"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Tuesday, April 20, 2010

Logic Puzzle: falsification optimization

Below we have four cards, where each has a number on one side and a letter on the other.

Now suppose we make a conjecture about those four cards: "Every card that has P on one side has 3 on the other". Now the question is what's the minimum number of cards we have to turn over in order to check the truth of the conjecture?

Some interesting statistics, concerning the number of people getting this right:
10% - General public
29% - Undergraduates
43% - Mathematicians

So if you got this correct, it means that you did better than an average mathematician :)


There's a family of interesting theories concerning the reasons why people get this wrong, and why a more practical set up of this "Watson selection task" reduces the percentage of people who get it wrong.


Mariusz Popieluch said...

Two :)

Iterate said...

Excellent, I got this right. The statement concerns only cards that have a P on one side. The first card clearly has a P, so we must check if there is a 3 on the other side. The second has an M, so the statement does not pertain to it and we don't care what's on the other side. The third card has a 3, so either it has a P on the other side and the statement is true, or it has some other letter and we don't care, so we need not check. The last card has a 5 so we must check that the letter is not a P, as that would invalidate the claim.

Iterate said...

P.S: Mathematician.

Mariusz Popieluch said...

You keep the "Mathematicians that answered correctly" statistic up:)

I got this one from YouTube:

Posted by a Cambridge graduate under the alias "singingbanana" - he has a whole collection of other interesting puzzles, mainly from the fields of combinatorics and logic.

mark said...

I like your blog, man. Michael Vincent, housemate and co-blogger, pointed it out to me.

I actually just lectured on this problem a few weeks ago (at UQ for Cogs1000).

If you haven't already, you should really read the famous Samuels, Stich and Tremoulet paper which discusses it. It's plenty interesting.

It turns out people do much better at this test (The Watson Selection Task, it is called) when it involves what has come to be known as "cheater detection".

For example, people almost always get the problem correct when it is re-framed in the following way:

Paricipants are told they are bouncers at a night club and need to know who is breaking the law. The cards have the ages of customers on one side, and their choice of beverage on the other. (I.e. '16' and 'Vodka' would be an instance of a law breaker.)

Why are people better at this version?

Well, many people think it is because our rationality is very specialised, and moveover that the way it is specialized should be understood with regard to the evolved modules of our brains. "Cheater detection", so the argument goes, is something that our ancestors needed.

It's a good problem.

-mark hooper

Mariusz Popieluch said...
This comment has been removed by the author.
Mariusz Popieluch said...

Hey Mark!
Thanks for the comment. I'll add a link to the The Watson Selection Task.
By the way, I had I like your blog too! I cought myself posting its entries on Facebook.

I can understand the improved performance in a more realistic/practical scenario. A reflection: could we generalise the "cheater detection" hypothesis, to a "tangible incentive" hypothesis? It would still be consistent with the evoulutionary expanation.

mark said...


I think something like your tangible incentive hypothesis would do better.

I'm a skeptic about the 'cheater detection' module.

My only worry about the 'tangible incentive' hypothesis is that it would be too broad.

There's always a danger when providing evolutionary explanations of not providing an explanation at all.

The quesiton is: "Why are we good at some puzzles but not others?"

Answer: "Well, we are good at the puzzles that our ancestors more commonly encountered."

No doubt. But this explanation isn't very deep.

Surely what we want to know is the specifics about the kinds of enctountered scenarios that are correlated with the evolution of the abilities we possess.

"Cheater detection" has a crack at answering this question. It says we bacame good at these puzzles at about the same time we developed our human-ish sorts of morality. We needed, they say, a way to spot free riders.