Mathematics Magazine - October 2017

The old real estate saying of location, location, location gets a math take in the October issue of Mathematics Magazine. Baeth, McKee, and Luther revisit a 1989 Putnam problem that focuses on “downtown” when the notion of distance changes. Kaplan considers the best place to open a restaurant in two-dimensional lattice, based on counting lattice paths.

Two articles also deal with location, but from a different perspective: Letson and Schwartz use parameteric curves and differential geometry to determine where one should stand to view a painting while Futamura and Lehr determine where one should stand in front of an image in two-point perspective to view it correctly.

Unrelated to the theme, Aboufadel explains how the Shazam app works, describing a wavelet-based method to search signals from glucose monitors. The issue includes proofs without words, a crossword and Pinemi puzzle, the announcement of the Allendoerfer awards, and problems, solutions, and results from the 46th USAMO and 8th USAJMO, as well as the regular department of Reviews and Problems.

—Michael A. Jones, Editor

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Vol. 90, No. 4, pp. 241 – 319

Articles

The Downtown Problem: Variations on a Putnam Problem

p. 243.

Nicholas R. Baeth, Loren Luther, and Rhonda McKee

The following problem appeared in the afternoon session of the Fiftieth William Lowell Putnam Mathematical Competition in 1989. A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the centerthan to any edge. In general, which points are closer to the center of a square (or cube) than to any of the four edges (six faces)? And what do we mean by “closer”? Using a variety of distance functions (metrics), we will investigate these questions and pose several additional open problems for the reader to explore.

To purchase from JSTOR: 10.4169/math.mag.90.4.243

Pinemi Puzzle

p. 257.

Lai Van Duc Thinh

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Proof Without Words: Triangular Sums and Perfect Quartics

p. 258.

Charles F. Marion

It is well known that the sum of two consecutive terms from the sequence of triangular numbers is a perfect square. We show that the sum of two consecutive terms from a subsequence of that sequence is a perfect quartic.

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The Regiomontanus Problem

p. 259.

Benjamin Letson and Mark Schwartz

The Regiomontanus problem is that of determining the optimal place to stand to view a painting. The problem, which predates calculus by some 200 years, appears in most modern calculus books, though isn’t known by that name. The original solution was purely geometric, the optimality following from the well-known intersecting secants theorem. Modern treatments are analytic, using calculus, but none of the underlying geometry(or its original solution) is apparent. In this paper, we generalize the problem and unite the two solutions using parametric curves and results from differential geometry.

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A New Perspective on Finding the Viewpoint

p. 267.

Fumiko Futamura and Robert Lehr

We take a fresh perspective on an old idea and create an alternate way to answer the question: where should we stand in front of an image in two-point perspective to view it correctly? We review known geometric and algebraic techniques, then use the cross ratio to derive a simple algebraic formula and a technique that makes use of slopes on a perspective grid.

To purchase from JSTOR: 10.4169/math.mag.90.4.267

Where Should I Open My Restaurant?

p. 278.

Nathan Kaplan

We answer a question in real estate that is a variation of a problem that arises in many combinatorics and discrete mathematics courses.

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Proof Without Words: Series of Perfect Powers

p. 286.

Tom Edgar

We wordlessly show that the sum of reciprocals of perfect powers (with duplicates included) is 1.

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Imitating the Shazam App with Wavelets

p. 287.

Edward Aboufadel

With the Shazam smartphone app, a listener captures a short excerpt of a recorded song with the smartphone’s microphone, and in a matter of moments the app reports the name of the song and the artist. Fourier analysis is a key mathematical tool that powers the app. In this paper, we describe a wavelet-based method that captures the basic process used by the Shazam app to search a database of number sequences (signals) to find those that are similar to a test signal. We will describe our implementation with a different source of signals: continuous glucose monitor data from the management of type-1 diabetes.

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Geometry

p. 296.

Maureen T. Carroll

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Proof Without Words: Sums of Odd Integers

p. 298.

Samuel G. Moreno

The number of unit-squares in the left figure equals the sum of the area of the rectangle (of length n), plus the area of the big triangle (of height 2n − 2), plus the area of the n − 1 small 2 × 1 triangles.

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Problems and Solutions

p. 299.

Proposals, 2026-2030

Quickies, 1073-1074

Solutions, 1996-2000

Answers, 1073-1074

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Reviews

p. 306.

Only 15; sabbatical at the NSA; Noether’s theorem; cryptocurrencies

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News and Letters

Carl B. Allendoerfer Awards

p. 308.

To purchase from JSTOR: 10.4169/math.mag.90.4.308

46th United States of America Mathematical Olympiad and 8th United States of America Junior Mathematical Olympiad

p. 311.

Gabriel Carroll and Doug Ensley

To purchase from JSTOR: 10.4169/math.mag.90.4.311