Challenge Level

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

Challenge Level

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Challenge Level

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.

Challenge Level

This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.

Challenge Level

Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?

Challenge Level

Three of Santa's elves and their best friends are sitting down to a festive feast. Can you find the probability that each elf sits next to their bestie?

Challenge Level

Explore the strange geometrical properties of the Koch Snowflake.