Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
An introduction to proof by contradiction, a powerful method of mathematical proof.
The symbol [ ] means 'the integer part of'. Can the numbers [2x];
2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three
Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic
Show that there are infinitely many rational points on the unit
circle and no rational points on the circle x^2+y^2=3.
Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
Can you make a square from these triangles?
Show that it is rare for a ratio of ratios to be rational.