I've been reading Daniel Kahneman's phenomenal *Thinking Fast and Slow*, and it has been blowing my mind. Not just a little--huge, messy chunks of my brain are splattered on the wall. Metaphorically, of course.

Kahneman has a large, audacious brain (he was awarded the Nobel Prize in Economics, if you need some evidence beyond my lavish praise), and he's spent decades thinking about how it works. He writes poignantly about his longtime collaboration with Amos Tversky and the great joy of their work together before Tversky's death in 1996 (which left Kahneman to collect the Nobel without his intellectual partner).

Over the years that Kahneman and Tversky worked together, they were fascinated by the errors that our minds make...errors that we leap to...that come easily and quickly to us. They would hatch up tests that illustrated this, with statistically significant, demonstrable results. I can imagine the film version--a montage scene starting in an Israeli cafe at noon. Time would whiz by as a tea cup stood out against the backdrop of dusk, then night. Then, smash cut to the test subjects walking into the university, the test proctor nodding, and Kahneman and Tversky behind a glass watching it all.

Kahneman starts his book by praising the mind that works intuitively. We see a photo of an angry person, and immediately we understand this information. Right? It's not a drawn-out process of conscious data collection and decision-making. You do not think, "the lines surrounding the mouth are at angles, what does that indicate? Wait--look at the eyes and nose, are there any clues there?" Rather, it's a single, immediate understanding: this person is livid. Our mind can access large pools of knowledge this way, by seamlessly integrating small, almost imperceptible data points. The abilities that work in this way - fast - are considered in a single category: "System 1." And System 1's power feels magical. Kahneman writes that it's the "hero of the story." It's the creative brain, the associative thinker. But, Kahneman warns, it's also got its weaknesses. System 1 has biases, and messes up dramatically. Frequently. It doesn't like to work hard, and it doesn't do well under duress. And, it can't be shut off to let System 2 do its thing in peace.

Meet System 2. It does the math. It can reason out tough problems, it can handle statistical thinking. But System 2 requires time and attention. Credit for the articulation of this two system scheme goes to Keith Stanovich and Richard West, and Kahneman marvels at the genius of this operating system...writing that the division of labor is brilliantly efficient, that it "minimizes effort and optimize[s] performance."

But sometimes these systems clash, and ensnare us in dangerous mental traps.

## A Few Examples

### 1. Which line is longer?

Look at the two illustrations above. Each shows two horizontal lines stacked on top of each other. In each picture, which horizontal line is longer? Top or bottom? If you know this one, your systems are aligned in their rapid response. If not, you might have had System 1 say: "Same length!" and System 2 say "Wait! Let's measure." (Or, System 1 might have noted the word "illusion" and quickly provided the answer: it's a trick!)

### 2. How many widgets?

This test was devised by Shane Frederick:

"If it takes 5 machines 5 minutes to make 5 widgets, how many minutes does it take 100 machines to make 100 widgets?"

Did your System 1 brain blurt out an answer? If it happened quickly, it's probably wrong. If you haven't seen this before, your system 2 brain needs to puzzle it out. (The correct answer is at the bottom of the post.)

### 3. A bat and a ball cost $1.10

The bat costs $1.00 more than the ball. How much does the ball cost? (Scroll to the end for the answer.)

Just like in example 2, to get this right takes a bit of attentive thought. And that's how you know the System 2 brain is working.

## Systems in Cahoots

System 2 is the skeptical and diligent editor to System 1. System 2 is the CFO, asking the hard questions and demanding evidence. It's a powerful counterpoint to System 1's leaps. But System 2 is also fallible. It has a tendency to believe System 1, and even become to its accomplice...by supplying evidence for System 1's quick conclusions.

One thing both systems of my brain appreciate is that Kahneman's book is littered with examples of fascinating studies. He references many that have fascinated us at Radiolab...the fruit and cake experiment, hot and cold coffee, the marshmallows... The skeptical mind loves all the supportive documentation. But then, the self-aware skeptical mind wonders if it's just that the evidence in the studies lets the lazy System 1 mind feel confident about the results--even when System 2 hadn't quite totally understood the information which was cited (as my tired brain sometimes didn't).

I think part of why this book is blowing my mind is that I've been reading it in the infrequent (and short), precious moments after my toddler falls asleep, before I'm too tired to read anymore. She's 2 1/2 now and is a marvel of language acquisition. Every day she surprises me with the new words she's using. A couple of days ago, she asked what the smear on the car window was. "Oh, that's bird poop," I told her. "Bird poop?" she repeated, and stared at the window. The next day, I still hadn't managed to find time to wash the car, and my daughter surprised me: "Open my backpack and get me the wipes and I want to clean the window for you. I am gonna make you so happy." I didn't know she knew the word "window," and I didn't know she could conjugate her verbs like that. It's amazing what she observes and assimilates and recalls.

My daughter has a limitless appetite for books about counting and the alphabet. And I love to think that I'm helping to make her a strong thinker, by making those immediate associations strong. Memorization isn't so much working analysis as instinct. But you need to have a full catalog of associations, a brain that plays and understands, to have a mind that calculates well and is skeptical about the right things. I also think about how the stop-and-think training of System 2 is my job as a parent. I have to teach her to apply system 2: "Wait--look both ways before you cross." A kid is all raw energy and instinct at that age and it takes practice to learn how to be diligent and intentionally attentive and rigorous. And like Daniel Kahneman, I am marveling at how the brain works...how efficient it is, how magical when the two systems collaborate perfectly. How satisfying it is to see instinct supported by effort. Or, as Steve from Blue's Clues says (hey! quiet down there, my "what-have-I-become" self-loathing brain system...it's a valid reference!), "Cause when we use our mind, take a step at a time, We can do anything...that we wanna do."

#### Answers:

Example 2: ~~100 minutes~~ 5 minutes! (Thanks everybody, for catching our error. Go System 2!)

Example 3: 5 cents

## Comments [25]

my favorite is ask someone to say "silk" 10 times. Then ask, "what do cows drink" - the common answer is 'milk', but the correct answer is 'water.

Juan from ma, if the ball was free it could not also be $0.10, so the answer can only be $0.05, as

[x + (x + 1.00) = 1.10]

[2x = 1.10 - 1.00]

[2x = 0.10]

[x = 0.05]

The "system 1 answer" $0.10 is also true because the problem that was presented never said that the ball was not free.

#2 What if the ball cost, $.02 and the bat cost, $.08. If the bat was $1.00 more than the ball the bat would cost $1.08 and the ball, $.02, making $1.10.

...wait...than that means the bat would cost $1.06 more than the ball. nevermind ha.

As kids, when dupped by problems like these, we called them "trick" questions. We called them this because we hadn't paid enough attention to the symantics of the problem. AND we hadn't appreciated our own assumptions and experience we had brought to the problem. There was a feeling of clever unfairness when we reread such problems.

I fell for the 100 min. widget answer for a split second, then my experience kicked in, having worked on an assembly line.

I missed the bat/ball one because I was impatient. I knew 10 cents couldn't be the answer (too obvious) but didn't take the time to think it out.

I did get the snail one right off . . . a very visual problem . . . and slow. Thanks, FUN!

#3: what if the ball cost $.01?

Acronyms [vowels: mathematical operations] by Shannon Schunicht

Wed 08/01, 10:20 AM - 10:30 AM session: FG12

Grand Ballroom: Ben Franklin V (seats 100)

Sheriton University City Hotel

3549 Chestnut St.

Philadelphia, PA 19104.3390

When instructing physics, formulas are continually espoused with applications, historical highlights, and derivatives in the same orderly fashion. Students have other classes and assignments. Physics now becomes second, if not discarded altogether. While in the Army, Mr. Schunicht was involved in a mid-air collision rendering three weeks of unconsciousness. Pragmatic discoveries were made to compensate for the residual memory deficits. The most valuable was having each vowel represent a mathematical operation, i.e. "a" multiplication to imply "@", "o" for division to mean "over", "i" for subtraction to signify "minus", "u" for addition to symbolize "plus", and "e" for "equals". Most constants, and variables are indeed consonants, e.g. "c" = "speed of light" & "z" = "altitude”. ADDITIONAL LETTERS may be inserted for intelligibility, but need be CONSONANTS An acronym for The Quadratic Equation is; exCePT i buiLD rabbiTS 4 caTS oN 2 HaTS. Everyone remembers Dr. Seuss?? The possibilities of this mnemonic technique are limitless as ∆ X => 0

Gaynell

There's two, no?

I've been slowly working my way through this book, and it is amazing. I'm astounded by the breadth of this book. Sure System 1 and System 2 are a big part of it, but there is so much more than that. I hope that Radiolab considers the topics in this book for future episodes. For instance, I would find an episode on Bayesian reasoning fascinating. Even an short about regression to the mean would be awesome, since it's so counterintuitive even though it is a fundamental part of nature. Thanks for turning me on to this book!

About the ball and the bat, there's a typo at the end with the answer.

@James

# 3 is not missing a point. Consider if the ball costs 10 cents and the bat costs a dollar more, then the bat will cost $1.10 and together with the ball the total will come to $1.20 not $1.10. Therefore the ball costs 5 cents, and the ball $1.05 :)

Excellent article, cant wait to read the book. Thanks for posting Radiolab, you are all thigs awesomeness as usual.

@ Leonardo Garcia from Lima Peru

No, your assumption incorrect. For 1 machine to produce 1 widget in 1 minute, then 5 machines would produce 25 widgets in 5 minutes. You are forgetting to account for the fact that the machines produce simultaneously rather than one after the other. 1 machine takes 5 minutes to produce 1 widget, therefore 100 machines takes 5 minutes to produce 100 widgets.

You've also screwed up the interpretation of #1. You say "In each picture, which horizontal line is longer? Top or bottom? If you know this one, your systems are aligned in their rapid response. If not, you might have had System 1 say: 'Same length!'"

No, the usual answer from someone who is not familiar with these illusions (and relies on "system 1" for a quick answer) is that the top line is longer in the Ponzo illusion, and the lower line is longer in the other illusion. In fact, the two lines on the left are the same length, and so are the two on the right.

I think you were thinking so much about "mental error" while writing this that the concept took root in your brain and caused you to screw up all three answers. Maybe you need to beef up your system 2?

It's fascinating how children learn language. It's at least as fascinating how they can learn to become kind and thoughtful as your daughter clearly is. Thank you for a wonderful story.

In question 2 there are two correct ways to get to the answer:

1) If 5 machines work on one widget at the same time, then the output is 1 widget/minute/5 machines (or 1/5 widget/minute/machine). In this case 100 machines will output 20 widgets/minute. 5 minutes yields 100 widgets (20*5 = 100).

2) If 5 machines each work on one widget apiece, then the output is 1 widget/machine/5 minutes (or 1/5 widget/machine/minute). In this case, 100 machines output 100 widges/minute, but every widget is 1/5 complete (equivalent to 20 full widgets, 100*1/5 = 20). In 5 minutes, they still have completed all the widgets.

I came about the answer to puzzle 1 differently. I reasoned that if 5 machines can make a widget every minute than 100 machines could make 20 widgets a minute. 40 in two minutes, 60 in three minutes 80 in four minutes and 100 in five minutes and so on...hence in 5 minutes these 100 machines made 100 widgets. But what could 1 machine in 1 minute produce? It would seem that 1 machine in 1 minute would produce 1 widget. If this is true that 100 machines in 1 minute would produce 100 widgets?

The answer to puzzle 2 is wrong. The correct answer is 5 minutes. 5 machines/ 5 widgets = 1 widget every 5 minutes per machine. At a rate of 1 widget every 5 minutes, 100 machines would create 100 widgets in 5 minutes.

Puzzle 3 was stated incorrectly. One must give the amount more than the ball, for example "The bat costs !!! $1.00 !!! more than the ball. How much does the ball cost?" Your immediate response might be that the ball cost $0.10 and the bat $1.00. This is incorrect because the bat would only be $0.90 more than the ball. So the correct answer is $0.05 for the ball, the price bat would then be exactly $1.05 ($0.05+$1.00=$1.05) and the total would be $1.10 ($0.05+$1.05=$1.10).

Question:

A snail is trapped at the bottom of a pit. Every day it climbs up 5 metres and every night it slides down 4. The pit is 20 metres deep. How many days does it take for the snail to escape?

It takes 16 days.

The snail climbs up 5 meters and slides down 4 meters every night. So it climbs up 1 meter per day. In 15 days it goes up 15 meters.

On the 16th day it goes up 5 meters and gets out of the pit.

It cant go down because it is out of the pit.

Question 2: If it takes 5 machines 5 minutes to make 5 widgets, how many minutes does it take 100 machines to make 100 widgets?

Answer: Based on the information in the question, it takes one machine 5 minutes to make one widget. To make 100 widgets using 100 machines it would take 5 minutes.

<B>The author has made a mistake by saying it takes 100 minutes.</B>

Question 3: A bat and a ball cost $1.10

The bat costs more than the ball. How much does the ball cost?

<B> The author does not even know how to ask a question or even copy some one elses question? What a joke?

The 3 original, unbroken questions:

-A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball.

How much does the ball cost?

-If it takes 5 machines 5 minutes to make 5 widgets, how long would it take

100 machines to make 100 widgets?

-In a lake, there is a patch of lily pads. Every day, the patch doubles in size.

If it takes 48 days for the patch to cover the entire lake, how long would it

take for the patch to cover half of the lake?

Answers here: http://www.sjdm.org/dmidi/files/Frederick%20%282005%29%20CRT.doc

as described in the original paper by Shane Frederick here, http://www.sjdm.org/dmidi/Cognitive%20Reflection%20Test.html

Another, similar question which I like is: A snail is trapped at the bottom of a pit. Every day it climbs up 5 metres and every night it slides down 4. The pit is 20 metres deep. How many days does it take for the snail to escape?

I, too, as a Momma, am fascinated on a daily basis by the acquisitiin and development of the language and criti al thinking of bothmy daugter who are 5 and 6 yrs. old. I relish when theyuse a new word or phrase. Last night my 5 yr. old used "perhaps" and it made me smile. It sounds like such a funny word for a five yr. Old to use. Enhoyed reading your post (5 a.m. is my preciou reading time.)

It honestly as if I was really pretendibg not to know where my mody was spacially. The three guyes were right about it as well. That was the strangest feeling.

James and Jordan are on the ball. These aren't really baffling tests.. at least when they are staged correctly. #2 isn't descriptive enough to come to single, definitive answer and #3 is just missing a part.

I think there are some mistakes in the brain teasers. The answer to the widget one should be, as James says, 5 minutes (according to my similar reasoning and many other sources). And you're missing a bit in the bat/ball question--if you're going with the common one, it's "the bat costs $1 more than the ball". That "$1" is pretty crucial, otherwise it doesn't make sense.

#2: My "System 1" yells 100 (because it likes symmetry), which is what you say is the answer. My "System 2" says it could be 5 minutes. 5 machines take 5 minutes to make 5. If all machines are identical and independent, this means it takes 1 machine 5 minutes to make 1. So, 100 of those machines would make 100 in 5 minutes.

#3: 5 cents? Why? Consider: Bat: $1, Ball: $0.10. or Bat: $.60, Ball: $.50.

Maybe I just don't get it.

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