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Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Snail's Pace

Stage: 3 and 4 Short Challenge Level: Challenge Level:1
In one hour, the snail can reach points within 1m of the corner at which it starts. So it can reach some of the points on the three faces which meet at that corner, but none of the points on the other three faces.

On each of the three reachable faces, the points which the snail can reach form a quarter of a circle of radius 1m.

So the required fraction is $\frac{3 \times \frac{1}{4}\pi \times 1 \times 1}{6 \times 1 \times 1} = \frac{\pi}{8}$.

This problem is taken from the UKMT Mathematical Challenges.
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