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Tiling Rectangles with Gaps by Ribbon Right Trominoes
Department of Mathematics, West Chester University, West Chester, PA, USA
Department of Mathematics, West Chester University, West Chester, PA, USA
- 1 Department of Mathematics, West Chester University, West Chester, PA, USA
- 2 Department of Mathematics, West Chester University, West Chester, PA, USA
Open Journal of Discrete Mathematics·Volume 07 (2017)·Pages 87–102·Published 17 April 2017·DOI10.4236/ojdm.2017.72010
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Abstract
We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.
KeywordsTiling RectanglesRibbon TrominoRectangle with GapsGap NumberColoring Invariants
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