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?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Painted Purple

Age 14 to 16 Short Challenge Level:

If opposite faces are painted in different colours, three red faces will meet at one corner, and three blue faces will meet at the diagonally opposite corner.

The three cubes that are adjacent to these two corner cubes will also be painted in just one colour.

The six cubes at the centre of each face will also be painted in just one colour.

The cube at the very centre will not be painted.

The remaining $12$ cubes will have at least one face painted in each colour
(27 - 2  - 2x3 - 6 - 1 = 12)

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.