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.
Viète's Product Proved in the Finest Ancient Style
Òscar Ciaurri, Emilio Fernández, Rodolfo Larrea, and Luz Roncal
This paper gives a very elementary, essentially visual proof of Viète'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.