Abstract
Checkerboard colouring arguments for proving that a given collection of polyominoes cannot tile a finite target region of the plane are well-known and typically applied on a case-by-case basis. In this article, we give a systematic mathematical treatment of such colouring arguments, based on the concept of a parity violation, which arises from the mismatch between the colouring of the tiles and the colouring of the target region. Identifying parity violations is a combinatorial problem related to the subset sum problem. We convert the combinatorial problem into linear Diophantine equations and give necessary and sufficient conditions for a parity violation. The linear Diophantine equation approach leads to an algorithm implemented in MATLAB for finding all possible parity violations of large tiling problems, and is the main contribution of this article. Numerical examples illustrate the effectiveness of our algorithm. The collection of MATLAB programs, POLYOMINO_PARITY (v2.0.0) is freely available for download.
Highlights
A polyomino is constructed from a finite number of edge-connected cellsin the plane
The recreational mathematics of tiling with polyominoes became popular after a series of articles in Scientific
We describe a simple sufficient condition for a tiling problem to have no solution based on the parity of the target region and the parities of the tiles
Summary
A polyomino is constructed from a finite number of edge-connected cells (or ‘squares’). The n-ominoes are polyominoes with area n and we refer to the cases for n = 1, 2, 3, 4, 5, 6 as monominoes, dominoes, triminoes, tetrominoes, pentominoes and hexominoes, respectively. The recreational mathematics of tiling with polyominoes became popular after a series of articles in Scientific. Each domino placed on the chessboard covers exactly one black square and one white square. If the dominoes exactly tile the board they must cover a total of 31 black squares and 31 white squares. The polyominoes tiles a finite region of the plane called the ‘target region’. The general decision polyominoes tiles a finite region of the plane called the ‘target region’. Case,black blackand andwhite whitesquares squaresof ofthe the which checkerboardcoloured colouredregion regionto tobe betiled tiledare arerepresented representedby byblack blackand andwhite whitevertices verticesof of checkerboard graph.
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