### Tetra Inequalities

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

### Staircase

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

### Proof Sorter - the Square Root of 2 Is Irrational

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.

# Be Reasonable

##### Age 16 to 18 Challenge Level:

Try a proof by contradiction. Suppose that the three irrational numbers do occur in some arithmetic series. Can you then go on to reach a contradiction?

It helps to have seen a proof that $\sqrt 2$ is irrational and to appreciate how the logic of arguments by contradiction work.

Then you only need to know the definition of an arithmetic series to do this problem. If the difference between $\sqrt 2$ and $\sqrt 3$ is an integer multiple of the common difference in an arithmetic series, and the difference between $\sqrt 3$ and $\sqrt 5$ is also an integer multiple of that common difference, can you use these two facts to write down two expressions, eliminate the unknown common difference and then find an impossible relationship?