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# Cushion Ball

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Age 16 to 18

Challenge Level

A standard English snooker table is 6' x 12'. We know that the shortest path between any two points on the table is given by the straight line between them. How do we figure out the shortest distance between two points if the ball must bounce off one wall? What if it must bounce off 2 walls? 3 walls?

If you want to try one yourself, let the bottom left hand corner of the pool table be the origin. Consider the case when the cue ball is at (5,5) and the target ball is at (10,2). Which wall gives the shortest path to the target?

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.