### Tetra Inequalities

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?

### 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 18Challenge Level

Why do this problem?
It is an exercise in proof by contradiction.

Possible approach

Then discuss the proof that $\sqrt 2$ is irrational.

The students can work with the interactivity Proof Sorter and perhaps some of them might read this article which was written by two undergraduates.

Key Question

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?

Possible support

Possible extension

Can you prove that $\sqrt{1}$, $\sqrt{2}$ and $\sqrt{3}$ cannot be terms of ANY arithmetic progression?