An Experiment in Cognitive Reasoning

Among several things, these days, I have been doing some (serious) reading of literature on psychology, cognitive development, learning, linguistics, philosophy and a few other subjects. Well, the ones I just named happen to be parts of interdisciplinary areas, which are precisely the ones I am interested in. Of course, on many levels those parts also have a lot to do with mathematics, especially mathematical education. Ok, that was just a little background I wanted to provide for the content of today’s post.

I need to do a small (online) experiment in order to test a hypothesis, which will be the subject of my next post. Let me not reveal too much for now. The experiment is in the form of two puzzles that I ask readers (you!) to solve. They are both “multiple choice” puzzles with exactly two correct answers to each. Please bear in mind that this is NOT an IQ test. It is also not meant to test how good you are at solving puzzles individually. I am really interested in “aggregate” results. That is, for testing my hypothesis, I am only interested in what the majority thinks are the right answers. What is more, you won’t be graded, and no one (not even me) will ever know if you got the right answers. Please submit answers to both the puzzles.

Lastly, please don’t cheat or try to look for answers offline or online. As I said, this is NOT a test!

Let us now look at the puzzles.

Puzzle 1:

There are four cards, labelled either X or Y on one side and either 3 or 7 on the other. They are laid out in a row with their top (visible) sides shown like this: X Y 3 7. A rule states: “If X is on one side then there must be a 3 on the other.” Which two cards do you need to turn over to find out if this rule is true?

1) X

2) Y

3) 3

4) 7

Puzzle 2:

As you walk into a bar, you see a large sign that reads, “To drink alcohol here you must be over 18.” There are four people in the bar. You know the ages of two of them, and can see what the other two are drinking. The situation is: Alisa is drinking beer; Dymphna is drinking Coke; Maureen is 30 years old; Lauren is 16 years old. Which two people would you need to talk to in order to check that the “over 18 rule” for drinking alcohol is being followed?

1) Alisa

2) Dymphna

3) Maureen

4) Lauren

If you think you have the answers to those puzzles, then please click here Puzzles to submit your answers. (I couldn’t use PollDaddy to embed the above puzzles in this post because I am not allowed more than 160 characters in a single question. What a pain!) So, please go ahead and click the above link to submit yours answers.

Note: I will keep this “poll” open for a week to collect as much data as possible. Thanks!

Update: It seems many readers weren’t aware of the short duration of the “poll” and that they would have very much liked to participate. So, I am extending the poll till Jan 7, 2010 for them.  (Doing so would also help me in collecting more data.)

POW-2: Diagonal slicing through squares

Since the cat’s out of the bag and we’ve had some public discussion of our first Problem of the Week, I thought I’d officially kick things off with a new problem, this time under the ground rules discussed in our last post. This one comes from our regular reader John Smith, who wrote me this problem in email. It comes in two parts; here’s the first:

Consider a rectangle with sides of integer lengths and , subdivided into unit squares, and a diagonal line from corner to corner. Show that the number of unit squares that the line crosses is . (Count only those cases where the line crosses the interior of a square.)

A little while later, John wrote me a follow-up question, which will be our part two:

What would be the -dimensional analogue of the previous problem? How would you prove it?

Please submit your solutions through email to topological[dot]musings[At]gmail[dot]com, rendering the stuff in brackets in the obvious way; please do not reply in comments! Solutions should arrive by May 20 (11:59pm UTC time if you want to be exact). We will post one of the correct solutions (or otherwise our own), but all correct solutions will be acknowledged, and successful solvers will have their names entered into our Problem-Solving Hall of Fame. So get your answers in, and happy solving!

[Added May 15: if you just want to solve part one, that’s fine — credit will be awarded for correct solutions. In general for multi-part problems, we ask that you submit all parts you intend to solve in one mailing — thanks.]