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Mathematics Magazine - February 2006

ARTICLES

A Bent for Magic
Paul C. Pasles
3-13
The term “magic square” is commonly used to refer to a variety of related concepts. In each of these, a square matrix has constant row sum and constant column sum, as well as other properties that depend on the goals and personal preferences of the magician. One particularly interesting species of magic square was pioneered by Benjamin Franklin. We examine unusual characteristics of the Franklin magic squares and argue that Franklin must have grounded his early experiments in the four-by-four case. We examine some algebraic properties of the panfranklin magic squares (for small orders) and the vertically symmetric Franklin magic squares (for all orders). Franklin’s famous eight-by-eight square is shown to be minimal.

Packing Boxes with Bricks
Richard J. Bower and T. S. Michael
14-30
Can you tile a 39 x 16 rectangle with 3 x 3 and 2 x 2 squares? More generally, which rectangular boxes can be packed by translates of given rectangular bricks in d dimensions? We survey old and recent theorems that give necessary and/or sufficient conditions for such packings.

We emphasize direct constructions and elementary counting methods and present a unified approach to many known results (e.g. theorems of de Bruijn, Klarner, Straus, Fricke, and Kelly). We also provide accessible proofs for some asymptotic packing theorems.

A New Model for Ribbons in R 3
Tom Farmer
31-43
What might a ribbon look like if one edge is forced to follow a given curve in space? For example, if the given curve is a helix then the ribbon might look like the red strip on a barber pole. Such a helical band is used as the prototype to describe a model that can be applied to other curves. A goal is to produce computer-drawn ribbons that can make illustrations of curves more interesting and informative. The methods used involve calculus, linear algebra, and elementary differential equations.

PROOF WITHOUT WORDS

Every Fourth Power Greater than One is the Sum of Two Nonconsecutive Triangular Numbers
Roger B. Nelsen
44

A proof without words of the following statement: If tk denotes the kth triangular number, then n4 =tn2-n-1 .

NOTE

Another Step FurtherÂ…On a Problem of the 1988 IMO
István G. Laukó, Gabriella A. Pintér and Lajos Pintér
45-53
In this note we consider a problem of the 1988 International Mathematical Olympiad, and show how new questions about the solution sets lead to the discovery of the structure of all solutions. The structure turns out to be governed by second order linear recurrence relations, and thus shares a lot of properties that one typically expects to find in connection with the Fibonacci-sequence. We provide several ideas for the generalization of the problem and many exercises for the interested reader to complete.

PROOF WITHOUT WORDS

Padoa’s Inequality
Roger B. Nelsen
53
A proof without words of the following inequality: If a,b,care the sides of a triangle, then abc ≤ (a + b – c)(B + c –a)(c + a – b).

NOTE

The Cross Ratio Is the Ratio of Cross Products!
Leah Wrenn Berman, Gordon Ian Williams, and Bradley James Molnar
54-59
The cross ratio is an important invariant in projective geometry. Even though it shares the term “cross”, we wouldn’t think it would be related to the familiar cross product of vector calculus. However, it turns out that you can use ratios of cross products to compute cross ratios! The authors discuss how to do so, as well as how to use cross products to compute variants of the cross ratio. Finally, they discuss the historical background of the terms and their relationships.

PROOF WITHOUT WORDS

Right Triangles and Geometric Series
Roger B. Nelsen
60
A proof without words using right triangles to illustrate the sum of a geometric series for both first term and common ratio equal to 1/2, 1/3, 1/4 and 1/5.

NOTES

The Euler-Maclaurin Formula and Sums of Powers
Michael Z. Spivey
61-65
The Euler-Maclaurin formula has been called “one of the most remarkable formulas of mathematics,” as it shows how a finite sum relates to the corresponding integral. In this note we illustrate the use of the Euler-Maclaurin formula by utilizing it to prove results concerning the power sum. First we show that, for each integer m greater than 0, the simple term mm is larger than the sum 1m +2m + ...(m-1)m . Then we show that, as m approaches infinity, sum 1m +2m + ...(m-1)m converges to a fixed percentage of mm.

PROOF WITHOUT WORDS

Inclusion-Exclusion for Triangular Numbers
Roger B. Nelsen
65
A proof without words of the following theorem: Let tk denote the kth triangular number, and let 0 ≤ a,b,c n and 2 n a + b + c. Then

ta + tb + tc - ta + b - n - tb + c - n - tc + a - n + ta + b + c -2n = tn .

Illustrations

NOTES

Disjoint Pairs with Distinct Sums
Gerhard J. Woeginger
66
Klamkin and Newman proved in 1969 that there are at most (2k-1)/5 disjoint pairs of positive integers with the distinct sums less or equal to k. The paper shows that this bound is tight.

POEM

The Eightfold Way, Lie Algebra, and Spider Hunting in the Dark
J. D. Memory
74
This poem exhibits connections between Buddha’s eightfold path to enlightenment, the Lie Algebra of strongly interacting particles developed by Murray Gell-Mann, and the appearance of the numeral “eight” in the anatomy of spiders. The form of the poem is based on Williams Blake’s The Tyger.

Dummy View - NOT TO BE DELETED