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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.

# Pyramidal N-gon

##### Age 14 to 16 ShortChallenge Level The base of the pyramid has $n$ edges, so also has $n$ vertices around the base. This then means that there are $n$ edges around the base (in red) of the pyramid and $n$ that meet at the apex (in black).

Therefore there are $2n$ edges in total.

There are also $n$ faces that meet at the apex of the pyramid, and one more for the base, so a total of $n+1$ faces.

Therefore the difference is $2n - (n+1) = n-1$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.