### 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...

# Always Perfect

##### Stage: 4 Challenge Level:

Have you tried some simple cases to see if there is a pattern?

If you have a number "$x$" the next number is "$x +1$". How could you write four consecutive numbers?

If you can express $x^4 + \ldots + 16$ as a perfect square, the two factors will be of the form $(x^2 +\ldots + 4)$.