1. Activity 2-5: What are you implying? www.carom-maths.co.uk

2. 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?

3. 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. The right answer is to pick Card with 7 in addition to the A. 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! If the other side of the 7 is a vowel, there IS a problem.

4. In the statements below, a, n and m are positive integers 1. a is even 2. a 2 is even 3. a can be written as 3n + 1 4. a can be written as 6m + 1 Consider these four assertions:

5. If we make a card with one statement on the front and one on the back, there are six possible cards we could make.

6. 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’. Task: how many different types of card do we have with these definitions? We can see that the statements for Card 1 mean that this is of type (I, I): each side implies the other.

8. 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).

9. 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.

10. What mistake do people make with the four-card problem? They assume that A I B can be reversed to B I A. So Vowel I Even reverses to (Not Even) I (Not Vowel), which shows we need to pick the 7 card. if A implies B, then (not B) implies (not A). In fact, A I B CAN be reversed, but to (not B) I (not A) – Modus Tollens. Let’s try a variation on our initial four-card problem.

11. 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.

12. 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?

13. 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: x 2 = 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

14. Carom is written by Jonny Griffiths, hello@jonny-griffiths.nethello@jonny-griffiths.net 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.