Cylindrical Strata Tiling Of Hexominoes
In Math Magic April 2010 Erich Friedman defines a strata tiling of n-ominoes, as a tiling of a rectangle with n-ominoes, such that congruent tiles form connected groups, and each such group touches the top and bottom of the rectangle. Erich goes onto ask what strata tilings exist. When I worked on that problem, I was disappointed to discover that there is no strata tiling for n=6 (hexominoes). I decided to explore a variant of the problem.
We define a cylindrical strata tiling to be the tiling of the net of a cylinder with n-ominoes, such that congruent tiles form connected groups, and each such group ‘wraps around’ the cylinder. If you join the net together so the bottom and top edges touch, then the groups of congruent tiles each forms a connected ring around the cylinder.
Happily, I discovered that cylindrical strata tilings do exist when n=6.
One such tiling is shown below.
This tiling uses 174 tiles. I’m fairly confident that this is minimal, but would love to hear from you if you can improve it.