### 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

##### Stage: 4 Challenge Level:

In 2D, vectors may be thought of as arrows with a fixed length and direction. The place at which the arrow starts is not important, so they may be translated around the plane without affecting the value of the vector.

This allows us to add and subtract vectors visually: if the vectors can be translated so as to form a closed circuit then the vector sum is zero. If the vectors cannot be translated so as to form a closed circuit then their vector sum is not zero, as in this diagram in which the vectors on the left have zero sum, whereas the vectors on the right don't have zero sum.

In this problem we are given three grids of vectors. The vectors are only represented visually, but it is assumed that their x and y values are whole numbers (Remember: the x and y values represent the horizontal and vertical offsets from the start of the arrow, not the location of the start of the arrow itself).

Your task, in each case is to try to see which subsets of the vectors form closed loops.

In each grid can you find a closed loop of vectors?

In each case is the closed loop unique? You can prove your assertions using algebra or a convincing visual argument.

Try hard to solve using visualisation before resorting to algebra.

Grid 1:

Grid 2

Grid 3