### 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...

# Spotting the Loophole

### Why do this problem?

This problem encourages students to use visualisation to help them to spot a solution which can then be verified using algebra. It shows students the power of using visual representations to solve vector problems, often the quickest route to a solution.

### Possible approach

This could form a short introduction to work on vectors.
Display the example grids on the board, showing the three vectors forming a closed loop and the four vectors on the right which do not have a zero sum.

Then challenge students to find any closed loops in each of the three grids.
When they think they have spotted a closed loop by eye, they should verify algebraically that it is indeed a closed loop.

### Key questions

What can you say about the horizontal components of the vectors in a closed loop?
And what about the vertical components?

### Possible extension

Vector Walk begins to explore properties of vectors and combining two basic vectors to reach points on the coordinate grid.

### Possible support

Students may find it helpful to draw the vectors on squared paper.