### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Rati-o

##### Stage: 3 Challenge Level:

Correct solutions were received from Chen from The Chinese High School Singapore, Andrei of School 205 Bucharest, Mary of Brirchwood High School (whose solutions is used as the basis of the one below) and Sana, Jenny, Chris and Rosion of Madras College, St. Andrews.

l is a 1/3 the length of the rectangle

w is a1/3 the width of the rectangle

area of the rectangle = 3l x 3w = 9lw

Area triangle A = Area of triangle C = lw

Area triangle B = Area of triangle D= lw

Therefore area of the parallelogram = 9lw - 4lw = 5lw

Therefore ratio of the area of the parallelogram to the area of the rectangle is 5lw : 9lw = 5:9