Mathematical Explorations

This page will be expanding as I put up a set of problems which I have enjoyed working on over the years.
In many cases there will be links to reports of my explorations.
In some cases there is a collection of problems, constituting a domain of exploration.
Domain Titles Initiating Problem(s)
Core Area & Core Perimeter What is the ratio between the area of a shape and the are of the curve enveloped by the lines which bisect the area of the shape? Same for perimeters
Animation for triangles
Polynomial Explorations
A collection of explorations of properties and features of polynomials as introduction to functions and the calculus, but going beyond what is normally included in getting to grips with polynomials
Intermediate Curvature Theorem If a function is twice differentiable on an interval, and if three points are chosen in that interval, is there a point on the curve at which the curvature is the same as the curvature of a circle through those  three points (the Menger curvature)?
Mean and Intermediate Value Theorems
Petrol Pump Problem A collection of generalisations of a problem in Krutetskii: if there are petrol pumps placed around a circular track, with just enough petrol in total for a car to make a complete circuit, is there somewhere on the track where a car can start and make it all the way, picking up petrol at the pumps?
Fractional Difference In how many ways can the fraction 1/n be written as the difference of two unit fractions?
Least Common Multiples In how many ways can n be the least common multiple of two (positive) numbers?
What connections if any are there with Fractional Difference?

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