The singapore maths question

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Here is the problem “for school children” that it is claimed has “stumped the world”. I wanted to write here not just a solution, but also a clear (I hope) line by line explanation of the reasoning which repeats itself multiple times as this is how people best learn this kind of thing. It's easy to miss just one line of reasoning. I also want to make comments about how being able to do this, or not do this question - or this type of question is a measure of nothing more than how interested you are in doing this question - or questions like it. Just a quick thing about what this question is all about: it’s called logical deduction. That’s worth saying because some articles are claiming this is about something called “induction”. Induction, as a method of reasoning, doesn’t exist. The best that can be said for “induction” is that it is a fancy way of referring to “guessing” - but that’s being too generous. Logic is actually a subject all its own. It is taught in some places, in some schools, as a part of mathematics (which it is) but it is also part of philosophy. At universities with philosophy departments you can take subjects in logic. Sometimes logic is also taught in schools of mathematics as well at universities. It is a learnable thing, taught at schools and universities. Like French. Or Chinese. Or the piano. If you can’t understand this question, or the solution, it’s kind of like not understanding Chinese. Or how to play the piano. It means: you’ve not learned how. And not having ever learned how is no reflection on your intelligence. And yet, for some strange reason, some people (usually teachers and academic administrators) see questions like this one as a measure of "IQ" in a way that knowing how to speak Chinese, or play the piano is not.

So to the question. The first thing is that, like many of these sorts of questions, they are made up by people who like to do these kinds of questions to test other people who like to do these kinds of questions. Like I said: if you don’t get it: don’t worry. It just means - you’re not practicing doing these sorts of things much. That is entirely normal - does not mean you are less smart than anyone else. Think of something you are good at and do often. Okay - now imagine there are people who do these kinds of logic questions as much as that, and enjoy it just as much.

In other words - to do this question is just another kind of skill - like understanding a language, drawing portraits or playing a musical instrument.

There are some “useful hints” you need before you start to learn a language or play a piano (we can call these kind of hints "implicit" knowledge - things that "go without saying" (unlike "explicit" - which is stated up front, clearly). People who are skilled at certain things know the implicit knowledge (subconscious hints) and sometimes are so familiar with these things they don’t know how others could not be. For example, a useful "implicit" rule some language learners know is: if you find a certain sound in a particular foreign language is very difficult to make - did you know that sometimes in English some combination of syllables or words sounds remarkably like a really foreign word? With a good teacher, if you struggle with a certain sound in one language, they might say ‘well we have exactly the same sound in English...it sounds like the first one and a half syllables in the word Hawaii” (or something like that). Once you know that kind of thing - it makes it easier to learn. So what is the implicit knowledge in this logic problem? Before I get to that - why don't they state it so everyone is on an equal playing field? Obviously it is so they (the teachers, the exam writers, whatever) can weed out the ‘smart people who know how to do this because they practise’ from the ‘normal person not interested in doing this kind of thing’. It's like a pre-test. If people haven't practiced this sort of thing, they are immediately eliminated and we can move on to testing out those who have practiced. A person who has not practiced might still get the answer - but they will usually take longer - so there's an inherent penalty in the test. So here’s what you need to know:

Everyone involved in the question (Albert, Bernard and Cheryl (A, B & C) are perfectly rational people who think absolutely logically all the time. If that seems strange, it is. And it's not stated in the problem.
The problem involves A knowing what B is thinking and B knowing what A is thinking. And they can do this because they all think exactly the same way. (Namely, absolutely logically, all the time). And this is not stated.
They are speaking out loud and can hear one another. This is crucial - but not stated in the problem. Again - you’d know this if you’d done things like this before. Namely in situations (like classrooms) where you can ask questions of someone else who already knows. But if you read this on a website, or worse, in an exam situation, somewhere - there’s no one to ask questions to! People who do well at these things, do lots of them.

Note in the question the word “respectively” - A knows the month. B knows the day. This is a key point, easily overlooked. Again - if you don't do these kind of questions much - you will not realize that there are certain absolutely crucial logic-type words.

So now - what is written in the problem: both A and B have the 10 dates in front of them.
A gets told the month, but not the day. And B he gets the day...not the month.

Bored yet? If you are, then notice that feeling for a moment: it’s not that you are less intelligent. You’re just bored by this. Imagine someone not bored - who is as interested in this as you are in your favorite thing. That's all so-called "intelligence" amounts to: other people naming certain kinds of interests as being those constituting intelligence. So that’s that. Let’s keep going:

Let’s say C said to B “the date is the 18th”. Well immediately he knows - because there is only one “18th” there. So her birthday is June 18th. The same goes for May the 19th. So if either 18th or 19th is used, the game is over. So we are now down to 8 possibilities.

But now A (who has the month) says that he doesn’t know and that he knows B doesn’t know too. So because B thinks perfectly rationally this means he knows A can’t have June at all. Why? Well because if A had June 17, then being perfectly rational, he would have said he knew because otherwise B would have spoken up immediately (as A must, in that case have had June). There’s only 2 June dates on the list! So if A sees “June” then B either says “I know” (namely he sees an 18 and there’s only one 18th) or is silent. If he’s silent then A knows it must be the other June date. Again, this issue of who is silent and who is saying what, when - is something that comes up in a very formulaic way with these puzzles. It's in the same class of problems as others where people are only allowed to say certain things like the person who always lies or always tells the truth and is guarding some door or prisoners forced to wear silly hats (http://en.wikipedia.org/wiki/Prisoners_and_hats_puzzle).

Anyways - let's keep going. We've eliminated June and we've eliminated May 19th. Notice how this goes? We are eliminating dates one by one. It’s just a puzzle.

Remember, A has the month, B has the day and both have now eliminated 3 dates like we have because 19 and 18 are unique numbers, and the other June date through logical deduction is eliminated on the assumption one of them would have spoken up if this were the month. Because B knows the day, but A says that B “does not know” then this means that A does not have May at all (because if he is so sure B “does not know” this means he thinks that May 15 and 16th are possibilities). You might have to read that bit of reasoning again - it can be tricky.

So all of June is gone. And now all of May is gone because A says B doesn't know. So we whittle it down to this:

July 14, 16
August 14, 15, 17

Keep in mind now that A and B think perfectly rationally - so they have thought much the way I have presented this and eliminated things, just like I have.

So they can start the problem from scratch at this point - and use the same reasoning as before:

“We both have the same list so August 17 is eliminated.” After all - if B had “17” he would speak up and say so. He'd say "I know it! I've got 17. That means it's August 17!"

But he does not. 17 is a unique number on that list now. He should have said something! But he doesn't. So by similar reasoning to before, August dates are all eliminated in one go (As A has said that he thinks “B does not know”). Progress. We (by which I mean A and B and you and I) are down to:

July 14 or July 16

Now because A says “B does not know” then this means A is not able to eliminate one of these dates. But B can! Because B has only one number in front of him. Which one? The  one that does not appear on the August list. Namely 16.

So when A says “B does not know”, B is able to reason “This means A thinks that my number could still appear in both the July and August lists - and so I could be confused. He thinks I am thinking that it could be any of:
July 14, 16
August 14, 15
But I’m not confused. I know August is out entirely and I can't be confused because I have only one number. If I had the number 14, then A would think it possible for me to still be confused (and I would be). But I'm not and so I am able to say "I know now". And so he does!

And A, who is an equally boring person (who thinks in an identical way (probably why they are friends)) says “Then I also know”. Because he understands B has reasoned that he has the unique number among the options for July, given that it was possible to rule out August altogether.

Still don’t get it?

Don’t worry - you would if you were really, genuinely interested. Which means - you would sit down for (literally) many minutes - probably hours - doing this problem and ones like it until it just became second nature. People do this! And weirdly lots of the people that do it say "I find logic puzzles so easy" and they do well at them. Because they have fun doing them. And so do well in tests with logic puzzles. And they think, and other people think, it is somehow an inherent (i.e: not learned) capacity. That it is something about their brains. But it's not. It's just a skill of their minds. Like speaking Chinese. Or playing the piano. Or dancing. Or any one of a million other things human beings can do. It's not a measure of intelligence. It's just a certain kind of learned skill. But it is a learned skill that is quickly, and highly, rewarded in schools because schools value this kind of thing and some students value getting gold stars or good marks or whatever. And so they do well and the cycle continues.

Postscript: I have looked at other solutions. They are different to mine in a way. Typically much shorter. And I think for someone who doesn’t know how to do these things - almost entirely unhelpful. Mine is maybe a little bit more or less confusing because I try to explain line by line my reasoning and try to explain how it's just something to learn. There are things you need to know (tricks, implicit knowledge, whatever). People who solve these things are typically good at solving them but not necessarily at explaining why the answer is what it is. And why should they? That is a whole other skill entirely - also equally (un)important for almost all of us almost all of the time. So if you didn’t understand my solution - no worries. If you don’t understand other solutions out there (like the so-called “official” answer) even more no need to worry. It’s like if you cannot read music and don’t really want to learn how. It probably won’t help if someone really articulate tries to explain it to you: you probably still won’t understand because you won’t really pay attention. But that doesn’t mean you lack some capability someone else has, or have a lower IQ - it means you are not interested. Truly, some people are interested in, for fun, doing lots of logic type puzzles of the sort they ask in IQ tests. That doesn't make these people smarter than someone who teaches themselves how to speak Mandarin or Russian - it just gives them different interests. Would you think someone who learned how to speak Mandarin completely fluently was really smart? Well every child in mainland China (more or less) learns this to a very high level of proficiency. 

If people were rewarded socially for doing IQ tests in the way they were for learning languages, we would all do really really well in IQ tests. We'd all be "fluent" in IQ testing questions and their answers. Instead, only some of us are ever interested in in doing lots of silly little mathematical and logical puzzles - because we find it fun and rewarding. 

PPS: This is why I think IQ should stand for “Interest Quotient” - it’s a measurement of how interested people are in trying to do well at answering questions on IQ tests. In other words: a completely useless number.