One of the biggest impediments to clear thinking is the fact that it's so hard for us to keep in mind that we could be wrong.
I asked you how it felt to be wrong, and you had answers like humiliating, frustrating, embarrassing, devastating. And those are great answers. But they're answers to a different question. Those are answers to the question, "How does it feel to find out you're wrong?" But being wrong? Being wrong doesn't feel like anything... You remember those characters on Saturday morning cartoons, the Coyote and the Roadrunner? The Coyote was always doing things like running off a cliff, and when he'd do that, he'd run along for a while, not seeing that he was already over the edge. It was only when he noticed it that he'd start to fall. That's what being wrong is like before you've realized it. You're already wrong, you're already in trouble... So I should amend what I said earlier. Being wrong does feel like something.
It feels like being right.
We cling desperately to the sense that we have it all figured out, that we're right about everything. Oh, in theoretical terms we realize we're fallible; all of us can remember times we've been wrong. But right here, right now? It's like my college friend's quip, "I used to be conceited, but now I'm perfect."
The trouble with all this is that it blinds us to the errors that we do make, because if you don't keep at least trying to question your own answers, you won't see your own blunders. It's why lateral thinking puzzles are so difficult, but so important; they force you to set aside the usual conventions of how puzzles are solved, and to question your own methods and intuitions at every step. This was the subject of a study by Andrew Meyer (of the Chinese University of Hong Kong) and Shane Frederick (of Yale University) that appeared in the journal Cognition last week. They looked at a standard lateral thinking puzzle, and tried to figure out how to get people to avoid falling into thinking their (usually incorrect) first intuition was right.
The puzzle was a simple computation problem:
A bat and a ball together cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost?
The most common error is simply to subtract the two, and to come up with ten cents as the cost of the ball. But a quick check of the answer should show this can't be right. If the bat costs a dollar and the ball costs ten cents, then the bat costs ninety cents more than the ball, not a dollar more (as the problem states). The correct answer is that the ball costs $0.05 and the bat costs $1.05 -- the sum is $1.10, and the difference is an even dollar.
Meyer and Frederick tried different strategies for improving people's success. Bolding the words "more than the ball" in the problem, to call attention to the salient point, had almost no effect at all. Then they tried three different levels of warnings:
- Be careful! Many people miss this problem.
- Be careful! Many people miss the following problem because they do not take the time to check their answer.
- Be careful! Many people miss the following problem because they read it too quickly and actually answer a different question than the one that was asked.
All of these improved success, but not by as much as you might think. The number of people who got the correct answer went up by only about ten percent, no matter which warning was used.
Then the researchers decided to be about as blatant as you can get, and put in a bolded statement, "HINT: The answer is NOT ten cents!" This had the best improvement rate of all, but amazingly, still didn't eliminate all of the wrong answers. Some people were so certain their intuition was right that they stuck to their guns -- apparently assuming that the researchers were deliberately trying to mislead them!
If you find this tendency a little unsettling... well, you should. It's one thing to stick to a demonstrably wrong answer in some silly hypothetical bat-and-ball problem; it's another thing entirely to cling to incorrect intuition or erroneous understanding when it affects how you live, how you act, how you vote.
It's why learning how to suspend judgment is so critical. To be able to hold a question in your mind and not immediately jump to what seems like the "obvious answer" is one of the most important things there is. I used to assign lateral thinking puzzles to my Critical Thinking students every so often -- I told them, "Think of these as mental calisthenics. They're a way to exercise your problem-solving ability and look at problems from angles you might not think of right away. Don't rush to find an answer; keep considering them until you're sure you're on the right track."
So I thought I'd throw a few of the more entertaining puzzles at you. None of them involve much in the way of math (nothing past adding, subtracting, multiplying, and dividing), but all of them take an insight that requires pushing aside your first impression of how problems are solved. Enjoy! (I'll include the answers at the end of tomorrow's post, if any of them stump you.)
1. The census taker problem
A census taker goes to a man's house, and asks for the ages of the man's three daughters.
"The product of their ages is 36," the man says.
The census taker replies, "That's not enough information to figure it out."
The man says, "Okay, well, the sum of their ages is equal to the house number across the street."
The census taker looks out of the window at the house across the street, and says, "I'm sorry, that's still not enough information to figure it out."
The man says, "Okay... my oldest daughter has red hair."
The census taker says, "Thank you," and writes down the ages.
How old are the three daughters?
2. The St. Ives riddle
The St. Ives riddle is a famous puzzle that goes back to (at least) the seventeenth century:As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven kids,
Each kid had seven cats,
Each cat had seven kits.
Kits, cats, kids, and wives, how many were going to St. Ives?
3. The bear
A man goes for a walk. He walks a mile south, a mile east, and a mile north, and after that is back where he started. At that point, he sees a large bear rambling around. What color is the bear?
4. A curious sequence
What is the next number in this sequence: 8, 5, 4, 9, 1, 7, 6...
5. Classifying the letters
You can classify the letters in the English alphabet as follows:
Group 1: A, M, T, U, V, W, Y
Group 2: B, C, D, E, K
Group 3: H, I, O, X
Group 4: N, S, Z
Group 5: F, G, J, L, P, Q, R
What's the reason for grouping them this way?
6. The light bulb puzzle
At the top of a ten-story building are three ordinary incandescent light bulbs screwed into electrical sockets. On the first floor are three switches, one for each bulb, but you don't know which switch turns on which bulb, and you can't see the bulbs (or their light) from the place where the switches are located. How can you determine which switch operates which bulb... and only take a single trip from the first floor up to the tenth?