An Exercise in Elementary Logic

Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

You are presented with four cards, with the face-up side on display, showing either a letter or a number. You are promised that each has a letter on one side and a number on the other.

Red Card displays the letter D

Orange Card displays the letter N

Blue Card displays the number 21

Yellow Card displays the number 16

You are now presented with the following statement: Every card with D on one side has 21 on the other side.

The Question is: What is the minimum number of cards needed to determine whether this statement is true? What are the colours of the cards you need to turn over to determine this?

Think about it: Do you need to turn over the Red Card? Do you need to turn over the Orange Card? Do you need to turn over the Blue Card? Do you need to turn over the Yellow Card?

Spoiler Alert (Solution).

When given this puzzle to solve, the great majority get it wrong.

You must turn over the Red Card to see if it has 21 on the other side. If it does not, the statement is false.

You must turn over the Blue Card to see if it has N on the other side. If it does, the statement is false.

Turning over the Orange Card does not help you verify or falsify the statement.

Turning over the Yellow Card does not help you verify or falsify the statement.

So the minimum number of cards need to determine whether the statement is true is two, and they are the Red Card and the Blue Card.

Bonus Question (The Tyre Problem)

Two employees turn up late to an important meeting. They claim that one of the tyres on their car had a puncture, but it is a lie.

Their suspicious boss send them to separate rooms and asks each of them to write down which tyre was punctured.

The Question: Assuming they have not colluded beforehand, and have no particular reason to think that one tyre is more likely to have been punctured, what is the likelihood that they will randomly name the same tyre?

Is it (1/4)²         = 1 in 16?

Think about it: There are four tyres. Each employee is choosing a tyre randomly and independently of one another. Maybe it is easier to think of a two-wheeled vehicle. In the same scenario, what is the likelihood they will randomly name the same tyre if they arrived on a motor bike? Is it (1/2) x (1/2) = 1 in 4?

Spoiler Alert (Solution)

Once the first employee randomly chooses a tyre on the car, there is a 1 in 4 chance that the other employee will choose the same one, e.g. if employee 1 chooses front left tyre, employee 2 has a 1 in 4 chance of randomly selecting the same one. Similarly, if the first employee randomly chose the back tyre on the motor bike, the chance that the second employee would come up with the same tyre is 1 in 2.

Further Reading and Links

https://selectnetworks.net