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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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Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

Which Solids Can We Make?

Age 11 to 14 Challenge Level:
Sometimes it helps to think about two dimensions to get a better understanding of what happens in three dimensions:
 
In two dimensions, the interior angles of convex polygons are always less than $180^{\circ}$. Can you explain why?
In three dimensions, the sum of the interior angles at each vertex of convex polyhedra must be less than $360^{\circ}$. Can you explain why?

The exterior angles of polygons are a measure of how far short the angles are from $180^{\circ}$.
The angle deficit at a vertex of a polyhedron is a measure of how far short each angle sum is from $360^{\circ}$.
 
The sum of the exterior angles of a polygon is always $360^{\circ}$.
What do you notice about the total angle deficit of a solid?
This table might help:
 
  Angle Sum  Angle Deficit  Number of Vertices  Total Angle Deficit 
Cube  270 90 8 720
Tetrahedron         
Octahedron         
Icosahedron         
Dodecahedron