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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.R. Brooks, C. Smith, A. Stone and W. Tutte. The dissection of rectangles into squares.

*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.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.N. de Bruijn. Filling boxes with bricks.

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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.N. Elkies, G. Kuperberg, M. Larsen and J. Propp. Alternating sign matrices and domino tilings I, II.

*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.M. Fisher and H. Temperley. Dimer problem in statistical mechanics—an exact result.

*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.C. Freiling and D. Rinne. Tiling a square with similar rectangles.

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*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.P. Hall. On representatives of subsets. J. London Math. Soc 10 (1935), 26-30. Given

*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.W. Jockusch, J. Propp and P. Shor P. Random domino tilings and the Arctic Circle theorem, preprint, 1995, arXiv:math.CO/9801068. In a domino tiling of an Aztec diamond, the diamond is partitioned into five regions: Four outer regions near the corners where the tiles are neatly lined up, and one central region where they do not follow a predictable pattern. The authors prove the Arctic circle theorem: In a random tiling of a large Aztec diamond, the central region is extremely close to a perfect circle inscribed in the diamond.

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*Theoret. Comput. Sci.***303**(2003), 303-331. Given a finite set of tiles*T*, the group of invariants*G*(*T*) consists of the linear relations that must hold between the number of tiles of each type in tilings of the same region. This paper surveys what is known about*G*(*T*). These invariants are shown to be much stronger than classical coloring arguments.M. Paulhus. An algorithm for packing squares.

*J. Combin. Theory Ser. A***82**(1998), 147–157. Paulhus presents an algorithm for packing an infinite set of increasingly small rectangles with total area*A*into a rectangle of area very slightly larger than*A*. He applies his algorithm to three known problems of this sort, obtaining extremely tight packings.J. Propp. Lattice structure for orientations of graphs, preprint, 1994, arXiv: math/0209005. It is shown that the set of orientations of a graph that have the same flow-differences around all circuits can be given the structure of a distributive lattice. This generalizes similar constructions for alternating-sign matrices and matchings.

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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.W. Thurston. Conway’s tiling groups.

*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.S. Wagon. Fourteen proofs of a result about tiling a rectangle.

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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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DOI: https://doi.org/10.1007/s00283-010-9160-9