### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

### Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

### The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

# Dodgy Proofs

### Why do this problem?

Proof is, of course, a central part of mathematics. However constructing proofs is often difficult for novices. This problem provides a bridge through the device of classic faulty proofs of 'obviously' wrong results. Analysing these faulty proofs will provide training in reading proofs, raise awareness of mathematical hazards, such as division by zero, and provide motivation for rigour. You might wish to use some of these proof every so often throughout the year.

### Possible approach

Make it a race to find errors in as many proofs as possible in the available time. Split the class into small groups and give them one of the first three proofs to work on. Let everyone know that the next proof will be given out when the group has convinced you of the location of the error.

When a group wishes to try to convince you of an error choose one member of the group at random: only this member is permitted to speak or write. If you are convinced, give the group the next dodgy proof; otherwise give minimal feedback such as "Sorry, I'm not convinced" or "That explanation didn't seem clear to me" and leave them to attempt to tighten up their argument.

Use common sense to judge the acceptable level of rigour.

Be on the lookout for the tell-tale signs of 'um', 'er', 'the thing' and 'it' which so often indicate confused thinking.

### Key questions

Does that argument sound clear to you? (students will probably know when they are bluffing!)

### Possible extension

Students might be asked to invent their own dodgy proofs.

### Possible support

You could simply discuss the proofs as a group or you could draft in some helpers who assist you in deciding if explanations are acceptably clear.