### Consecutive Numbers

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

### 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.

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?

# Powerful Order

##### Stage: 3 and 4 Short Challenge Level:

By comparing b and c first, we have $b= 8^8= (2^3)^8=2^{24}=(2^2)^{12}=4^{12}> 3^{11} = c$

so $c< b$.

But also $b= 2^{24}< 2^{25}=a$ so $b< a$.

Together these give $c< b< a$.

This problem is taken from the UKMT Mathematical Challenges.
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