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Explore the properties of combinations of trig functions in this open investigation.

Chance of That

What's the chance of a pair of lists of numbers having sample correlation exactly equal to zero?

A Different Differential Equation

Explore the properties of this different sort of differential equation.

The Best Square

Age 16 to 18
Challenge Level

This is an open investigation. It can be taken to various levels of complexity and note that the construction of clear, algorithmic procedures is difficult but fascinating.

Imagine that you have been asked to write a computer program to determine how accurately people can draw, freehand, a square of side $50$cm on an interactive whiteboard.
How would you mathematically judge the accuracy of such a drawing? Create a well-defined process by which the computer would be able to compute your measure of accuracy. You can assume that the whiteboard stores the freehand square internally as a set of pixels described by pairs of Cartesian coordinates.

Extensions: Consider the implementation of a similar process to judge the perfection of a free-hand circle; How might you tackle the problem of deciding if an image is a square or a circle?
Did you know ... ?

Image recognition is big business and very difficult: it involves cutting edge mathematics and computing. Imagine the intricacies involved in programming a computer to recognise typeface, handwriting or even human faces. Some progress is now being made into recognising images of faces from the binary encoding of photographs. We wonder how rapidly this development will proceed and what the resulting technological and sociological implications will be.