# Squaring the plane

Year of Award: 2014

Award: Robbins

Publication Information: American Mathematical Monthly, 115 (2008), no. 1, 3-12.

Summary: (Adapted from the JMM 2014 Prizes and Awards booklet) The problem is simple. You are supplied with infinitely many square tiles, but they all have different sizes, in fact, there is exactly one $n$-by-$n$ square for each positive integer $n$. Your task is to use these squares to tile the plane, no overlaps or gaps allowed, and you must use all the squares. This article gives a complete description of such a tiling of the plane.

About the Authors: (From the JMM 2014 Prizes and Awards booklet)

Frederick V. Henle received his baccalaureate from Harvard University in 1992 and his master's degree from Dartmouth College in 1997. He has taught mathematics and computer science at Mercersburg Academy, played in the first violin section of the Maryland Symphony Orchestra, and is now a lead developer at athenahealth, Inc. Work on this paper and subsequent papers with his father, Jim, has been both personally and professionally fulfilling, and an experience that he hopes one day to share with his children.

James M. Henle earned his baccalaureate degree from Dartmouth College in 1968 and his Ph.D. from MIT in 1976. Early in his career he taught at the University of the Philippines and at Burgundy Farm Country Day School, but for most of his professional life he has been a member of the faculty at Smith College.

Jim credits his mathematical awakening and development to his high school teacher Richard Jameson, to Dartmouth logician Donald Kreider, to his thesis advisor Gene Kleinberg, to his Smith colleagues Marjorie Senechal and Joe O'Rourke, to the columns of Martin Gardner, and most important, to his brother Michael Henle. Jim counts over two dozen collaborators on his research papers. The most frequent have been his academic siblings, Carlos Di Prisco, Arthur Apter, and Bill Zwicker; the most significant is his son, Fred.

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Summary:

(Adapted from the JMM 2014 Prizes and Awards booklet) The problem is simple. You are supplied with infinitely many square tiles, but they all have different sizes, in fact, there is exactly one $n$-by-$n$ square for each positive integer $n$. Your task is to use these squares to tile the plane, no overlaps or gaps allowed, and you must use all the squares. This article gives a complete description of such a tiling of the plane.