12 ms·
Improbably Right
- dns_snek 3y agoThe cards example doesn't work in Firefox, it reveals the opposite side of the card :/
- tzot 3y agoAh, so it's my Firefox :)
- Kim_Bruning 3y agoConcur. I was very confused until I decided to reopen in chromium. %-)
- mlajtos 3y agoOh, this browser again. It seems that Firefox might have a problem with `mix-blend-mode: multiply;` and 3D CSS transforms. Cards are now correctly behaving in Firefox, but across the board they are visually less appealing. Thank you for bringing this up.
- moffkalast 3y agoAbsolutely hilarious how people claim that Gecko isn't half broken anymore.
- glofish 3y agoI believe that the author misremembers the common solution for the first question > 3 hens lay 3 eggs in 3 days. How many eggs do 12 hens give in 12 days? > [...] When we got to it, everybody shouted "Three!" I don't see why the "common" wrong answer would be 3. Why would anybody think that? The problem looks like this 3, 3 -> 3 When we see the 12 and 12 the intuitive, common, wrong answer should be 12 eggs. That's what makes sense IMHO.
- menzoic 3y agothe article says everyone shouted 12 not Three
- forbiddenlake 3y agoIt said 3 originally. https://news.ycombinator.com/item?id=35688034 https://news.ycombinator.com/item?id=35688034
- mlajtos 3y agoOf course it was "Twelve!". Thank you for bringing this up. It is fixed now.
- moondrek 3y agoI read it as the students being familiar with a similar "trick" question and erroneously pattern matching on that. Another similar riddle goes like: "If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?" The correct answer there being "5 minutes", and the intuitive wrong answer being "100" as per your post. That would make sense as to why the teacher also expected "3", since they should be familiar with the existence of their non-intuitive questions, even if they misremember the specific non-intuition.
- Cthulhu_ 3y agoOr as to why adding more people to a late project makes it later: 1 woman takes 9 months to grow a baby, how long does it take 9 women to grow a baby?
- thechao 3y agoI took me surprisingly long to convince myself the author was correct: the answer is 48 because there's 4x as many hens and 4x as many days: 3 x 4 x 4 = 48. The "wrong" way to solve this problem is to compute hen-laying in terms of hen-egg-days, and then scaling.
- 3y ago
- tzot 3y ago1. I had a similar issue in the fifth grade of elementary school (in my country, elementary school is 6 years with an extra year at age about 5 before pre-elementary school, then three years of middle high, then three years of optional high school before university, but almost everyone does the full 12/13 years.), when we discussed Fahrenheit (foreign) vs Celsius (domestic). The teacher said that 50°C is 106°F because 212°F is 100°C and 212/2 = 106. At the time I said “this doesn't feel right, 106 is not in the middle between 32 and 212” but I couldn't phrase it mathematically yet, and the teacher didn't take my objection in a good manner. Afterwards I learnt that he understood his error but never said a word to me about it. 2. I don't see the colours on the “back sides” of the cards. I just see the cards flipped/rotated 180° around the Y-axis (probably through CSS), so all numbers are visible, just in cards 3 & 4 the numbers are mirrored.
- eviks 3y agoHow did you learn it about the teacher?
- tzot 3y agoFrom his nephew, one year older than me, in the sixth grade with whom we played football (soccer). He said that he heard his uncle talking to his mom.
- mlajtos 3y ago2. Firefox user, right? It should be fixed now. Thank you for reporting this issue. 1. You reminded me of another nice example of wrong shortcuts. Average speed of car from city A to city B is 120km/h. Average speed back (B -> A) was 80km/h. What was the average speed of the car on the whole path (A -> B -> A)? It's not 100km/h.
- firstlink 3y ago> It's not 100km/h. Of course it is... if you're averaging over distance. There's no reason that's less valid than whichever average you have in mind.
- roenxi 3y agoInteresting. So we can tell from the last example that humans are perfectly good at doing this reasoning in theory and that therefore the trouble is actually in interpreting what the question is asking. Put another way, if the question could be beamed into someone's head they'd get it right. The flaw is in the part of the mind that loads the words into their brain and not the reasoning after that.
- mlajtos 3y agoI think it shows that we suck at generalization. Even though the task is the same, we are not able to see the same pattern in an arbitrary framing of the same problem. Beaming directly might help.
- travisjungroth 3y agoI don’t think this is the most accurate way to describe it. It’s considering the mind as separate from the brain. There’s also no true version of the problem to “beam” into the brain. That would just be a third version. Our brains aren’t purely general purpose problem solvers. There are parts of the brain that are specifically related to facial recognition versus all images. We’re better at recognizing differences in faces versus buildings. There seems to be a similar “hardware acceleration” for problems about people following rules (can’t remember the source for this). People generally do better on problems if you can frame them as who is breaking a rule, who is lying, etc. I think the last example would still be easier to solve on average even if it was a rule unfamiliar to people.
- eviks 3y ago> task is presented in context of social relations (or more broadly "evolutionarily familiar problems"), people tend to get it immediatelly, or rather intuitivelly. Bonkers, right? That's obvious, not bonkers - if you have experience in some class of problems, it's easier to solve the same kind of problem, the patterns match in your head, so the solution pattern is used right away. Just like you can solve these numeric problems quickly if you're trained on them
- moffkalast 3y agoSo in a sense we have "hardware acceleration" for solving certain types of problems?
- mlajtos 3y agoBut the task is exactly the same in both situations – only the symbols on the cards have less/more familiar meaning. I see this as being not able to generalize from known examples.
- eviks 3y agoIt's not exactly the same exactly because the difference in symbols maps to different patterns - one is a pattern from school, which you forgot since it hasn't been used for many years (or never learned properly). Another is a pattern that keeps getting refreshed throughout your life. It's a very different cognitive workload, leading to difference off required effort/correctness, especially in these simplistic low value tests And why is your assessment of the ability to generalize limited to only this math representation which people don't use?
- yaakov34 3y agoCan't pass by without making a public service announcement about the average speed example: don't compute integrals using the calculus definition of Sum(f(x_i) * delta). Look up quadrature or numerical integration methods instead. Although it is true that you can approximate the average speed by taking an average of instantaneous speed measurements, that's usually a very bad way to do it in any real world situation. Numerical difference values are always noisier than the underlying quantity, sometimes to the point of being unusable, so of course if you can just read off the quantity you want directly (total difference over time), you should do that. But even if you can't, you should use a proper integration method instead of the calculus definition. I have seen the Sum(f(x_i) * delta) calculation in a lot of real-world code. It has bad convergence properties, bad errors when the function has large derivatives, and bad performance when the data has noise. Some of the code I've seen produces garbage results, or has thousands of function evaluations when you need, like, four. "Quadrature? I think I heard that before, but I don't remember what it means." In summary, please don't compute derivatives as (f(x_i+1)-f(x_i))/delta, or compute integrals as Sum(f(x_i) * delta), and especially, please don't do the first immediately followed by the second. Which also happens. Look up numerical methods instead. This has been a public service announcement.
- hgsgm 3y agoHuh? Quadrature is a general term for "measuring area". In this context it's a synonym for integration. I think you are trying to say that it's better to do weighted sums of fewer samples, instead of uniformly weighted Reimann sum. Both are "calculus definition" integration, of course, since calculus is true.
- ouid 3y agoWhen mathematicians say Quadrature, they mean that if your function is suitably approximated by projecting onto some orthogonal basis functions, you can get very cheap approximations by cleverly expressing those integrals exactly as a linear combination of their values at certain points along the interval. You need very few. https://en.m.wikipedia.org/wiki/Gaussian_quadrature https://en.m.wikipedia.org/wiki/Gaussian_quadrature It is significantly more subtle than what you are thinking.
- Jiro 3y ago>If a card shows an even number on one side, then its opposite side is blue. As the problem fails to specify that the card has a number on one side and a color on the other, you need to check three cards.
- mpeg 3y agoNo, you don't. You still only have to check those 2 cards. If you're asking about card 3 then that's the common mistake – it doesn't matter whether that card has a number or an emoji on the other side, because the only logical test here is that if it has an even number on one side, the other side must be blue. It doesn't matter what blue cards have on the other side, we don't need to test them. edit: sorry, I just realised you're referring to card 1 and not 3, yeah I suppose you'd need to check it too unless there's a rule about always needing to match a color to a number.
- quirino 3y agoWhat they mean is that the card that has a 3 on the front might have an even number on the back.
- stared 3y agoWhy would anyone answer 12? Unless... most human thinking is this lower-end language model, which looks at word patterns without giving much thought to them. "12" is not intuition. It is repetition.
- hgsgm 3y agoIt's misapplied intuition for double ratios. If a triangle has sides 3:3:3 in feet, in inches it has sides 12:12:12.
- btbuildem 3y agoWhy? Because if three hens laid three eggs in three days, then it means one hen lays one egg per day - so twelve hens, given twelve days, will lay twelve eggs. That's the immediate reasoning, right? It's very easy to fall into that trap, and miss the fact that two of the variables were changed.
- pajko 3y agoAnother tricky question is the one about 3 doors, two with goats behind them, and the 3rd one is hiding the prize. The host opens door #1 with a goat behind it, the doors #2 and #3 are still closed. Now what's the chance of the door #2 hiding the prize behind it? The intuitive answer is 50%, but it's wrong.
- anamexis 3y agoYou're not fully describing the Monty Hall problem - as you've described it, the answer is indeed 50%. In the Monty Hall problem: 1. You pick one of three doors. 2. The host opens one of the two doors you didn't pick to reveal a goat 3. You are given the opportunity to keep your original choice, or pick the other unopened door In this situation, there is a 1/3 chance that the prize is behind the original door, and a 2/3 chance it's behind the remaining unopened door.
- xiande04 3y agoThat's not how it goes. And, yes, the answer is 50% in this case. The problem relies on first choosing a door (1). The host then opens a door that you did NOT choose (3) and asks if you want to switch your choice to door #2. In this case, the probability that the goat is behind door 2 is not 50%. https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
- mlajtos 3y agoYou described a slightly wrong Monty Hall problem [0]. When I explain solution to this problem, I always just add more doors. 1 million doors, 1 car, 999999 goats. You pick one door, and I show you 999998 goats. Only two doors are left. What is the chance you picked car at the first guess? 0: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
- contravariant 3y agoThe answer is 0 because the prize is behind the third door #3, but I feel you're either missing a couple of details or you're trying to mess with people who know Monty-Hall.
- 3y ago
- mpeg 3y agoI love these kind of logic puzzles, if you're up for losing an evening the clips game [0] is one of my long time favourite ones. It starts easy but puzzles 6 and 7 are tricky to do within the assertion limit [0]: https://md5crypt.github.io/clipsgame/ https://md5crypt.github.io/clipsgame/
- mdonahoe 3y agoI can't even beat the first level.
- hgsgm 3y agoIs because the the puzzle has several facts that are relevant in the real world, but irrelevant in the puzzle, and the solution to the puzzle is a fact that is false in the real world. It's a terrible tutorial example.
- mlajtos 3y agoI also struggle. :D I think `(eats tiger plant)` would be fine, but it isn't.
- mdonahoe 3y agoAh, my smarter friend figured it out. Hint: what is eatable?
- mlajtos 3y agoI passed with `(is tiger eatable)`. I don't get it. :D
- olafalo 3y agoI'm not very good at this but I'll see if I can explain my thought process: (is tiger fed) requires (eats tiger ?food) which means there must exist ?what such that (is ?what eatable) and (is ?what ?food) But in the fact table, nothing is actually eatable! We could have our tiger eat plants by asserting (eats tiger plants) and (is grass eatable), but that's two assertions. Tigers eat animals, and tigers ARE animals, so since we need to make something eatable we might as well do it on the only animals we have: (is tiger eatable) I got level 2 as well but only after checking for hints in the source. I have been thoroughly nerd-sniped.
- simse 3y agoSorry, unrelated to the writing (which I enjoyed), but related to the horizontal scrollbar! If you change w-screen to w-full on both the navbar and footer, it goes away. 100vw -> 100%. Just a tiny enhancement to an otherwise good looking website.
- mlajtos 3y ago> unrelated to the writing (which I enjoyed) Aw, thank you. :) > related to the horizontal scrollbar Thank you very much, I was not aware of this issue. Changed both to `w-full`, hope scrollbar goes away. :) BTW As a pizza aficionado I deeply enjoyed "wood fired pizza". :D
- rcme 3y agoI'm always amazed that people can get so hung up on events that happen in elementary schools. This piece feels like it could be posted to /r/iamverysmart.
- mlajtos 3y ago> I'm always amazed that people can get so hung up on events that happen in elementary schools. This piece feels like it could be posted to /r/iamverysmart. Ages were 10, 15 & ~23 respectively. Yes, the post suffers from selection bias. I will write more about my failures in the future. And as per your request, I went and try post it to /r/iamverysmart, but only screenshot submissions are allowed and screenshot of the whole post wouldn't be very effective.
- munificent 3y agoThere's a certain set of people who by virtue of genetics upbringing or both, feel very strongly: 1. You should do your best to please authority figures. 2. You should do your best to be correct. It's really distressing the first time they run into a clear example where those two deep-seated needs are in conflict. For a lot of gifted teacher's pet kind of kids, this is their Kobayashi Maru moment. I still remember mine too. :-/
- Robotbeat 3y agoSo I know people are probably tired of posts that just plug something into ChatGPT, but I think it might be interesting to see how GPT3.5/ChatGPT might handle it. It got the answer to the hen problem wrong every time. I kept modifying the prompt to let it know it was a tricky problem, to think step by step, to self-critique and adjust, & tried at temperatures of 0, 0.7, and 1 multiple times and after a dozen tries it kept giving either 12, 36 (most common), or 144. (Starting a new session every time.) > This is a tricky arithmetic word problem. Think step by step. After having an initial answer, critique it, double check it, and change it to the correct answer if necessary: 3 hens lay 3 eggs in 3 days. How many eggs do 12 hens give in 12 days? Again, show your work and don’t just give the answer right away but reason through the problem. That is, UNTIL I gave it the specific advice that the article gave, adding this to the beginning of the above prompt (with another fresh session): >I know that humans are biased for intuitive understanding rather than logical one. Intuitions are shortcuts that are very valuable, but sometimes they are just plainly wrong. …and then it got the right answer immediately, first try. So this trick isn’t just a neat blog post, but empirically useful, and I’ll probably include it in prompts from now on for these kinds of problems… (And note this is a blog post from today, so the blog post was not in the training set.)
- mlajtos 3y agoI love your comment. I tried GPT-4 via ChatGPT and it nailed it without any prompt engineering. --- Prompt: 3 hens lay 3 eggs in 3 days. How many eggs do 12 hens give in 12 days? GPT-4: Let's first determine how many eggs one hen can lay in 12 days. We know that 3 hens can lay 3 eggs in 3 days. This means that each hen lays 1 egg in 3 days (3 hens / 3 eggs = 1 hen / 1 egg). To find out how many eggs one hen can lay in 12 days, we need to multiply the number of eggs laid in 3 days by 4 (since 12 days is 4 times longer than 3 days). So, one hen will lay 1 egg * 4 = 4 eggs in 12 days. Now, we have 12 hens. To find out how many eggs they can lay in 12 days, we just multiply the number of eggs one hen can lay in 12 days by the number of hens: 12 hens * 4 eggs per hen = 48 eggs in 12 days. --- To be completely fair, this math problem can be googled, so it is probably in the training set.
- gorpy7 3y ago
- atleastoptimal 3y agoThis entire post is "I am le smart, so smart that other people think I'm wrong" I think it would come across better if the author mixed times where their non-traditional intuition gave them the wrong answer.
- eyelidlessness 3y agoI don’t think that’s a remotely fair characterization of the author’s self-evaluation. A fair bit of the article is spent expressing embarrassment, presuming they were in fact wrong, and surprise at discovering that presumption was their error. What you’re describing would be an entirely different post, likely making a very different point. There’s value in exploring “sometimes your idiosyncratic intuitions turn out to be right”, without muddying that with the obvious intuition that this won’t always be the case. There’s value in exploring the idea through demonstrative anecdote without counter examples. If anything, counter examples might influence the kind of pattern matching biases which lead to such faulty intuitions. Anyway, I think the author has demonstrated that they’re fallible and happy to be corrected. They’ve acknowledged more concrete problems with the post and made changes to address them at least a couple times.
- mlajtos 3y ago> fallible and happy to be corrected Thank you, that is a very accurate description of how I operate. If you don't mind, I'll steal this phrase from you. :)
- eyelidlessness 3y agoJust saw this now but I absolutely don’t mind.
- mlajtos 3y agoThank you :)
- hgsgm 3y ago
- firstlink 3y ago> The teacher was perplexed of why I said such wrong answer to such simple question. This is extremely toxic on many levels. A teacher who doesn't know their subject matter or at least can't be bothered to refresh before a lesson. A teacher who submits to social pressure from their students when determining the answer. A teacher who shames students for being wrong. A teacher who shames students for being right! The students who were wrong initially will, thanks to this teacher, have completely failed to learn the lesson the problem was teaching. They will remember only that they got the problem right (the teacher said so) and only due to some technicality in the answer book were they subjected to an embarrassing "lesson" from the smart kid. Yes, this is making a mountain of a molehill, but molehills like this can have mountainous impacts on society.
- tshaddox 3y agoI mean, there’s a lot going wrong with this “lesson.” The exercise is pretty clearly designed to trick people, which is very counterproductive if the goal is to teach people about multiplication or fractions or whatever. It’s really not that damning that a teacher would be fooled into skipping the actual work, since that deceit is clearly the entire point of choosing those numbers. It’s a bit like if a proofreading exercise in an English class deliberately did that trick where you put two “the”s next to each other but across a line break. That will often fool a skilled English writer, but that says nothing about the writer’s skill.
- rcoveson 3y agoWhile the teacher in the story obviously bungled it, I disagree with your interpretation of the lesson. The point is not to "teach people about multiplication or fractions or whatever". The point is to teach them to recognize dimensional parameters in a natural scenario. This can be counter-intuitive; we have a bias towards linear relationships. Somebody twice as tall is twice as heavy, thinks the child. Tricks ("surprises" is a less negative descriptor) are important because they wake your brain up. Most people don't like learning arbitrary things for their own sake, but they don't like being fooled either. Describe the base rate fallacy to a room of college students and it might stick for the rest of the semester, but give them this problem[0] and tell them how many licensed doctors it tricked (after you let them answer incorrectly, of course) and they might remember it for a while longer. 0. http://pi.math.cornell.edu/~mec/2008-2009/TianyiZheng/Bayes.html http://pi.math.cornell.edu/~mec/2008-2009/TianyiZheng/Bayes....
- deepsun 3y ago> That is literaly measuring car's speed at every instant! Well, there's no way to measure to measure speed instantly, it's always over some period even if small (take some time interval and see how much distance have been covered). For example, at time zero you cannot know what speed the car has. Similar like when I was working with audio, producing frequency diagram -- the frequency cannot be gathered from a single timestamp on the waveform, you always need some interval, like 20ms to get an FT.
- mlajtos 3y agoSpeedometer should be enough. :)
- denton-scratch 3y agoI didn't get why he deprecated distance/time as a direct way of determining average velocity. Is this pedantry about "average" vs. "mean"? As I understand it, "average" is a vague word, one of whose meanings is "mean". And I imagine most of the other meanings would have distance/time as their value as well.