September 2012 Contents
In the September issue of The College Mathematics Journal, John Dodge and Andrew Simoson enumerate Ben-Hur staircase climbs to ensure that left and right legs are exercised equally, Reuben Hersh explains Faulhaber polynomials, Óscar Ciaurri and his colleagues prove Vietè’s product in the finest ancient style, Joe DeMaio counts triangles to count squares, Brian Thomson looks at Riemann integrability, and much more.
Ben-Hur Staircase Climbs
John Dodge and Andrew Simoson
How many ways may one climb an even number of stairs so that left and right legs are exercised equally, that is, both legs take the same number of strides, take the same number of total stairs, and take strides of either 1 or 2 stairs at a time? We characterize the solution with a difference equation and find its generating function.
The Hyperbolic Sine Cardinal and the Catenary
The hyperbolic function sinh(x)/x receives scant attention in the literature. We show that it admits a clear geometric interpretation as the ratio between length and chord of a symmetric catenary segment. The inverse, together with the use of dimensionless parameters, furnishes a compact, explicit construction of a general catenary segment of given length hanging from supports of different heights.
Vičte’s Product Proved in the Finest Ancient Style
Ňscar Ciaurri, Emilio Fernández, Rodolfo Larrea, and Luz Roncal
This paper gives a very elementary, essentially visual proof of Vičte’s product. We employ only the Pythagorean theorem, similarity of triangles, and exhaustion.
Counting Triangles to Sum Squares
Counting complete subgraphs of three vertices in complete graphs, yields combinatorial arguments for identities for sums of squares of integers, odd integers, even integers and sums of the triangular numbers.
Teaching Tip: Are You Changing the Rules? Again?
Students often complain that the rules of mathematics are being changed. A short conversation between a professor and a class of college algebra students dramatizes this in the realm of complex numbers and the legal realm of speed limits.
On the Steiner Minimizing Point and the Corresponding Algebraic System
Ioannis M. Roussos
We summarize the most important facts about the Steiner point of a triangle and find formulas for its distance to each vertex in terms of the side-lengths of the triangle.
Viviani Polytopes and Fermat Points
Given a set of oriented hyperplanes in , define by v(X) = the sum of the signed distances from X to p1,…, pk , for any point . We give a simple geometric characterization of P for which v is constant, leading to a connection with the Fermat point of k points in . Finally, we discuss the full content of Viviani’s theorem historically.
A Strong Kind of Riemann Integrability
Brian S. Thomson
The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and students of elementary integration theory a curious and illuminating detour from the usual Rieman integral.
Proof Without Words: Partial Sums of an Arithmetic Sequence
Anthony J. Crachiola
A visual proof that a partial sum of an arithmetic sequence equals the number of terms times the average of the first and last term.
Why the Faulhaber Polynomials Are Sums of Even or Odd Powers of (n + 1/2)
By extending Faulhaber’s polynomial to negative values of n, the sum of the p’th powers of the first n integers is seen to be an even or odd polynomial in (n + 1/2) and therefore expressible in terms of the sum of the first n integers.
Extending the Alternating Series Test
Alternating series have the simplest of sign patterns. What about series with more complicated patterns? By inspecting the alternating series test closely, we find a theorem that applies to more complicated sign patterns, and beyond.
Series that Converge Absolutely but Don’t Converge
Robert Kantrowitz and Michael Schramm
If a series of real numbers converges absolutely, then it converges. The usual proof requires completeness in the form of the Cauchy criterion. Failing completeness, the result is false. We provide examples of rational series that illustrate this point. The Cantor set appears in connection with one of the examples.
Two Semigroup Elements Can Commute with Any Positive Rational Probability
Vadim Ponomarenko and Natalie Selinski
We construct semigroups with any given positive rational commuting probability, extending a Classroom Capsule from November 2008 in this JOURNAL.
PROBLEMS AND SOLUTIONS