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Trickier than it seems: Four puzzles

I enjoy a good puzzle every now and again. Particularly one that seems very easy... Until you try to solve it.

Below are a four classic questions/ puzzles/ problems. If you've read a bit about problem solving before, you might know some of them. But I hope that you will nevertheless find it interesting to revisit them. (And perhaps make the same mistakes as before!) And if you don't know them, you're in for a treat!

The ball and the bat

Let's start with an easy one. As you can tell by the prices, this question is from way back:

A bat and a baseball together cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?

This is just a warm-up exercise, and you will probably have answered correctly. (If not, shame on you!) But perhaps you noticed that you almost gave the wrong answer. Your intuition (almost) seduced you with an easy, but incorrect answer. Quite possibly, your train of thought was something like this:

Right, obviously the answer is $0.10. But wait... that's too easy. There must be more to it. Oh, I see now. How stupid of me, the correct answer is of course (...).

So the context in which the question was asked (a post about tricky puzzles) made you think twice before answering. And this dramatically increased your chance of answering correctly. But the amazing thing is that, when embedded in a stream of questions, almost everybody gives the wrong answer.

Click here to see the solution!

The nine dots puzzle

Let's move on to the next puzzle. A more visual one this time. Nine dots are laid out in a grid. How do you connect all dots, using four straight lines, without taking your pen of the paper?

Statistically, you are highly unlikely to figure this one out. So be proud if you do! This puzzle is really hard. Unless you've seen it before, in which case you'll immediately recognize the solution. The solution, by the way, is not lame. You don't need to fold the paper or anything.

This puzzle is often used to illustrate the concept of "thinking outside the box". Personally (and I'm sure you do the same), when I solve a puzzle like this, I have some vague notion of what the solution should look like, even though I don't know the actual solution. In other words, the 'box' is a space of candidate solutions that you mentally search through. What makes this puzzle so incredibly difficult is that the solution doesn't match most people's expectation of what a solution should look like. Even though, when you see it, you immediately recognize that it's a simple, valid, and non-lame solution.

Incidentally, this puzzle is popular among business/management/corporate-types, because they feel that solving it is a sign of intelligence. They feel that being able to think outside the box is about the best quality that one can have. But I'm not so sure about that. After all, most daily life issues are dealt with perfectly well by thinking inside the box. That's why the box is there: It provides us with an efficient, and generally effective, way to solve problems.

Click here to see the solution!

The four cards problem

The next puzzle is always good for some confusion. I first heard it when I took a course called Thinking and Deciding. I don't remember whether my initial response was correct. Probably not.

You have four cards, as shown below. Each card has a number on one side and a letter on the other. John tells you that, if a card has a vowel ('A', 'E', etc.) on one side, it always has an even number on the other side. Which two cards do you need to turn over to see if John is right?

This puzzle is tricky. And it remains tricky, even when you know, in principle, how to approach the problem. What makes it so difficult is that we are inclined to seek evidence that confirms John's rule, rather than disproves it. Furthermore, the question primes us to think about vowels and even numbers, because these are explicitly mentioned. But this is not a fruitful way to think about the problem!

Click here to see the solution!

The Monty Hall problem

A final puzzle. This is a really, really difficult one. I'm not sure I have a full grasp on it myself. Sometimes I think I do.

The problem, which is known as the Monty Hall problem, was made famous by a 70s game-show called Let's Make a Deal. Hosted by, yes, Monty Hall. In this show, contestants played the following game:

There are three identical doors. Behind one door is a car (indicated by green in the image below). Behind the other two doors are goats (indicated by red). The goal for the contestant is to select the door with the car behind it. If successful, he or she can take the car home. The host and the contestant play a game. The contestant first selects one door. Next, the host opens one of the remaining doors. The host, who knows where the car is, always opens a door with a goat behind it. Now there are only two doors left, and the contestant is asked to stay or switch. That is, he or she can stick with the originally selected door, or switch to the remaining one (the one that wasn't opened by the host).

And the question is: Should the contestant stay or switch!?


Click here to see the solution!