A book of puzzles collected by Ivan Moscovich includes the six circles theorem (first announced in 1974 by Evelyn, Money-Coutts, and Tyrell) as a remarkable fact, but no explanation is given. Diagrams and geometry software seem to confirm it, but no proof could be found in the accessible literature. It took months of thinking time and the pursuit of several ideas that led nowhere before the following proof materialised.

​A teacher, Tom-P, with an interest in preparing students for Mathematical Olympiad competitions, published the following problem.

Pierre de Fermat, among other things, enunciated a principle of least time for the behavior of nature, regarding light being refracted at the boundary between two transparent media. The principle was a physical hypothesis that led Fermat, via some ingenious mathematics, to a prediction of the observed law of refraction. As it turned out, the prediction matched the known results, thus supporting the proposed principle.
In this piece, we invoke the same least-time idea but in the well-known context of a lifeguard running across sand at a beach. We show with a little calculus how the law of refraction, Snell’s law, emerges. Then, we consider what a least-time path should look like if the sand were to vary in its softness between the lifeguard station and the water.
Finally, we see that the same mathematical ideas can be applied to the brachistochrone problem that Johann Bernoulli proposed and we give a relatively easy solution.

​If you could put all of the mathematics of the Fibonacci and related sequences into one document you would probably end up with something very much like this particular blog. We think this special Fibonacci compilation deals with it all. See what you think.  

​We have compiled some of our favourite mathematical puzzles in this blog. They have come from a range of sources and solutions for each puzzle are provided at the end.  Enjoy.

​A piano is as much a mathematical instrument as it is a musical instrument. The history and the various logics that have been applied to its tuning is a fascinating one. This blog introduces the some of the reasoning that has led us from the days of perfect harmony to a modern day tempered pragmatism.  

​Getting a refreshing geometrical bent to the algebraic problem of finding quadratic roots provides fertile ground for upper secondary students. This
blog investigates geometric methods that were developed by the 17th century French philosopher, mathematician and Scientist Rene Descartes and
the 19th century philosopher, essayist and mathematician Thomas Carlyle. This is great material for classroom investigations or project work.

​Echoing the intent of some articles recently published in the Australian Mathematics Education Journal, we encourage the use of puzzles in the classroom as an effective pedagogical strategy. This blog traces  our own experience of discovery and of being led to make connections between the problem at hand and some half-forgotten pieces of mathematical learning. We argue that students are likely to feel motivated just as we were, given the stimuli of suitable puzzles. 

​George Gamow’s book One Two Three…Infinity – Facts and speculations of Science published in 1947 by Viking Press and reprinted by Mentor Books for the New American Library in 1953, included a problem about a lost treasure.
 
According to Gamow’s original story, there was a young and adventurous man who found, among his grandfather’s papers, a parchment that revealed clues to the location of a great treasure. According to the parchment, the treasure was buried on a certain island, and using two prominent trees and an old broken Gallows the exact location could be obtained. Alas the Gallows were gone, and despite the young man digging holes all over the island, the treasure remained hidden.
 
While the authors in this blog have added their own colour to the story the original puzzle remains completely intact. Three separate proofs are given that show that the location of the Gallows was irrelevant to finding the treasure, but please don’t tell that young man.

​Any sketch of a cubic function will show the curve crossing the  axis at least once. If it has exactly one real root then it must have two complex conjugate roots as well. While the real root is easily seen as the  intercept, the complex roots are not so obvious. This blog however illustrates a simple technique that enables an estimate to be made on them. Given a scaled sketch, nothing more than a ruler and a sharp pencil is required to obtain the estimate.