It could have been .04 0r.03 and the bat would still cost more than $1. Abstract. Interesting: these could cover a couple of misunderstandings, one is that B>=100, the other that "The bat costs $1.00 more than the ball" does not mean B-b=100, but that B-b>=100. Ah yeah, I meant to make this bit clearer and forgot. I visualise an amount (represented by a length on the number line), I visualise a dollar higher than that amount on that number line, and then move them around so that they're not overlapping, then see that the sum is $1.10. My suspicion is still that very few people solve this problem with a fast intuitive response, in the way that I rapidly see the correct answer to the lilypad question. Reading through the comments I count four other people who explicitly agree with this (1, 2, 3, 4) and three who either explicitly disagree or point out that they find the widget problem hardest (5, 6, 7). A bat and ball cost $1.10 If the bat costs $1 more than the ball then what does the ball cost? I've referenced the cognitive reflection test as one of those litmus tests of rationality, where I feel like any decent practice of rationality should get people to reliably answer the questions on that test. This is a really interesting point. Even the more 'intuitive' responses, like Marlo Eugene's, seem to rely on some initial careful reflection and a good initial framing of the problem. _____ days. Imagining the “ten cents” answer doesn't actually feel compelling; it just feels wrong. I hear your pain. Plenty of people noped out of mathematics long before they got to simultaneous equations, so they won't be able to solve it this way. Out of the people who agreed with that the bat and ball is different, this comment from @awbery does a particularly good job of giving a potential explanation for why: The problem is a ‘two things’ problem. Also, there is a kind of problems like "one wallet contains ten coins, another one contains twice more, and the total is twenty; explain" that get asked much earlier than kids learn algebra, if I remember right. In reality most collisions between bat and ball (especially the ones I am able to make) are glancing collisions which require a two-dimensional analysis. I'll only summarise the bat-and-ball-related parts of the comments here. I have the same experience as you, drossbucket: my rapid answer to (1) was the common incorrect answer, but for (2) and (3) my intuition is well-honed. How many seats have I booked? Writing out the basic problem again, we have: Now, instead of immediately jumping to the standard method of eliminating one of the variables, we can just look at what these two equations are saying and solve it directly 'by thinking'. However, getting people to actually answer the question correctly was a much more difficult problem. They apparently find cognitive effort at least mildly unpleasant and avoid it as much as possible. The correct answer to this problem is that the ball costs 5 cents and the bat costs — at a dollar more — $1.05 for a grand total of $1.10. I think this is why Kyzentun and Ander’s methods help get at the bat and ball problem intuitively – because they bypass the conflict between object level and abstract and translate it into the formal algebra realm. The correct answer is 5¢. This means that they are unable to immediately tell that the problem has a unique solution. Further, these solutions don't all split neatly into 'System 1' 'intuitive' and 'System 2' 'analytic'. Ethnomethodology. Participants solved the bat-and-ball problem and were allowed to make a second guess after they had entered their answer. David Chapman recommended Formal Languages in Logic by Dutilh Novaes for more background on this. I created the machine.minute unit (equivalent to the kWh unit) that allowed me to understand that a widget is made in 5 machine.minutes. In the original problem, the 110 units and 100 units both refer to something abstract, the sum and difference of the bat and ball. For the second one though, I was too careful: I immediately started transcribing the problem in a pertinent format, i.e. Connect more specifically to Stanovich's idea of cognitive decoupling. However, after more than a little head scratching I’ve gained an understanding of this puzzle. Further out, it could be interesting to actually test some theories by trying alternative, disguised versions of the question, on Mechanical Turk or something. Nevertheless, the ball is hit for a home run, demonstrating in dramatic fashion that the batter's grip plays no role in the ball-bat collision. “Bat & Ball” Cognitive Reflection Test. Meyer, Spunt and Frederick tested this hypothesis by getting respondents to recall the problem from memory. The lily pads question takes me a conscious time-step longer to answer than either of the other two; the initial flash is “inconclusive”, and then I see myself rechecking the part where the quantity doubles every step before answering “47”. The Obvious Answer is (click to reveal) That is because only 32% of people get this simple math problem correct. The language correctly reflects there are two things we should consider. We generally learn arithmetic as young children in a fairly concrete way, with the formal numerical problems supplemented with lots of specific examples of adding up apples and bananas and so forth. I'm not really sure what to make of that statement you put in italics. One is that 5 cents and 10 cents both just register as 'some small change', whereas 24 days and 47 days feel meaningfully different. At this point there's not really any room to escalate beyond confiscating the respondents' pens and prefilling in the answer 'five cents', and I worry that somebody would still try and scratch in 'ten cents' in their own blood. Similarly, anders works the problem by 'getting rid of the 100 cents', and splitting the remainder in half to get at the price of the ball: I just had an easy time with #1 which I haven’t before. yup that's better" is all done on system 1. “The bat-and-ball problem is our first encounter with an observation that will be a recurrent theme of this book: many people are overconfident, prone to place too much faith in their intuitions. The simplest 'analytical', 'System 2' solution is to rewrite the problem as two simultaneous linear equations and plug-and-chug your way to the correct answer. I learned algebra, fortunately, not by going to school, but by finding my aunt's old schoolbook in the attic, and understanding that the whole idea was to find out what x is - it doesn't make any difference how you do it. Here are questions which might be similar to (I): (4a) I booked seats J23 to J29 in a cinema. Seconding Habryka. I'd kind of assumed that there'd be some kind of serious-business Test Creation Methodology, but for the CRT at least it looks like people just noticed they got surprising answers for the bat and ball question and looked around for similar questions. Split the difference evenly for 5 cents? Here's a few I wanted to highlight: Is the bat and ball question really different to the others? I can't quite remember my strategy for bat & ball, but I think I generated the $0.1 ball, $1 bat answer, saw that the difference was $0.9 instead of $1, adjusted to $0.05, $1.05, and found that that one was correct. Instead they seem to focus mainly on the first condition (adding up to $1.10) and just use the second one as a vague check at best ('the bat would still cost more than $1'). The impact between bat and ball is a collision between two objects, and in its simplest analysis the collision may be taken to occur in one-dimension. There’s no object level mirror trick in the other two problems, they’re straight forward maths mapping an object level visual representation. Probably nowhere much for a while, as I have other priorities. In ordinary language, "that costs $1.00 more than the other one" is not incorrect if the difference is $1.01. endstream endobj startxref This simple problem does have 'ten cents' as the answer, so it's very plausible that people are getting confused by it. I’d really like to see this reviewed. �Loڀ47��Uؙ�� %�]�2����dl�r���0����|0K�H/���D -��Pv There are a couple of possible reasons for this. When you first read the problem and hear that the bat is a dollar more than the ball, and the bat and the ball cost a dollar and ten cents, your brain assumes that the ball is automatically 10 cents. How much does a bat cost? The answer is .05.’ And then I checked my answer by doing 1.05 + .05 and 1.05 - .05. I'm reminded of a story in Feynman's The Pleasure of Finding Things Out: Around that time my cousin, who was three years older, was in high school. This suggests enough half-remembered mathematical knowledge to find a sensible abstract framing, but not enough to solve it the standard way. For the bat and the ball, I do something similar to the margins example. So there's an extra 10 cents--oh, of course, the difference between $1 and $1.10 has to be distributed evenly between both items, so the answer is 5 cents. I can also imagine (c) I'm leaping to the “wrong” answer, then trying to verify it, noticing it's wrong, and correcting it, all in the same subconscious flash, but that feels off. This is where you really get to see the variety of ways that people tackle the problem. What I did was take away the difference so that all the items are the same (subtract 100), evenly divide the remainder among the items (divide 10 by 2) and then add the residuals back on to get 105 and 5. Or something else. The bat and ball still gets me, though. This isn't at all my field and I'd be interested in any good sources. There's a kind of natural inertia to this kind of puzzles. So you divide the remainder equally (assuming negative values are disqualified) and get 0.05. The Bat and Ball problem has been upheld as a thin slice measure of an individual’s disposition or ability to engage in reflective thought, and is now included as a covariate in many studies. h�bbd``b`�$ׂm7Dd �Pa⺂X� �D���2��Sg&F�@#:��G�� ~4 For the bat to cost $1 more than the ball, the ball has to cost 5 cents and the bat $1.05. I'm only just getting round to reading some ethnomethodology, and I haven't got my bearings yet. In the case of the OP, system 1 has been trained to really understand exponential growth and ratios. It’s a different type of problem to the other two in this sense, because the objects they present can be used as given in the solution. It is safe … Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at a dedicated player of the attacking team, who tries to hit it with a bat and run between various safe areas in the field to score points, while the defending team can use the ball in various ways against the attacking … Do they all do a different bit of the job? After this, they ditched subtlety and resorted to pasting these huge warnings above the question: These were still only mildly effective, with a correct solution jumping to 50% from 45%. I must have missed this comment before, sorry. At this point they completely gave up and just flat out added “HINT: 10 cents is not the answer.” This worked reasonably well, though there was still a hard core of 13% who persisted in writing down 'ten cents'. Yes, the most common answer for that for what the ball costs is 10 cents. If you haven't, it's only three quick questions, go and do it now. What I'd really like is some insight into what individual people actually do when they try to solve the problems, rather than just this aggregate statistical information. This is exactly what bothers me and resulted in me wanting to look up the question online. The surprisingly high rate of errors in this easy problem illustrates how lightly System 2 monitors the output of System 1: people are not accustomed to thinking hard, and are often content to trust a plausible judgment that quickly comes to mind. X + Y = $1.10. I find it interesting that he frames mental processes as being inherently effortless or effortful, independent of the person doing the thinking. Ooh, I'd forgotten about that test, and how the beer version was much easier - that would be another good one to read up on. While researching the problem I came across this article from the online magazine of the Association for Psychological Science, which discusses a variant 'Ford and Ferrari problem'. Impact of feedback on the bat-and-ball problem. There's a variant 'Ford and Ferrari' problem that is somewhat related:> A Ferrari and a Ford together cost $190,000. I'm going to continue reading Dutilh Novaes and some ethnomethodology. The language correctly reflects there are two things we should consider. (6) You are in a race and you just overtake second place. Example given: 'I see. The first sentence presents two things, a bat and a ball. The categories are: Correct answer, correct start. Frederick's original Cognitive Reflection Test paper describes the System 1/System 2 divide in the following way: Recognizing that the face of the person entering the classroom belongs to your math teacher involves System 1 processes — it occurs instantly and effortlessly and is unaffected by intellect, alertness, motivation or the difficulty of the math problem being attempted at the time. With the total cost of bat & ball at $1.10, and the difference between the two being $1, the ball couldn’t cost 10¢ because that’d make the bat cost $1.10, which would bring the combined price to $1.20. This is wrong because if the ball costs 10 cents then 1 dollar more then 10 cents would be $1.10. Then I realise there's two identical bits that are added to the $1, which means they're 10/2 each. Click on the link to see learn about why the batter's grip doesn't matter during the baseball-bat collision, including a more detailed discussion of the Todd Frazier "no-hands" home run. But especially in the context of expecting a trick question, I second-guess it and come up with the correct answer fairly quickly. The text is 100 columns wide. A baseball bat is $3.00 more than the price of the ball. I'd hear him talking about x. I said to my cousin, "What are you trying to do? I just saw the answer to the bat and ball problem within a few seconds. I've come nowhere near to doing a proper literature review. I think I figured out and verified the answer to all 3 questions in 5-10 seconds each, when I first heard them (though I was exposed to them in the context of "Take the cognitive reflection test which people fail because the obvious answer is wrong", which always felt like cheating to me). The ‘obvious wrong answers’ for 2. and 3. are completely unappealing to me (I had to look up 3. to check what the obvious answer was supposed to be). So, here is the problem: A bat and ball cost $1.10 The bat costs one dollar more than the ball How much does the ball cost ? The correct answer to the bat & ball question is 5¢. This question first turns up informally in a paper by Kahneman and Frederick, who find that most people get it wrong: Almost everyone we ask reports an initial tendency to answer “10 cents” because the sum $1.10 separates naturally into $1 and 10 cents, and 10 cents is about the right magnitude. We have a bat, B. I would speculate, in decreasing order of intuitive probability, that in order to get the answer, either (a) I've seen an exactly analogous “trick” problem before and am pattern-matching on that or (b) I'm doing the algebra quickly using my seemingly well-developed mathematical intuition. (I notice I didn't remember that the steps were days, only remembering that there was a time unit; I don't know if that's relevant.) Success in the bat and ball problem seems to involve decoupling from the noisy wrong answer. In the experimental condition order, participants solved a simpler isomorphic version of the problem prior to solving a standard version that, critically, had the same item-and-dollar amounts. This showed a clear difference: 94% of 'five cent' respondents could recall the correct question, but only 61% of 'ten cent' respondents. They apparently find cognitive effort at least mildly unpleasant and avoid it as much as possible. Rather than making any big conclusions, the main thing I wanted to demonstrate in this post is how complicated the story gets when you look at one problem in detail. I'm not sure how to visualise machines taking 5 mins to make 5 things. A possible reason for this is that the intuitive but incorrect answer in (1) is a decent approximation to the correct answer, whereas the common incorrect answers in (2) and (3) are wildly off the correct answer. In the course of our back and forth I switch my phrasing to the form "You have seven apples and you take away five, how many left?" That's where they left it. I have a vague suspicion that Frederick trawled through something like 'The Bumper Book of Annoying Riddles' to find some brainteasers that don't require too much in the way of mathematical prerequisites. How many fence posts are there? Many people yield to this immediate impulse. Kahneman's examples of system 1 thinking include (I think) a Chess Grandmaster seeing a good chess move, so he includes the possibility of training your system 1 to be able to do more things. I like to think this dismissal bites people in the backside when they learn Mendelian genetics (more easily seen when the genes in question interact hierarchically) or, Merlin forbid, mass-spectrometry, where the math difficulty is complicated by the chem difficulty of molecules not dividing into usual subunits. The three items on the CRT are “easy” in the sense that their solution is easily understood when explained, yet reaching the correct I've also written a bit more about cognitive decoupling and the history of the term here. If you're able to rapidly 'just see' the answer to the bat and ball question, how do you do it? I haven't thought about the bat and ball question specifically very much since writing this post, but I did get a lot of interesting comments and suggestions that have sort of been rolling around my head in background mode ever since. Instead, it gives a aggregated overview of types of responses, which doesn't go into the kind of detail I'd like. Here’s the solution: Although $1.00 + $0.10 does equal $1.10, if you take $1.00 – $0.10 you get $0.90, but the problem requires that the bat costs $1 more than the ball. In Kyzentun's version these become much more concrete objects, the width of the text and the total width of the margins. If the ball costs 10 ¢, then the total cost will be $1.20 (10¢ for the ball and $1.10 for the bat), not $1.10. This problem is taken from the UKMT Mathematical Challenges. It correctly sounds like a + b; two things. No, that adds up to $1.20. Conversely, finding √19163 to two decimal places without a calculator involves System 2 processes — mental operations requiring effort, motivation, concentration, and the execution of learned rules. I should actually go through them! Notice that this is missing the 'more than the ball' clause at the end, turning the question into a much simpler arithmetic problem. My tentative guess is that the bat and ball problem is close to being this kind of efficient tool. The intuitive-but-wrong answer turns out to be extremely sticky, and the paper is basically a series of increasingly desperate attempts to get people to actually think about the question. She was very frustrated by it, and if I verbally asked "What's seven minus five?" Then I looked at the question, and computed that you’d need 500 machine.minutes to make 100  widgets, so 5 minutes with 100 machines. He was having considerable difficulty with his algebra, so a tutor would come. In response, another commenter, Tony, suggests a correct solution which is an interesting mix of writing the problem out formally and then figuring out the answer by trial and error:\. I don't really share drossbucket's intuition - for me the 100 widget question feels counterintuitive the same way as the ball and bat question, but neither feels really aversive, so it was hard for me to appreciate the feelings that generated this post. Marlo Eugene's solution, for instance, is a mixed solution of writing the equations down in a formal way, but then finding a clever way of just seeing the answer rather than solving them by rote. My original post on the problem was a pretty quick, throwaway job, but over time it picked up some truly excellent comments by anders and Kyzentun, which really start to dig into the structure of the problem and suggest ways to 'just see' the answer. For the first one the bat had to be one dollar MORE than the ball so, if the ball was 10 cents the bat had to be $1.10cents that plus another 10 cents is $1.20. But I also share your sense that the answer to (3) is 'wildly off', whereas the answer to (1) is 'close enough'. (Why the hell would the lily pads take the same amount of time to cover the second half of the lake as the first half, when the rate of growth is increasing?). A baseball bat and a ball total cost is $8.50. These are designed to be cognitively unpleasant in the same way as the bat and ball, so I keep putting them off. The jump in success rate could be down to better trained intuition. In (1), 5 and 10 are both similarly small compared to 100 and 110. I'm done.'. In (3), 24 is small compared to 48, but 47 isn't. Finally some examples of how the problem is solved in the wild! Thus if the ball equals to x, the bat equals to x plus 1... ', Correct answer, incorrect start. If the bat costs $1.00 more than the ball and the total is $1.10, then the ball must cost 5 cents and the bat must cost $1.05.