A relation between Catalan Numbers and World Series type problems

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The May issue of *The College Mathematics Journal* begins provocatively with the article "Quiz Today: Should I Skip Class?" by Peter Zizler, exploring the impact on course grades of selective assessment (e.g., counting only the best 3/5 quizzes). Weird fractions return in "How Weird Are Weird Fractions?" by Ryan Stuffelbeam. There is also a new solution to the Basel problem by David Benko and John Molokach in "The Basel Problem as a Rearrangement of Series."

As always, Problems and Solutions challenge readers and Media Highlights keep them well-informed.—*Michael Henle*

Vol. 44, No. 3, pp.166-255.

**Journal subscribers and MAA members:** Please login into the member portal by clicking on 'Login' in the upper right corner.

Peter Zizler

How does selective assessment, wherein one counts only the best $$k$$ out of $$n$$ quizzes set, impact grade inflation? Based on our analysis, a specific quantitative answer can be given to a student who plans to skip a quiz, depending, of course, on the student's quiz writing consistency—or inconsistency.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.166

Joe DeMaio

A visual proof that alternating some of the first $$n$$ squares counts the number of edges in the complete graph on $$n+1$$ vertices.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.170

David Benko and John Molokach

We give an elementary solution to the famous Basel Problem, originally solved by Euler in 1735. We square the well-known series for $$\arctan(1)$$ due to Leibniz, and use a surprising relation among the re-arranged terms of this squared series.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.171

Paul Bruckman, Joseph B. Dence, Thomas P. Dence, and Justin Young

Reciprocal triangular numbers have appeared in series since the very first infinite series were summed. Here we attack a number of subseries of the reciprocal triangular numbers by methodically expressing them as integrals.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.177

Gordon A. Swain

We show that inside every triangle the locus of points satisfying a natural proportionality relationship is a parabola and go on to describe how this triangle-parabola relationship was used by Archimedes to find the area between a line and a parabola.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.185

Yukio Kobayashi

A visual proof that the triangular numbers are sums of every other odd number.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.189

Jeremiah William Johnson

We count the number of group homomorphisms between any two dihedral groups using elementary group theory only.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.190

James Sandefur and John Mason

Given a family of $$p\geq3$$ points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all $$p$$ points. Similarly, we show that for $$p\geq3$$ circles, there are three of them such that the smallest circle encompassing them encompasses all the circles, and that there are three circles for which the smallest circle encompassing and tangent to them encompasses all the circles.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.193

Ryan Stuffelbeam

A positive rational is a weird fraction if its value is unchanged by an illegitimate, digit-based reduction. In this article, we prove that each weird fraction is uniquely weird and initiate a discussion of the prevalence of weird fractions.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.202

Anurag Agarwal, James E. Marengo, and Likin C. Simon Romero

A $$k$$-out-of-$$n$$ system functions as long as at least $$k$$ of its $$n$$ components remain operational. Assuming that component failure times are independent and identically distributed exponential random variables, we find the distribution of system failure time. After some examples, we find the limiting distribution of system life when $$n$$ approaches infinity, and apply these results to evaluate two challenging integrals.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.210

David James and Cynthia Botteron

A certain weighted average of the rows (and columns) of a non-negative matrix yields a surprisingly simple, heuristical approximation to its singular vectors. There are correspondingly good approximations to the singular values. Such rules of thumb provide an intuitive interpretation of the singular vectors that helps explain why the SVD is so effective in analyzing large data sets.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.220

Fuchang Gao

A simple proof of the right-hand rule of the cross product is presented.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.227

Walden Freedman

$$N$$ people take turns pulling a heavy rope hanging off the edge of a building. How much of the rope should each pull in order to share the work equally? Which person's fair share is closest to 1/$$N$$th of the rope's length? We answer these questions using basic calculus and suggest some related problems for student exploration.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.229

Problems 1001-1005

Solutions 976-980

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.233

*Journey through Mathematics: Creative Episodes in Its History*

by Enrique A. González-Velasco

reviewed by Robert E. Bradley

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.241

*The Manga Guide to Linear Algebra*

by Shin Takahashi, illustrated by Iroha Inoue

reviewed by Susan Jane Colley

*Math Girls*

by Hiroshi Yuki (translated by Tony Gonzalez)

reviewed by Susan Jane Colley

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.244

To purchase the article from JSTOR: http://dx.doi.org/10.4169/college.math.j.44.3.248