Age 14 to 16
Challenge Level

Why do this problem?
The result can be proved by an argument using proof by contradiction and it is a useful example of this type of reasoning. It also calls for visualisation and to clearly explain the reasoning provides learners with another challenge.

Possible approach
You might discuss arguments by contradiction with the class first. We often use arguments by contradiction in ordinary conversations that are nothing to do with mathematics.

You might want the class to realise that a statement and its contrapositive are always logically equivalent. If we consider a statement and its contrapositive, and we can prove one of them, then we have also proved the other statement.

One approach is to ask the learners to make up 'If ...then... " statements of their own and write down their contrapositives. (See Possible support below).

Invite the class to try to prove the Proximity result using an argument by contradiction. A good strategy in cases like this is to ask the learners to work individually for a short time, then to work in pairs and explain their arguments to their partner and agree on the best argument, then to work in fours so that each pair explains their argument to the other pair and they try to get the best argument possible between them. Then invite groups to come to the board and try to convince the whole class that their argument works.

An alternative approach to consider is to work with your learners on the NRICH resource Contrary Logic first and then to tackle the Proximity problem.

Key Questions
If you turn the statement you are trying to prove round to use a proof by contradiction what would you start by assuming?
If the icosahedron has 3 red vertices is there any loss of generality in taking the top vertex to be red?
If the top vertex is red, what can you say about the other vertices around it?

Possible extension
Learners might try Proof Sorter to prove that the square root of 2 is irrational.

A natural follow up, and re-inforcement for confidence in using this sort of argument, would be to work on the resource Contrary Logic.

Read the article on Proof by Contradiction .

Possible support
Even small children will understand the logical equivalence of the following statements, supposing the only money they have to spend is the pocket money they get on Saturdays:
(1) If you spend all your money on Saturday you will have none to spend for the rest of the week.
(2) If you have money to spend in the rest of the week you did not spend it all on Saturday.

These two statements are contrapoitives of each other. For another example, if we are talking about polygons, then the two statements (3) and (4) below are logically equivalent.
(3) If this figure is a triangle then it has three sides
(4) If this figure does not have three sides then it is not a triangle.