98% of people can’t solve it

…and I show you how this may paradoxically prove that almost anyone can learn mathematics if they practice enough.

There are four cards lying on a table. You know that each card has a letter on one side and a number on the other side. Your task is to check whether the above cards also satisfy the rule that “if a card has a vowel on one side, then it has an odd number on its other side”. Which of the four cards do you necessarily have to turn over to be sure that the rule indeed holds?

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Give the puzzle a try and once you are done, continue reading.

How hard really is the puzzle? Well, the title isn’t clickbait. It indeed turns out that only a tiny proportion of people can solve the puzzle correctly. That is also why the puzzle became popular among some high-profile universities that used to pose it as part of their interviews (true story :)).

So… are you sure that your solution is correct? One last chance to check it!

Here is the solution:

To check whether it is indeed true that “if a card has a vowel on one side, then it has an odd number on its other side”, you must turn over two cards: the first card (A) and the fourth card (6).

Are you wondering why? Rather than explaining the details of the solution and using fancy words like ‘implication’ or the ‘contrapositive’, let us consider another variation of the same puzzle: the post office plot.

The post office plot

Assume that sending a standard letter from a local post office costs 3 euros. The post office, additionally, also offers the following service. They noticed that people are lazy to properly glue the envelopes and so the customers can bring the letter in an open envelope and they will glue the envelope on customer’s behalf before sending it off. In this case, however, the customer must pay 4 euros.

Imagine that you work at the post office. There are four letters in front of you: one with an open envelope, one with a closed envelope, one with a 4 euro stamp and one with a 3 euro stamp.

You know that each of them already has either a 3- or 4- euro stamp on one side and has an open or closed envelope on the other side. Your job is to check whether every customer who handed in a letter in an open envelope has also paid the higher postage. Which of the envelopes do you necessarily have to turn over to find out?

Solution:

You have to turn over the first and fourth envelope. Here is why: the first envelope is open so you have to turn it over to check that it has a 4-euro stamp on the other side. The second envelope is closed. You do not know whether the other side has a 3-euro or a 4-euro stamp but in this case either stamp suffices, so no need to turn this one over.

The third letter has a 4-euro stamp. You do not know whether its envelope is open or closed on the other side, but either way, the higher postage has been paid so no need to check this one either. Finally, the fourth letter has a 3-euro stamp. If its envelope was open on the other side, that would be a problem because that way the customer did not pay enough postage. So, you must turn over this letter and check that its envelope is indeed closed.

The same, but different

Did you find the second version of the puzzle easier? The truth is, the two represent exactly the same puzzle except that the second version revolves around a post office story. (To see that the puzzles are indeed equivalent, just substitute ‘If the letter has an open envelope, then the postage must be 4 euros’ for ‘if there is a vowel on one side, there must be an odd number on the other side of the card’.)

Many people find the second version substantially easier and are able to solve it even though they were not able to correctly solve the original puzzle.

So, why is the second version so much easier? One possible explanation is that it is set in a very familiar setting. We all have experience with posting a letter and paying different amounts for different services. It is much more tangible and requires less mental effort than the abstract exercise with the cards in the first version.

Just get used to it

Taking the argument one step further, this example illustrates how mastering logical reasoning may be dependent on the context. The more familiar context, the easier it seems to master the skill.

I personally have always believed that practice makes perfect. The more one trains the motor skills, for example when dancing or playing an instrument, the better one gets. Similarly, the more one exposes themselves to a foreign language, the better one gets. And, based on this puzzle, I find it fascinating that the same may also hold for learning logical reasoning and mathematics. Maybe becoming good at math is also, to some extent, simply a matter of getting used to it? The more you expose yourself to the abstract concepts and practice with them, the better you get.

What do you think? Do there really exist people who ‘have no brains for math’ and nothing can help them? Or is it much more about the motivation to get exposed and used to the abstract concepts, thus making them more relatable, and hence easier to handle?

P.S. References wanted. I got to know about the first version of the puzzle from various places at school and the university. Later on, I remember reading about the second version with the post office in a psychology of education textbook. However, I have been unable to trace back which textbook it was. If you know the source, please let me know.

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