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Two dominoes within D can be found to cover any boundary square within S, and four dominoes within D can be found to cover any non-boundary square within S.
With these definitions, the task of tiling the Aztec diamond may be reduced to a constraint satisfaction problem formulated as a binary integer program:.
This formulation can be solved with standard integer programming packages. Additional constraints can be constructed to force placement of particular dominoes, ensure a minimum number of horizontal or vertically-oriented dominoes are used, or generate distinct tilings.
An alternative approach is to apply Knuth's Algorithm X to enumerate valid tilings for the problem. There are many tools used throughout tiling projects, but two useful ones are GeoGebra and program created by Jim Propp , Greg Kuperberg , and David Wilson in SageMath to count the tilings of a shape.
From Wikipedia, the free encyclopedia. GeoGebra category. I", Journal of Algebraic Combinatorics , 1 2 : —, doi : Archived PDF from the original on Retrieved 22 April An Aztec diamond of order is the region obtained from four staircase shapes of height by gluing them together along the straight edges.
It can therefore be defined as the union of unit squares in the plane whose edges lie on the lines of a square grid and whose centers satisfy.
The first few are illustrated above. The number of squares in the Aztec diamond of order is , giving for , 2, OEIS A The number of domino tilings of an order Aztec diamond is , where is the triangular number Elkies et al.
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