### 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?

# Three Cubes

##### Stage: 4 Challenge Level:

1. The distance between centres of two adjacent faces of a cube is $8$cm.

What is the side length of the cube?

2. I can make an equilateral triangle by cutting off the corner of a cube.

If the area of the largest equilateral triangle I can create in this way is $140$cm$^2$, what is the side length of this cube?

3. A third cube has an edge length of $12$cm.

At each vertex, a tetrahedron with three mutually perpendicular edges of length $4$cm is cut away.

Can you find the volume and the surface area of the remaining solid?
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