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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

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?


Age 16 to 18
Challenge Level
$$(x + x^2 + x^3 + x^4)^2 = x^2 (1 + x)^2 (1+x^2)^2 =[x + 2x^2 + x^3][x + 2x^3 + x^5].$$ corresponds to the re-labelling $1, 2, 2, 3$ and $1, 3, 3, 5$ .