Games, Logic, and Language..

A bit more than a month ago, there was a math (actually logic) problem that went viral and that generated a lot of discussion.   This was a bit like the famous blue/black vs. gold/white dress phenomenon–See here:

Remember this?  Now what colors do you see?

Remember this? Now what colors do you see?

that got everyone in a tizzy.

In this case, however the so-called math problem that generated all of the controversy was focused on a kind of logic game.  The hook of this problem was that it was supposedly given to 5th graders in Singapore as a test problem–and therefore if you couldn’t figure it out, it must mean that you were not that smart In reality, it came from a Math Olympiad test for high-school age children–so it was a bit trickier than that..

The problem can be found at this link–but–here’s the original image also:

math3-superJumbo-v2

I can say that when I went through the problem–which I think is pretty hard ( and I took a college level logic course in Germany, was a big old Math nerd in high school, and an engineering graduate to boot)–I got a little flustered.   My best guess was for Aug 17, but when I went to look at the linked page above’s solution–they laid out a logic that said it was July 16th.
Their logic seemed odd to me, however, and so I went reading through the comments of the article and found that there was a pretty strong dispute between Aug 17 and July 16th.

Now–the logic of the article went as follows:
#1. First statement by Albert is analyzed: “I don’t know when your birthday is, but I know Bernard doesn’t know, either.”

This statement–especially the second half of it–is parsed to mean that the months of May and June are not the months given to Albert–because both of those months have a unique date in them–namely June 18th and May 19th–and if Bernard had either of those numbers, then he would know the birthday immediately. Therefore, those months couldn’t be possible months that Albert was given to have the 2nd half of the statement be true.

#2. Second statement by Bernard is analyzed: ” I didn’t know originally, but now I do.”

From this statement, it is argued that Bernard has deduced from Albert’s statement that May and June are not the months.  Bernard also claims to know the answer. Now.. with only July and August as possible months, Bernard could not know the answer if he had been told the #14–since there are 14’s in both July and August in the list.  So the only possibilities for him are July 16, August 15 or August 17.

#3. Final statement by Albert is analyzed: “Well, now I know too!”

This statement is analyzed to say that August could not have been the month that was told to him–because there would still be August 15 & August 17 left.  Therefore, he must have been told July–and with the previous knowledge that July 14 & August 14 were eliminated–July 16th must the answer.

That, at least, is the logic presented in the solution.

But there’s something odd going on here if you think about human social relations and how people do things.  If you think of the situation with Cheryl, Albert, and Bernard–then some of the mathematical logic that is employed here is actually in conflict with how humans would act.

Specifically–I’d point to the analysis above in #1.  In specific, the question–in my mind–was whether Cheryl’s birthday could ever have been on the 18th or 19th of any month.  I say this because if either of those dates were true–then this game was entirely rigged.  Cheryl messed with Albert and told Bernard the answer outright–because with those dates, he also already knows the month too.

That’s not generally how any kind of fair game works.  Games tend to involve competition–and to stage a “game” and give one person the winning answer up front would constitute cheating.

This was part of the logic that I–and a number of other people in the comments I read–went from.

If you do this–then you get the following logical progression:
A) Assuming the point that the game is not going to be given away outright by a date of 18th or 19th–a point that Albert & Bernard could each reach on their own (besides being told a different date in Bernard’s case),  Albert’s first statement that he does not know the answer eliminates the month of June–because with the 18th and 19th gone–being told that it was June would have told him the answer.  All other months, however are possibilities that would make the second half of the statement true–namely that Bernard also doesn’t know.
B) With Albert’s statement, however, Bernard can see that June is eliminated.  Now Bernard claims that he knows the date–with May July and August as possible months–the only way this is possible is if that # is unique and only occurs in one of the months. The only date that fits these criteria is August 17th– because all other remaining dates 14, 15, & 16–have two possible month options.
C) This statement by Bernard that he knows the answer eliminates all possible dates that occur in two or more months for Albert.  Thus–August 17th must be the answer.

Now–if you look at this situation–the problem really occurs because of the linguistic assumptions and social assumption that one makes with regard to this situation.  Really–it goes into what actually makes up a game and how do people behave–or at least–how should they behave if they are going to act in such a situation.

This is the fuzziness of human behavior and language that can sometimes make it really problematic to map it directly onto logical or mathematical problems.

It is also, I would argue, one of the problems that is at the root of creating true artificial intelligence–namely that activities as simple and basic logical guessing games are not easily or necessarily accurately extracted from the richer and complex social embeddedness of human behavior.  Yes–it’s possible to choose the assumptions that make the math and computer logic work out easier–but when your goal is to teach computers to “be intelligent” by having them imitate a reality that is easier to program but less conforming to the messiness of the world–well.. you’ll achieve something much more trite and much less impressive than three people talking to each other competitively.

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About Prof. Woland

I contain multitudes. Come meet us.
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