Friday, April 10, 2009

Two Puzzles

First scenario: You are presented with four cards and are told that the cards follow two simple rules.

1. On one side of each card is a letter and on the other side is a number.
2. Cards with vowels on one side must always have even numbers on the other.

Rule #1 is always true. On the sides of the cards you can see are printed:

A 18 C 57

First question: How many cards do you need to turn over in order to check rule #2 above?

Take a moment to think about your answer before moving on to the next scenario. The immediate answer that many people give is the wrong one.

Second scenario: You work as a manager at a popular bar/restaurant. Because you have an attached game room, the bar has become a popular hangout spot for students from the local community college. One of your duties as manager is to periodically make a sweep through the game room to make sure none of the students under the age of 21 (the legal drinking age in the U.S.) have procured beer from the bar. For the sake of simplicity, the bar only serves beer and cola (no mixed drinks), so you can tell by simply looking in a patron's glass what they are drinking.

On one such sweep through the game room you see only four customers.
  • A man of indeterminate age holding mug of beer.
  • A woman drinking a glass of cola.
  • A man with a long gray beard whose glass you cannot see.
  • A young teenage girl whose glass you also cannot see.
Second question: For which of the patrons above do you need to check identification and/or beverage choice?

The second question isn't intended to be as tricky as the first. Your first instinctive answer is probably the correct one.

Note that the second question is simply a more concrete version of the first. I've replaced even numbers with ages, but that's really the only difference. Now that you know this, go back and read the first question again. If you thought you needed to check all four cards in the first question, would you like to take another guess after reading a concrete analogue? Now what about if you thought you needed to check three cards?

Spoilers Ahead!

Don't continue reading unless you want to see the solutions.

Solution to the first scenario

Many people mistakenly think, when presented with the first scenario, that you have to turn over all four cards to check the reverse. The problem is in the details of the wording of the second rule.

2. Cards with vowels on one side must always have even numbers on the other.

This in no way implies that a card with an even number on one side must have a vowel on the other, but this is the mistaken assumption that many people make. The fact is, if a card has an even number on one side, you don't need to care what kind of letter is on the reverse. A consonant printed on the other side doesn't break the second rule.

Based solely on rule #2, you shouldn't care what's printed on the other side of the card marked '18' or the card marked with the letter 'C'. You do need to flip over the cards marked 'A' and '57'. If the card marked 'A' has an odd number on its reverse, then rule #2 is broken. Similarly, rule #2 will be broken if the card marked '57' has a vowel on its reverse side.

Solution to the second scenario

The more concrete (but still somewhat contrived) second scenario is answered correctly much more often than the first. Let's go through each patron and decide if we need to check their age or their beverage.
  • A man of indeterminate age holding mug of beer. Clearly you need to check his age to see if he's above the legal drinking age.
  • A woman drinking a glass of cola. You don't need to check her age because she's drinking cola, which patrons of any age are permitted.
  • A man with a long gray beard whose glass you cannot see. If a man has a long gray beard then he is presumably of a legal drinking age, so you don't need to see what's in his glass.
  • A young teenage girl whose glass you also cannot see. You need to check the girl's glass to see what she is drinking.
So, just as in the first scenario where we had two cards we needed to check, here we have the same number of patrons who we need to check.

As I said before, the two puzzles are virtually the same in every way, except one is a more concrete (and for some puzzling reason easier) version of the other. When you have an abstract problem that you just can't seem to get to the heart of, sometimes it will help to assign concrete values to the abstract variables of the problem. Once you've solved a concrete instance of the problem, go back and see if this solution has shed some light on the original problem.

If you like logic puzzles of the sort I presented here, I highly recommend the book Conned Again, Watson! Cautionary Tales of Logic, Math, and Probability by Colin Bruce, from which I adapted these puzzles.


Niyaz said...

Malcolm Gladwell argues (in his book - The tipping point ... ) that the reason people can answer the second puzzle is not because it is explained in concrete terms, but because it uses people as the elements.

For example, even if you give the question with taking some cars/colors as example, most people will have difficulty in answering it. When you take real people as examples, we can easily digest it and we can process the information easily.

Bill the Lizard said...


The Tipping Point! Thank you! I knew I had read about this somewhere else relatively recently, and I just couldn't figure out where. I've been going through all my old math/puzzle books looking for it.

Anonymous said...

I read about this in Pinker's book (and then Toby & Cosmides paper), so I knew what was coming. So I mentally framed the first variant as "which cards could be cheating", and found it easier. (T&C's hypothesis being that our minds have evolved specific processes to detect cheating, but that logic relies on more general cognition.)

Asmor said...

In the first example, we're given a rule that has to do with assigning one half of a common pair (vowels/consonants) to one half of another common pair (odds/evens).

I thought I'd read, though in retrospect realize I just assumed, that there was a similar rule that consonants must be paired to odd numbers. When I could see no way to check fewer than four cards, I re-read the rules more carefully to see if there was some trick I missed, and sure enough I noticed my mistake.

In the second example, it's a very intuitive situation based on rules that all people have internalized.

It's not an issue of the second one being more concrete; it's an issue of the first one preying on our intuition (assigning a single rule based off of an obvious dichotomy, but not completing that dichotomy) and the second situation does not have that issue.

Anonymous said...

Personally, I didn't initially want to check the first for all four. I was thinking to check only 'A' and '18' until I reread #2 (and stopped committing to a logical fallacy :)). #2 specifically is an implication: vowel => even. Thus, I had to check '57' to see if the contra-positive holds, and it doesn't matter if a consonant is on the back of '18' because no "promise is broken".

Having solved the first, the second was trivial.

Therese said...