### 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?

# Fruit Line-up

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

Each fruit can have at most two fruits next to it but each type of fruit must be next to three other types of fruit so there are at least two of every fruit. This means there are at least $8$ fruits in total. In fact $8$ are sufficient, as shown in the arrangement

OABPOBAP (O for Orange, P for Peach, A for Apple, B for Banana)

This problem is taken from the UKMT Mathematical Challenges.
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