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Cushion Ball

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?


Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?


Age 14 to 18 Challenge Level:

Why do this problem?
The problem develops visualisation. Students may be interested because snooker is a popular game and they may be impressed that expert players can judge the angle of the shot accurately to bounce of two, three or even 4 walls and hit the target. One way to solve the problem is to use coordinates, similar triangles and gradients.

Possible approach
Work through one of the cases as a class together and then suggest that students draw a diagram. Then discuss the diagram as a class before the students calculate the various paths.

Key question
How can we turn into a straight line the path and the reflected path after the ball bounces off a cushion?

Possible support
Initially ignore the pink and blueballs which might be in the way of the shot.
See also Cushion Ball which has an interactivity.

Possible extension
See the article In Space Do All The Roads Lead Home?