BH) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (RK) (MR) (MR) (GS) (GS) (GS)
 (GS) (GS) (GS) (MR) (MM) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (MR) (GS) (BH) (MR) (GS) (MM)
 (GS) (GS) (GS) (RK) (RK) (MR) (RK) (GS) (MR) (MR)
 (GS) (GS) (GS) (GS) (RK) (RK) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (MR) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (MM)
 (BH) (BH) (RK) (RK) (RK) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (MR) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS)
 (GS) (GS) (GS) (GS) (GS) (GS) (MR)

Jeremy Galvagni wondered what the results would be for polyhexes instead of polyominoes.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/20/10.