Gerald L. Alexanderson

The William Lowell Putnam mathematical competition

The following results of the thirty-first William Lowell Putnam Mathematical Competition held on December 5, 1970, have been determined in accordance with the regulations governing the Competition. This competition is supported by the William Lowell Putnam Intercollegiate Memorial Fund left by Mrs. Putnam in memory of her husband and is held under the auspices of the Mathematical Association of America. The first prize, five hundred dollars, is awarded to the Department of Mathematics of the University of Chicago, Chicago, Illinois. The members of the team were Robert Israel, Robert A. Oliver, and Robert Tax; to each of these a prize of one hundred dollars is awarded. The second prize, four hundred dollars, is awarded to the Department of Mathematics of Massachusetts Institute of Technology, Cambridge, Massachusetts. The members of the team were David M. Christie, Don Coppersmith, and Steven Winker; to each of these a prize of seventy-five dollars is awarded. The third prize, three hundred dollars, is awarded to the Department of Mathematics of the University of Toronto, Toronto, Ontario, Canada. The members of the team were Daniel Gautreau, Daryl Geller, and Joseph S. Repka; to each of these a prize of fifty dollars is awarded. The fourth prize, two hundred dollars, is awarded to the Department of Mathematics of the Illinois Institute of Technology, Chicago, Illinois. The members of the team were George F. Cornelius, Zbigniew Friedorowicz, and Wayne F. Mroz; to each of these a prize of fifty dollars is awarded. The fifth prize, one hundred dollars, is awarded to the Department of Mathematics of the California Institute of Technology, Pasadena, California. The members of the team were Leonidas Guibas, Andrew M. Odlyzko, and David J. Smith; to each of these a prize of fifty dollars is awarded. The six persons ranking highest in the examination, named in alphabetical order, are Jockum Aniansson, Yale University; Don Coppersmith, Massachusetts Institute of Technology; Jeffrey Lagarias, Massachusetts Institute of Technology; Robert A. Oliver, University of Chicago; Arthur Rubin, Purdue University; and Steven K. Winker, Massachusetts Institute of Technology. Each of these has been designated as a Putnam fellow by the Mathematical Association of America and is awarded a prize of two hundred and fifty dollars. The five persons ranking second highest in the examination, named in alphabetical order, are Daryl Geller, University of Toronto; Zbigniew Friedorowicz, Illinois Institute of Technology; Dale H. Peterson, Yale University; Joseph S. Repka, University of Toronto; and Jonathan Rosenberg, Harvard University. To each of these a prize of one hundred dollars is awarded. The following teams, named in alphabetical order, won honorable mention: University of California at Davis, the members of the team were Dean Hickerson,

Mathematical people : profiles and interviews

This unique collection contains extensive and in-depth interviews with mathematicians who have shaped the field of mathematics in the twentieth century. Collected by two mathematicians respected in the community for their skill in communicating mathematical topics to a broader audience, the book is also rich with photographs and includes an introduction by Philip J. Davis.

The William Lowell Putnam mathematical competition: problems and solutions: 1965-1984

1. Dedication 2. Preface 3. Recollections of the first Putnam examination by Herbert Robbins 4. List of problems 5. Solutions to the list of problems in the various competitions Twenty-sixth Twenty-seventh Twenty-eight Twenty-ninth Thirtieth Thirty-first Thirty-second Thirty-third Thirty-fourth Thirty-fifth Thirty-sixth Thirty-seventh Thirty-eighth Thirty-ninth Fortieth Forty-first Forty-second Forty-third Forty-fourth Forty-fifth 6. Appendices Winning teams Winning individuals 7. Index of problems.

Arrangements of planes in space

This paper presents a variety of formulas for the number of cells, faces, and edges, bounded and unbounded, that are formed by an arbitrary set of planes in R^3. Using an elegant geometric method described in 1966 by Brousseau, we first prove a version in R^3 of the general partition formulas established algebraically by Zaslavsky in 1975. From these formulas we deduce two families of inclusion-exclusion formulas for the counters, the first similar to formulas outlined by Roberts in 1889, the second related to formulas given by Steiner in 1826. We conclude with some non-trivial new bounds for the counters of an arbitrary arrangement in R^3 and two specific examples.

DISSECTIONS OF A PLANE OVAL

This paper is concerned with the general plane dissection problem of which this problem is a prototype. In section 3 we establish general formulas for the number R of regions and the number S of segments in a dissected oval in terms of the intersection structure of the dissecting chords. The formula for R is the analog for ovals of a formula for lines in the plane found in 1966 by Alfred Brousseau [6]. In section 4 we deduce general formulas for R and S that depend on different data. These formulas are analogs for ovals of formulas for lines in the plane given in 1889 by Samuel Roberts [15]. Section 5 concludes with a number of examples, including a third set of general formulas for lines in the plane based on formulas proved in 1826 by J. Steiner [16]. In the Klein-Beltrami model of the hyperbolic plane, points are represented by points of the open unit disk and lines by chords of that disk (see, for example, Wylie [17, pp. 290-331]). Ovals in the hyperbolic plane are represented by ovals in the disk. It follows that our formulas are correct for dissected ovals in the hyperbolic plane as well as in the Euclidean plane. We shall, however, confine the exposition to the Eucidean plane.

The Sixty-Third William Lowell Putnam Mathematical Competition

The following results of the fifty-third William Lowell Putnam Mathematical Competition, held on December 5, 1992, have been determined in accordance with the governing regulations. This annual contest is supported by the William Lowell Putnam Prize Fund for the Promotion of Scholarship, left by Mrs. Putnam in memory of her husband, and is held under the auspices of the Mathematical Association of America.

Dissections of a tetrahedron

Abstract A formula for the maximum number of cells formed by planes that dissect a tetrahedron through its edges is established in four ways: by recursion, by a computation using Eulers formula V − E + F − C = 1, as a special case of a general partition formula due to S. Roberts, and as a consequence of a formula of Steiners on the partition of space by families of parallel planes.

A simplical 3-arrangement of 21 planes

Abstract A study of certain projective 4-arrangements associated with the cross-polytope has turned up a sporadic simplicial 3-arrangement A31(21) of 21 planes that is not included in the list recently compiled by Branko Grunbaum and G.C. Shephard. In this paper we describe this arrangement and determine the values of its various numerical counters.

The Sixty-second William Lowell Putnam Mathematical Competition

The results of the Seventy-Second William Lowell Putnam Mathematical Competition, held December 3, 2011, follow. They have been determined in accordance with the regulations governing the Competition. The contest is supported by the William Lowell Putnam Prize Fund for the Promotion of Scholarship, an endowment established by Mrs. Putnam in memory of her husband. The annual Competition is held under the auspices of the Mathematical Association of America. The first price,

The Sixty-Fourth William Lowell Putnam Mathematical Competition

25,000, was awarded to the Department of Mathematics of Harvard University. The members of the winning team were Eric K. Larson, Evan M. O’Dorney, and Alex (Lin) Zhai; each was awarded a prize of