When Should We Flip a Card?

Exploring the Four Card Problem

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

The Four Card Problem

The Four Card Problem, also known as the Wason selection task, is a captivating puzzle that tests our logical reasoning abilities. Invented by Peter Cathcart Wason, this task challenges us to determine the minimum number of cards required to verify or falsify a given statement. Let’s look deeper into this intriguing problem.

The Scenario: Card Setup

Imagine being presented with four cards, each displaying either a letter or a number. These cards lay the foundation for the puzzle, providing the information necessary to reach a conclusion. Let’s examine an example:

The face-up sides of the cards show: 23; 28; R; B

Each card has a letter on one side and a number on the other side.

Alongside these cards, you are given a statement: ‘Every card with 28 on one side has R on the other side’.

Determining the Minimum Number of Cards

Now, the crucial question arises: How many cards must you turn over to determine the truthfulness of the given statement? And which specific cards should you investigate?

Common Misconceptions

At first glance, the task might appear deceptively simple. Many individuals are inclined to turn over the R card, assuming it holds the key to verifying the statement. However, this line of thinking is misguided. Regardless of what is on the other side of the R card, it does not contribute to determining whether every card with 28 on one side has R on the other.

Similarly, the inclination to turn over the 23 card is also misleading. Even if the 23 card reveals an R on its other side, it does not provide any insight into the truthfulness of the statement. The existence of R on the opposite side of the 23 card merely confirms that the statement ‘Every card with 23 on one side has B on the other side’ is false. It does not shed light on the validity of the statement regarding the 28 card and R.

The Key to Solving the Puzzle: Logical Analysis

To arrive at the correct solution, we must identify the cards that have the potential to disprove the given statement. The crucial observation lies in recognising that only a card displaying 28 on one side and something other than R on the other side can invalidate the statement.

In this scenario, the cards we need to focus on are the 28 card and the B card. Let’s explore the reasoning behind this.

The Correct Solution: Minimum Number of Cards

The Card with 28 on Its Face-Up Side: This is the most direct test of the statement. If the other side is not R, the statement is false.

The Card with B on Its Face-Up Side: This card needs to be checked because if the other side is 28, it would contradict the statement. The statement only mentions what is on the other side of 28, not what is on the other side of R.

The cards with 23 and R on their face-up sides do not need to be checked. The card with 23 is irrelevant to the rule, which only concerns 28. The card with R does not need to be checked because the rule does not specify what should be on the other side of R.

So, you only need to turn over two cards: the one showing 28 and the one showing B.

Conclusion: Thinking beyond Initial Assumptions

The Wason selection task, or the Four Card Problem, immerses us in the intricacies of logical analysis and conditional reasoning. By identifying the two necessary cards to flip, the 28 and the B, we confront the task’s real challenge, and learn the importance of testing for falsification rather than confirmation.

The puzzle serves as a powerful reminder of the complexities that lie beneath seemingly simple tasks and the importance of careful analysis when engaging in logical problem-solving. It challenges us to think beyond initial assumptions and consider the logical implications hidden within the given information. As such, it is a clear reminder of the complexities hidden within seemingly straightforward problems and the value of meticulous analysis in navigating the world of logic.