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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?
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?
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.
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 |
NRICH is part of the family of activities in the Millennium Mathematics Project.