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# In Constantly Passing

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Age 14 to 16

Challenge Level

- Problem
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A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same direction.

Buses leave the depot at regular intervals; they travel along the dual carriageway and back to the depot at a constant speed.

At what interval do the buses leave the depot?

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?