### 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?

# Regional Division

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

Some trial and error will produce a solution like that on the right, where there are $9$ different areas enclosed.

To see that this is indeed the maximum, there is always one central region ($9$ on the diagram), and then the others must be separated from this by one of the sides of one of the rectangles. Each side can only separate one region, and as there are a total of $8$ sides, this means at most $9$ regions in total.

This problem is taken from the UKMT Mathematical Challenges.
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