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## References

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*Comput. Geom.***5**(1995), 1–25. The authors consider the problem of tiling a region with horizontal*n*× 1 and vertical 1 ×*m*rectangles. Their main result is that, for*n*≥ 2 and*m*> 2, deciding whether such a tiling exists is an*NP*-complete question. They also study several specializations of this problem. - [2]
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*Duke Math.*J.**7**(1940), 312–340. To each perfect tiling of a rectangle, the authors associate a certain graph and a flow of electric current through it. They show how the properties of the tiling are reflected in the electrical network. They use this point of view to prove several results about perfect tilings, and to provide new methods for constructing them. - [3]
J. Conway and J. Lagarias. Tiling with polyominoes and combinatorial group theory.

*J. Combin. Theory Ser. A***53**(1990), 183–208. Conway and Lagarias study the existence of a tiling of a region in a regular lattice in \({\mathbb{R}}^2\) using a finite set of tiles. By studying the way in which the boundaries of the tiles fit together to give the boundary of the region, they give a necessary condition for a tiling to exist, using the language of combinatorial group theory. - [4]
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*Amer. Math. Monthly***76**(1969), 37–40. The author studies the problem of tiling an*n*-dimensional box of integer dimensions*A*_{1}× ... ×*A*_{ n }with bricks of integer dimensions*a*_{1}× ... ×*a*_{ n }. For a tiling to exist, de Bruijn proves that every*a*_{ i }must have a multiple among*A*_{1}, ...,*A*_{ n }. The box is called a*multiple*of the brick if it can be tiled in the trivial way. It is shown that, if*a*_{1}|*a*_{2},*a*_{2}|*a*_{3}, ...,*a*_{n−1}|*a*_{ n }, then the brick can only tile boxes that are multiples of it. The converse is also shown to be true. - [5]
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*J. Combin. Theory Ser. B***25**(1978), 240–243. The unique perfect tiling of a square using the minimum possible number of squares, 21, is exhibited. - [6]
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*J. Algebraic Combin***1**(1992), 111–132, 219–234. It is shown that the Aztec diamond of order*n*has 2^{n(n+1)/2}domino tilings. Four proofs are given, exploiting the connections of this object with alternating-sign matrices, monotone triangles, and the representation theory of*GL*(*n*). The relation with Lieb’s square-ice model is also explained. - [7]
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*Philosophical Magazine***6**(1961), 1061–1063. A formula for the number of domino tilings of a rectangle is given in the language of statistical mechanics. - [8]
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*Math. Res. Lett***1**(1994), 547-558. The authors show that a square can be tiled with rectangles similar to the 1 ×*u*rectangle if and only if*u*is a zero of a polynomial with integer coefficients, all of whose zeros have positive real part. - [9]
B. Grünbaum and G. Shephard.

*Tilings and patterns*. W.H. Freeman and Company, New York (1987). This book provides an extensive account of various aspects of tilings, with an emphasis on tilings of the plane with a finite set of tiles. For example, the authors carry out the task of classifying several types of tiling patterns in the plane. Other topics discussed include perfect tilings of rectangles and aperiodic tilings of the plane. - [10]
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*m*subsets*T*_{1}, ...,*T*_{ m }of a set*S*, Hall defines a*complete system of distinct representatives*to be a set of*m*distinct elements*a*_{1},...,*a*_{ m }of*S*such that*a*_{ i }∈*T*_{ i }for each*i*. He proves that such a system exists if and only if, for each*k*= 1,...,*m*, the union of any*k*of the sets contains at least*k*elements. - [11]
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*n*-ominoes.*J. Combin. Theory***7**(1969), 107-115. Klarner investigates the problem of tiling a rectangle using an odd number of copies of a single polyomino. He also characterizes the rectangles that can be tiled with copies of an*a*×*b*rectangle, and the rectangles that can be tiled with copies of a certain octomino. - [14]
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*Algebra and tiling. Homomorphisms in the service of geometry*. Mathematical Association of America: Washington, DC, 1994. This book discusses the solution of several tiling problems using tools from modern algebra. Two sample problems are the following: A square cannot be tiled with 30°–60°–90° triangles, and a square of odd integer area cannot be tiled with triangles of unit area. - [19]
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*Amer. Math. Monthly***97**(1990), 757–773. The author presents a technique of Conway’s for studying tiling problems. Sometimes it is possible to label the edges of the tiles with elements of a group, so that a region can be tiled if and only if the product (in order) of the labels on its boundary is the identity element. The idea of a height function that lifts tilings to a three-dimensional picture is also presented. These techniques are applied to tilings with dominoes, lozenges, and tribones. - [20]
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This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.

Contributions are most welcome.

This paper is based on the second author’s Clay Public Lecture at the IAS/Park City Mathematics Institute in July, 2004

Supported by the Clay Mathematics Institute

Partially supported by NSF grant #DMS-9988459, and by the Clay Mathematics Institute as a Senior Scholar at the IAS/Park City Mathematics Institute

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Ardila, F., Stanley, R.P. Tilings*.
*Math Intelligencer* **32, **32–43 (2010). https://doi.org/10.1007/s00283-010-9160-9

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### Keywords

- Side Length
- Mathematical Intelligencer
- Small Rectangle
- Large Rectangle
- Tiling Problem