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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Some experimentation with some integers might help.
The factorisation is not straight forward but the first and last terms are easy to establish, then the middle terms follow quite quickly.