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A Mean Tetrahedron

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?

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Tetra Square

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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Icosian Game

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Tetrahedra Tester

Age 11 to 14 Challenge Level:

What are the limitations on the lengths of the sides of any triangle?

Look at different cases.

If the base is a 4 by 5 by 6 triangle, what size can the other faces be?

How many different tetrahedra can have as its base a 4 by 5 by 6 triangle?

Can you find 8 or more possible tetrahedra?

How will you know when you have them all?