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?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Instant Insanity

Stage: 3, 4 and 5 Challenge Level:

The edges in these 4 graphs show the colour pairings of opposite faces of the cubes.

To solve the problem combine all 4 graphs then look for 2 subgraphs, one representing the colours on the front and back walls of the tower and the other representing the colours on the left and right hand walls of the tower, such that each contains all 4 colours and precisely one edge of each numbered cube.