, , , , , , , ,

The previous post noted that a puzzle given to some Singapore high school students about Cheryl’s birthday has been bothering people around the world. The puzzle was a classroom exercise and I noted what the bother revealed about the different types of puzzles. This posting focuses on what the public discussion about solutions reveals.

First, the answer to the puzzle is: July 16. People may not understand why but the correct answer is not in dispute.

Second, I remember solving similar kinds of problems in high school. It is a tight logic puzzle in the sense that the problem gives enough information to solve it, but just enough. It takes practice to learn to infer what you need to know from the information given to you, especially when it is not expressed directly. Inference is an extremely important life skill because evidence and information are not always given in the form that makes its implications obvious.

(I am busy marking final exams at the moment and an inability to infer is evident in some of the answers. Instructors often complain that students expect to be “spoon-fed” with the obvious data that can be dropped into a well-known formula in the obvious place. In the opposite direction, I am occasionally surprised at how easy it is to mislead students by inserting irrelevant facts and red herrings. In the last couple of years, the test commonly used to decide if somebody is good enough to pursue an MBA added a section called “Integrative Reasoning” to measure how well people reason. So, I think that this kind of puzzle is very instructive even if it is not realistic.)

Third, many of the complaints about the puzzle’s solution use the idea that such tight puzzles are unrealistic in the sense that the information is limited but known precisely. The classic problems of real estate market analysis are examples of a loose problem where a lot of information is available but it is imprecise or noisy. The point that confuses many people is that a tight puzzle leads to a tight solution whereas the solution to a loose puzzle cannot be equally tight. The birthday puzzle is so tight that an arbitrary change to almost any sentence is likely to make it impossible to solve. That also means that any attempt to “reinterpret” the puzzle (in the same way that politicians try to rationalize a preconceived answer) is likely to produce an impossibility rather than a different answer. The solution to a loose puzzle is a range of answers or, better, a probability distribution of possible solutions which shows the more likely solutions as well as the less-likely-but-still-possible solutions. In a loose puzzle, an arbitrary change in the available information changes the magnitude of answer but does not make an answer impossible.

People think that they can find a tight solution to loose puzzle, yet we live in an imperfect world. This conflict is serious in the real estate business. All sorts of people (developers, home buyers, mortgage lenders, …) gather information in order to predict the future (i.e. to infer what will happen 10 or 30 years from now based on the imperfect information that is available). They think that they can summarize that prediction by a single number and act on that number. Rather than complain about the impossible, we try to teach our students a more realistic approach so that they can offer practical answers.

Fourth, I find the responses to the puzzle reveal a lot about how others think of a solution.  The journalists find the puzzle hard, but journalists are famous for not being good at math. Journalists are supposed to good at asking questions, and I have yet to see a journalist ask a local math teacher. (As I said, I played with these kinds of puzzles in high school.)

Moving past the journalists, some people evade the pure logic. The most popular “creative” solutions are

  • The problem is Cheryl: she should be friendlier and give clearer information to her friends if she wants birthday presents. If this solution were stated differently, this is a popular and effective solution for many real estate businesses: i.e. become her friend and ask her directly. Or
  • That the question is poorly worded (In my opinion, this means that means that the people reading the puzzle were expecting the puzzle writer to be as sloppy as in real life.  If so, the lesson should be that we can learn to write more precisely and more efficiently to help others to better understand what we are saying.) Or
  • Some people did not need to know how to solve these kinds of logic problems, since they could look up the answer by checking Google. This cute answer ignores the fact that the initial reports on this puzzle stated, inaccurately, that the problem of Cheryl’s birthday was asked of Grade 5 students. Search engines may be able to find an answer quickly. They are less good at guaranteeing the quality of those answers and the results are occasionally dangerous: [1] [2].

So, the puzzle of Cheryl’s birthday is not really a math question. In so many ways, it is a Rorschach Test which reveals about how people think about problem solving and about solutions.

At graduation, parents thank me for teaching their son or daughter. I tell them that the student did the hard work; I merely asked questions designed to stretch the puzzle-solving muscles. To the students who are writing their last classroom exam ever as I write this, congratulations. We have tried our best to let you see the true nature of a puzzle and the possible solutions. It is not easy and, unfortunately, it does not become easier when you move into the real world. The good thing is that the new graduates are energetic and knowledgeable. There is a world of worthy puzzles waiting for them.