### 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?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Amazing

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

The rooms are labelled A, B, C, D, E, F, G, X, Y as shown.

We look first at routes which visit no room more than once. We need to consider only routes which go from X to A, since each of these routes has a corresponding route which goes from X to C. For example, the route X A D E Y corresponds to the route X C D G Y.

Routes which start X A then go to B or to D. There are three routes which start X A B, namely X A B E Y, X A B E D G Y and X A B E D C F G Y. There are also three routes which start X A D, namely X A D E Y, X A D G Y and X A D C F G Y.

The condition that a gap in a wall closes once a person has travelled through it means that it is not possible to visit a room more than once unless that room has at least four gaps leading into and out of it, and the only such room is D. There are two routes which start X A and visit D twice. These are X A D G F C D E Y and X A D C F G D E Y.  So there are 8 routes which start X A and there are 8 corresponding routes which start X C, so there are 16 routes in all.

This problem is taken from the UKMT Mathematical Challenges.

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