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The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

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This is a beautiful result involving a parabola and parallels.

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Square Pair Circles

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Napoleon's Hat

Age 16 to 18 Challenge Level:

Why do this problem?
Learners can investigate using the interactivity, make a conjecture and then try to prove their own conjecture.

It can be an exercise in coordinates.

It is useful for learners to appreciate that the symmetry of the result makes it unecessaryto work out all three distances.

Possible approach
Ideally learners would work independently on this problem.

Key question
How do we find the coordinates of the centroid of an equilateral triangle?

Possible support
GeoGebra is FREE educational mathematics software and your learners can experiment with it and draw their own dynamic diagrams.