Why do this problem?
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.
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
An alternative approach to consider is to work with your learners
on the NRICH resource
first and then to tackle the Proximity
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
Learners might try
to get the proof that the square root of 2 is
irrational in order/
A natural follow up, and re-inforcement for confidence in using
this sort of argument, would be to work on the resource
Read the article on
Proof by Contradiction
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