### 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?

### Grid Points on Hyperbolas

Find a condition which determines whether the hyperbola y^2 - x^2 = k contains any points with integer coordinates.

### Parabella

This is a beautiful result involving a parabola and parallels.

# Napoleon's Hat

##### Stage: 5 Challenge Level:

 The diagram shows three equilateral triangles $ABC, AYX$ and $XZB$. The point $X$ is a moveable point on $AB$. The points $P$, $Q$ and $R$ are the centres of the three triangles. Experiment with the dynamic diagram. What can you say about triangle $PQR$? Can you prove your conjecture?
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