There’s an intriguing little math puzzle that can be used to position yourself as smarter than everyone around you — even if you’re not good at math. It’s the Tuesday Child Puzzle, which I came across recently. The question is simple enough:

*If I have two children and one of them is a boy born on Tuesday, what are the odds that my other child is also a boy?*

This question is nice because most everyone gets it wrong initially, even people who are normally good at math. You can go to this blog posting and study the details. But here, in addition to giving an alternate way to see the answers, I’ll give important recommendations for the most effective way to present this puzzle to others, to maximize both your fun and their humiliation.

First, note that the answer is not 50% (it’s a little less, 13 out of 27) But it’s a very safe bet that 50% is what you will hear. Here’s an excel file that contains my solution — no real programming required in this case (it also lets you experiment with variations on the question, like allowing for a boy born on Tuesday, Wednesday, or Friday).

So, on to the recommended delivery. A meal setting is preferred, one that allows you access to something to drink.

You (with glass of water in hand): “Hey here’s a question. If I have two children, and one is a boy born on a Tuesday, what are the chances that the other child is also a boy?”

Quickly drink some water but do not swallow yet.

Them: “Um, it’s 50% isn’t it?” [or, “I don’t see what the day has to do with it. It’s 50%.”]

You: Spit out the water suddenly as you start and then stifle a rapid laugh of apparent surprise. “Oh sorry,” you say, gasping and wiping your mouth, hopefully also with them having to clean some of the water off themselves. “Yea, I guess that one was a little hard. Here’s the easier version for you: Let’s say I have two children and one is a boy born on *any *day of the week. What are the chances the other one is a boy this time?”

At this point you rapidly reload water into your mouth, and then look at them with an inquisitive look like you’re expecting another answer.

The answer to this question is again not 50% (it’s 1 in 3, 33%). But this time they may not answer so rapidly. If they (wisely) won’t answer, then if you don’t understand the mechanics of the puzzle, just tell your audience to think about it and you’re sure they’ll figure it out eventually. And say this with a careful delivery of humble smugness.

Enjoy this fun community puzzling activity. For extra credit, experiment with different liquids.

on July 27, 2010 at 4:58 am |JeffJoUnfortunately, these questions you ask are ambiguous, and it is the failure to recognize how they are ambiguous that causes the results to seem unexpected. Consider two versions of what led up to the first statement:

Case #1: A father is chosen at random. He is given a slip of paper as he is led onto a stage. The paper says “Pick one of your children. Tell the audience the number of children you have, the chosen child’s gender, and the day of the week on which it was born.”

Case #2: A father is chosen at random from all fathers who have two children, including one boy born on a Tuesday. He is also ushered onto a stage and given a slip of paper that instructs him to tell the audience the criteria used to select him.

Now shift scenes. You are in the audience when a man is ushered onto the stage. He looks at a slip of paper, thinks a moment, and says “I have two children and one of them is a boy born on Tuesday.” What is the probability that he has two boys?

First, notice that I changed the question slightly. You asked about the “other” child. In the case that the man has two Tuesday Boys, it isn’t always clear which is the “other.” And while the difference may seem trivial, the distinction is part of the confusion around these questions.

Anyway, the correct answer to the question depends on which case applies to the man you listened to. In Case #1, it is 1/2. In case #2, it is 13/27. Your simulation only covered the second case. To get the first, after you have two children, flip a coin to see which one the father will tell about. If it is not a Tuesday Boy, don’t keep that trial even if the other child is a Tuesday Boy. You will find that the 27 cases where you have a Tuesday Boy reduce to 14 (just over half, since one father didn’t need to flip the coin), the 13 where you also have two boys reduces to 7, and the answer is exactly 1/2.

If you simulate the simpler problem, where you don’t worry about the day of the week, the answers are 1/2 and 1/3 for the two cases, respectively. The reason 13/27 seems unintuitive, is because the fact that a Tuesday Boy was REQUIRED in the second case is not intuitively obvious from the statement “one of them is a boy born on Tuesday.” In fact, as the other blog points out, the puzzle could equally well be named after either of your two children, which is probably two different names. You choose one, just like the father in case #1, so the better answer to your question is 1/2, not 13/27. It is still ambiguous, but there is no valid reason to assume that case #2 applies.

on July 28, 2010 at 2:19 pm |BarbaraAs JeffJO says, the correct answer to this question depends on what you think the context is.

If the question is asking, Considering the set of people who have 2 children, and considering that you know one child is a boy, what’s the chance that the other child is a boy, the answer is (approximately) 33%.

If the question is asking, If a person has one child who is a boy, what is the chance that his next/other child is a boy, the answer is (approximately, as you say) 50%. No laughter appropriate.

Limiting the possible answers to the ambiguous question to the first alternative provides the opportunity for great slapstick effects.

Barbara

on September 17, 2010 at 8:04 pm |timwitI agree with jeffjo’s comment that the result depends on the ambiguously specified way that the person is selected. And I agree with Barbara that this ambiguity provides an arguable excuse for the fun part.