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

An "n" x "n" chessboard is called deficient if one square is missing from any spot on the board. Can all deficient boards with a number of cells divisible by 3 be tiled by bent (or L-shaped) trominoes? The answer is yes, with exception of the order-5 board. This paper deals with the general problem plus numerous related puzzles and proofs…

Motivated by Fitch Cheney's 5-card trick, we present and analyze several "n"-card tricks. We determine the minimum "n" to perform the trick, with and without timing.

Pompeiu's theorem states that if ABC is an "equilateral" triangle and M a point in its plane, then MA, MB, and MC form a new triangle. In this article, we have a new look at this theorem in the realm of arbitrary triangles. We discover what we call Pompeiu's Area Formula, a neat equality relating areas of triangles determined by the points A, B,…

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In…

The purpose of this article is to discuss specific techniques for the computation of the volume of a tetrahedron. A few of them are taught in the undergraduate multivariable calculus courses. Few of them are found in text books on coordinate geometry and synthetic solid geometry. This article gathers many of these techniques so as to constitute a…

The study of Kurt Godel's proof of the "incompleteness" of a formal system such as "Principia Mathematica" is a great way to stimulate students' thinking and creative processes and interest in mathematics and its important developments. This article describes salient features of the proof together with ways to deal with…

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

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

In this article, the author proves two conjectures of Thomas Dence.

We give an irredundant axiomatization of the complete ordered field of real numbers. In particular, we show that all the field axioms for multiplication with the exception of the distributive property may be deduced as "theorems" in our system. We also provide a complete proof that the axioms we have chosen are independent.

We state and prove the methods of False Position (Regula Falsa) and Double False Position (Regula Duorum Falsorum). The history of both is traced from ancient Egypt and China through the work of Fibonacci, ending with a connection between Double False Position and Cramer's Rule.

In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies…

Let I be a real interval. We call a continuous function [mu] : I x I [right arrow] [Bold R] a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I [right arrow] [Bold R} be a differentiable and strictly convex or strictly concave function. If a, b [image omitted] I with a [not equal to] b, then there exists a…

This article explores the time it takes for a liquid to drain from a cylindrical container through a hole in the bottom. Using dimensional analysis and some thought experiments this time is determined and Torricelli's law derived as a consequence. Finally, the effect of pouring liquid into the container as it drains is considered.