www.carom-maths.co.uk
Activity 2-5: What are you implying?
You are given these four cards:
Each of these cards has a letter on one side
and a digit on the other.
This rule may or may not be true:
if a card has a vowel on one side,
then it has an even number on the other.
Which cards must you turn
to check the rule for these cards?
This is a famous question
that has tested people’s grasp of logic for years.
Most people pick Card with A, correctly.
Many pick Card with 2 in addition.
But… if the other side of the Card with 2 is a vowel, that’s fine.
And if the other side of the Card with 2 is a consonant,
that’s fine too!
The right answer is to pick Card with 7 in addition to the A.
If the other side of the 7 is a vowel,
there IS a problem.
Consider these four assertions:
In the statements below, a, n and m are positive integers
1. a is even
2. a2 is even
3. a can be written as 3n + 1
4. a can be written as 6m + 1
If we make a card with one statement on the front
and one on the back, there are six possible cards we could make.
Define A I B to signify ‘A implies B’,
Define A RO B to signify ‘A rules out B’, and
A NINRO B to mean ‘A neither implies nor rules out B’.
We can see that the statements for Card 1
mean that this is of type (I, I): each side implies the other.
Task: how many different
types of card do we have
with these definitions?
We conclude we have four different types of card:
1 is (I, I),
3 and 5 are (RO, RO),
2 and 4 are (NINRO, NINRO),
while 6 is (I, NINRO).
Are these the only possible types of card?
Yes, since if A RO B, then B RO A,
so (RO,I) and RO, NINRO)
are impossible cards to produce.
Can we be sure that if A RO B, then B RO A?
We can use a logical tautology called MODUS TOLLENS:
if A implies B, then (not B) implies (not A).
Now A RO B means A I (not B).
If A I (not B), then by Modus Tollens,
not (not B) implies (not A).
Thus B implies (not A), that is B RO A.
if A implies B, then (not B) implies (not A).
What mistake do people make with the four-card problem?
They assume that A I B can be reversed to B I A.
In fact, A I B CAN be reversed,
but to (not B) I (not A) – Modus Tollens.
So Vowel I Even reverses to (Not Even) I (Not Vowel),
which shows we need to pick the 7 card.
Let’s try a variation on our initial four-card problem.
You are given four cards below.
Each has a plant on one side and an animal on the other.
Task: you are given the rule:
if one side shows a tree, the other side is not a panda.
Which cards do you need to turn over to check the rule?
This time the obvious reversal works,
for ‘Tree RO Panda’ is the same as ‘Panda RO Tree’:
you DO need to turn over the two cards named in the question.
Given two circles, there are four ways
that they can lie in relation to each other.
Task:
if you had to assign
(I, I),
(RO, RO),
(NINRO, NINRO),
and (I, NINRO)
to these,
how would
you do it?
Let’s invent a new word, DONRO,
standing for DOes Not Rule Out.
Task: is it true that
if A DONRO B and B DONRO C, then A DONRO C?
Pick an example to illustrate your answer.
A: x = 2
B: x2 = 4
C: x = 2
A: The shape S is a red
quadrilateral
B: The shape S is a
rectangle
C: The shape S is a
blue quadrilateral
A: ab is even
B: b is less than a
C: ab is odd
With thanks to:
The Open University and my teachers on the
Researching Mathematical Learning course.
Mathematics In School
for publishing my original article on this subject.
Carom is written by Jonny Griffiths, [email protected]