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The College Mathematics Journal Contents—January 2009

Vol. 40, No. 1, pp. 2-76

ARTICLES

Two Applications of the Hamming-Golay Code
By Andy Liu

In this paper, we give two unexpected applications of a Hamming code. The first one, also known as the "Hat Problem," is based on the fact that a small portion of the available code words are actually used in a Hamming code. The second one is a magic trick based on the fact that a Hamming code is perfect for single-error correction.—

Sledge-Hammer Integration
By Henry Ahner

Integration (here visualized as a pounding process) is mathematically realized by simple transformations, successively smoothing the bounding curve into a straight line and the region-to-be-integrated into an area-equivalent rectangle. The relationship to Riemann sums, and to the trapezoid and midpoint methods of numerical integration, is illustrated.

Trick or Technique?
By Michael Sheard

More often than one might at first imagine, a simple trick involving integration by parts can be used to compute indefinite integrals in unexpected and amusing ways. A systematic look at the trick illuminates the question of whether the trick is useful enough to be called an actual technique of integration.

Factoring Heron
By Vaughan Pratt

The inter-derivability of the Pythagorean Theorem and Heron's area formula is explained by applying Al-Karkhi's factorization to Heron's formula.

Diametric Quadrilaterals with Two Equal Sides
By Raymond A. Beauregard

If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider diametric quadrilaterals having two equal sides. They can be reduced to the two forms (a, b, a, d) (an isosceles trapezoid) and (b, a, a, d) (a skewed kite). It is shown that these quadrilaterals are diametric if and only if (r, a, d) is a right triangle satisfying d< √2a where r = √(d2 - a2), in which case the area of either quadrilateral is ((b+d)/4) √(d2- b2), where b = d - 2a2/d. The integer case (Brahmaguta quadrilaterals) is then considered. It is shown that each primitive Pythagorean triple (t, u, v) with v > √2u determines a Brahmagupta diametric quadrilateral with two equal sides uv, two other sides v2 – 2u2, v2, and area ut3. Moreover every primitive diametric quadrilateral with integer sides, two of which are equal, and with integer diagonals, arises in this way.

Solomon's Sea and π
By Andrew J. Simoson

This paper is a whimsical survey of the various explanations which might account for the biblical passage in I Kings 7:23 that describes a round object - a bronze basin called Solomon's Sea - as having diameter ten cubits and circumference thirty cubits. Can the biblical pi be any number other than 3? We offer seven different perspectives on this historical puzzle.

Evolutionary Stability in the Traveler's Dilemma
By Andrew T. Barker

The traveler's dilemma is a generalization of the prisoner's dilemma which shows clearly a paradox of game theory. In the traveler's dilemma, the strategy chosen by analysis and theory seems obviously wrong intuitively. Here we develop a measure of evolutionary stability and show that the evolutionarily stable equilibrium is in some sense not very stable. Understanding evolutionary stability on a continuum instead of as a yes-or-no definition helps to explain and partially resolve the paradox in the traveler's dilemma.

CLASSROOM CAPSULES

Series involving iterated logarithms
By J. Marshall Ash

The boundary between convergent and divergent series is systematically explored through sums of iterated logarithms.

Sums of Integer Powers via the Stolz-Cesàro Theorem
By Sidney H. Kung

The Stoltz-Cesàro Theorem, a discrete version of l'Hôpital's rule, is applied to the summation of integer powers.

Short Division of Polynomials
By Li Zhou

The familiar process of synthetic division is extended to much more complicated divisors and turns out, surprisingly, not to be as difficult as one might imagine.

MOODY’S MEGA MATH CHALLENGE: A MODELING COMPETITION

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