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.

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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?

Spotting the Loophole

Stage: 4 Challenge Level:

In order to solve this problem you will need to think about what makes a loop closed instead of open.

In two dimensions the overall changes in the x direction around the loop must cancel out and the overall changes in the y directions around the loop must also cancel out. You can think about these two directions separately if you cannot quickly spot how to form a closed loop.

Visually you can use lots of intuitive shortcuts, such as that 'long' arrows may need multiple 'short' arrows to close up into a loop.

Of course, when you think that you have spotted a loop you will need to check carefully to see if you are correct by adding up the vectors exactly.