Trying to be Smart: on bringing up extremely rare exceptions to prove forests don’t exist, only trees

When my kids don’t want to do their work (typically word problems in math,) they start coming up with all kinds of crazy scenarios to try to evade the question. “What if Susan cloned herself?” “What if Joe is actually the one driving the car, and he only saw the car pass by because he was looking at himself in a mirror?” “What if John used a wormhole to travel backwards in time and so all of the people at the table were actually Joe and so I only need to divide by one?” “What if Susan is actually a boy but her parents accidentally gave him the wrong name?” “What if ALIENS?”

After banging my head on the wall, I started asking, “Which is more likely: Sally and Susan are two different people, or Sally cloned herself, something no human has ever done before in the 300,000 years of homo Sapiens’ existence?” And sometimes they will, grudgingly, admit that their scenarios are slightly less likely than the assumptions the book is making.*

I forgive my kids, because they’re children. When adults do the same thing, I am much less sympathetic.

Folks on all sides of the political spectrum are probably guilty of this, but my inclinations/bubble lead me to encounter certain ones more often. Sex/gender is a huge one (even I have been led astray by sophistry on this subject, for which I apologize.)

Over in biology, sex is simply defined: Females produce large gametes. Males produce small gametes. It doesn’t matter how gametes are produced. It doesn’t matter what determines male or femaleness. All that matters is gamete size. There is no such thing (at least in humans) as a sex “spectrum”: reproduction requires one small gamete and one large gamete. Medium-sized gametes are not part of the process.

About 99.9% of people fit into the biological categories of “male” and “female.” An extremely small minority (<1%) have rare biological issues that interfere with gamete formation–people with Klinefelter’s, for example, are genetically XXY instead of XX or XY. People with Klinefelter’s are also infertile–unlike large gametes and small gametes, XXY isn’t part of a biological reproduction strategy. Like trisomy 21, it’s just an unfortunate accident in cell division.

In a mysterious twist, the vast majority of people have a “gender” identity that matches their biological sex. Even female athletes–women who excel at a stereotypically and highly masculine field–tend to identify as “women,” not men. Even male fashion designers tend to self-identify as men. There are a few people who identify as transgender, but in my personal experience, most of them are actually intersex in some way (eg, a woman who has autism, a condition characterized as “extreme male brain,” may legitimately feel like she thinks more like a guy than a girl.) Again, this is an extremely small percent of the population. For 99% of people you meet, normal gender assumptions apply.

So jumping into a conversation about “men” and “women” with “Well actually, ‘men’ and ‘women’ are just social constructs and gender is actually a spectrum and there are many different valid gender expressions–” is a great big NO.

Jumping into a discussion of women’s issues (like childbirth) with “Actually, men can give birth, too,” or the Women’s March with “Pussyhats are transphobic because some women have penises; vaginas don’t define what it means to be female,” is an even bigger NO, and I’m not even a fan of pussyhats.

Only biological females can give birth. That’s how the species works. When it comes to biology, leave things that you admit aren’t biology at the door. If a transgender man with a uterus gives birth to a child, he is still a biological female and we don’t need to confuse things by implying that someone gestated a fetus in his testicles. Over the millennia that humans have existed, a handful of people with some form of biological chimerism (basically, an internalized conjoined twin who never fully developed but ended up contributing an organ or two) who thought of themselves as male may have nonetheless given birth. These cases are so rare that you will probably never meet someone with them in your entire life.

Having lost a leg due to an accident (or 4 legs, due to being a pair of conjoined twins,) does not make “number of legs in humans” a spectrum ranging from 0-4. Humans have 2 legs; a few people have unfortunate accidents. Saying so doesn’t imply that people with 0 legs are somehow less human. They just had an accident.

In a conversation I read recently, Person A asserted that if two blue-eyed parents had a brown-eyed baby, the mother would be suspected of infidelity. A whole bunch of people immediately jumped on Person A, claiming he was scientifically ignorant and hadn’t paid attention in school–sadly, these overconfident people are actually the ones who don’t understand genetics, because blue eyes are recessive and thus two blue eyed people can’t make a brown-eyed biological child.  A few people, however, asserted that Person A was scientifically illiterate because there is an extremely rare brown-eyed gene that two blue-eyed people can carry, resulting in a brown-eyed child.

But this is not scientific illiteracy. The recessive brown-eyed gene is extremely rare, and both parents would have to have it. Infidelity, by contrast, is much more common. It’s not that common, but it’s more common than two parent both having recessive brown-eyed genes. Insisting that Person A is scientifically illiterate because of an extremely rare exception to the rule is ignoring statistics–statistically, the child is more likely to be not biological than to have an extremely rare variant. Statistically, men and women are far more likely to match in gender and sex than to not.

Let’s look at immigration, another topic near and dear to everyone’s hearts. After Trump’s comments about Haiti came out (and let’s be honest, Haiti’s capital, Port au Prince, is one of the world’s largest cities without a functioning sewer system, so “shithole” is actually true,) people began popping up with statements like “I’d rather a Ugandan immigrant who believes in American values than a socialist Norwegian.”

I, too, would rather a Ugandan with American values than a socialist Norwegian. However, what percentage of Ugandans actually have American values? Just a wild guess, but I suspect most Ugandans have Ugandan values. Most Ugandans probably think Ugandan culture is pretty nice and that Ugandan norms and values are the right ones to have, otherwise they wouldn’t have different values and we’d call those Ugandan values.

Updated values chart!

While we’re at it, I suspect most Chinese people have Chinese values, most Australians have Australian values, most Brazilians hold Brazilian values, and most people from Vatican City have Catholic values.

I don’t support blindly taking people from any country, because some people are violent criminals just trying to escape conviction. But some countries are clearly closer to each other, culturally, than others, and thus have a larger pool of people who hold each other’s values.

(Even when people hold very different values, some values conflict more than others.)

To be clear: I’ve been picking on one side, but I’m sure both sides do this.

What’s the point? None of this is very complicated. Most people can figure out if a person they have just met is male or female instantly and without fail. It takes a very smart person to get confused by a few extremely rare exceptions into thinking that the broad categories don’t functionally exist.

Sometimes this obfuscation is compulsive–the person just wants to show how smart they are, or maybe everyone around them is saying it so they start repeating it–but since most people seem capable of understanding probabilities in everyday life (“Sometimes the stoplight is glitched but usually it isn’t, so I’ll assume the stoplight is functioning properly and obey it,”) if someone suddenly seems incapable of distinguishing between extremely rare and extremely common events in the political realm, then they are doing so on purpose or suffering severe cognitive dissonance.

 

*Oddly, I solved the problem by giving the kids harder problems. It appears that when their brains are actively engaged with trying to solve the problem, they don’t have time/energy left to come up with alternatives. When the material is too easy (or, perhaps, way too hard) they start trying to get creative to make things more interesting.

 

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21 thoughts on “Trying to be Smart: on bringing up extremely rare exceptions to prove forests don’t exist, only trees

  1. Couple reactions.

    1. You’ve previously floated the idea that “liberals” are less into abstractions than “conservatives.” Can’t comment on “all liberals,” they’re a coalition of different groups after all–but when it comes to “blue tribers” I’m convinced this is the opposite of the truth. Blue tribers abstract and generalize everything. They *over*generalize. They abstract away nitty-gritty reality. They know this–have been burned by it in the past (see: communism). That’s why they’re so on guard against it. All debates are bravery debates.

    >But this is not scientific illiteracy. The recessive brown-eyed gene is extremely rare, and both parents would have to have it. Infidelity, by contrast, is much more common. It’s not that common, but it’s more common than two parent both having recessive brown-eyed genes. Insisting that Person A is scientifically illiterate because of an extremely rare exception to the rule is ignoring statistics–statistically, the child is more likely to be not biological than to have an extremely rare variant.

    I wouldn’t have accused A of scientific illiteracy, but indeed my mind did go straight to worrying about the poor person who really *wasn’t* unfaithful, but gets put through hell because no one even considered the possibility of that rare gene–or did consider it but decided they didn’t need to bother to *actually* check for it because statistics. See also, the woman who had her kids taken away because tests said they “genetically weren’t hers”–turned out she was a chimera. I mean, *she existed*. See also the Law of Very Large Numbers, fellow atheist. ;)

    This tendency to handwave away the rare exceptions, because they’re rare; to conflate “rare” with “nonexistent”; to *set up systems that make no provision for such things and thus crush actual people*–this is the flaw that *blue tribers* tend to have. And know they have (they might just think “people” have it). And be on guard against.

    >It takes a very smart person to get confused by a few extremely rare exceptions into thinking that the broad categories don’t functionally exist. Sometimes this obfuscation is compulsive–the person just wants to show how smart they are, or maybe everyone around them is saying it so they start repeating it

    It seems like you’re saying that in your experience, “very smart people” are especially likely to have this focus, but you can only guess as to why? If so, I’d bet it’s because being “very smart” is rare and causes unusual educational needs–so they’re more likely to have a *personal experience* of being a “rare exception” whose existence was handwaved away to their harm. Similarly, many second-wave feminists were civil rights activists first (even those who were white), and later said this was because they’d felt stereotyped and pre-judged as women, which made them want to fight stereotyping and prejudice in general.

    Is this another example of *over*generalization, do you think? Hey, just how much generalization is appropriate in life, anyway? ISTM this has traditionally been a major front in the “hyper-Protestant” vs. uh “Not” cultural war. (See also the “lizard people” who want abstract legibility, not metis…) …maybe I’ll leave a comment on your heuristics/”malevolent AIs” post going into more detail about this (wanted to before, didn’t have the time).

    2. Most of the time, when a conversation goes:

    A: “The forest is like this.”
    B: “Here’s an exception”–

    –most of the time, B’s goal is not “to prove…only trees [exist].” Instead, B is reacting to what they see as the subtext of A’s statement. They aren’t arguing with A’s literal statement, they are arguing with “what A obviously means by it.”

    Some people would like us all to just be literal–then we could dismiss the whole issue. But…well, Grice explained implicature: Leaving half of your point in the implication is *normal*. It’s to be expected. In that environment, inferring part of people’s meaning is only sensible.

    Freddie deBoer talked about this wrt HBD in a post called “The Right to Live in History” on his old blog, which has since been taken down. IIRC he defended people who interpret “strong emphasis on IQ tests” as a sign of racism–similarly, California’s Larry P. v. Riles ruling was roughly, “For black kids, IQ tests are just irremediably poisoned, just stop using them.” And for the reasons above, he had a point. It’s just that there were additional effects of this behavior that he (and the Riles court) hadn’t considered. Such as those brought up in Crawford v. Honig (black kids with suspected learning disabilities unable to get them evaluated because of California’s prohibition on using IQ tests for black kids).

    I don’t think this issue can be solved with “let’s all just be literal and leave information on the table.” It will only be solved with “here’s even more information.”

    So…maybe that clarifies my question from my other comment. Bravery debates: Someone’s from a community that needs to be reminded exceptions exist; they get into the habit of issuing constant reminders. They unknowingly drift into a community that needs to be reminded general tendencies exist. There, their constant reminders that exceptions exist have the effect of…

    Of? What happens?

    What even is “a community that needs to be reminded general tendencies exist”? How can you recognize it? What problems does it have?

    The one problem I’ve actually seen is the one I mentioned in my earlier comment: “All groups have the same means and distributions of $trait, therefore if groups are not present in placements that rely on $trait in the same percentages in which they appear in the general population, there must be something wrong.”

    Are there others? What are they?

    I mean, you said it: “None of this is very complicated. Most people can figure out if a person they have just met is male or female instantly and without fail.” Most people don’t seem to have any problem of undergeneralization. It’s *over*generalization that causes problems in everyday life. At least in my experience…which is why I’m so on guard against it…oh but bravery debates…

    *Is* undergeneralization actually a problem anywhere? What does that look like?

    >Oddly, I solved the problem by giving the kids harder problems.

    …when I read that anecdote, I literally typed, “Dude, give them harder problems.” I then deleted it because it was rude and off-topic and anyway I hadn’t finished reading the piece. But yeah: Not “oddly” at all. :)

    Like

    • Just to make sure I’ve gotten them all, is this your last major comment recently that I haven’t responded to, yet?

      Unfortunately, words like “liberal” and “conservative” feel increasingly useless. What was liberal back in 1990 doesn’t seem the same as “liberal” today. Unfortunately, I don’t have good words, yet, to replace these.

      This habit of not seeing the forest for the trees is probably something both sides do–conservatives probably do it to libs on the subject of gun control, where cons use details about specific kinds of guns to refute lib arguments that are aren’t really about specific kinds of guns. Like, when people say, “We need to ban assault rifles,” what they really mean is, “Let’s prevent people from getting their hands on big scary guns that can easily kill lots of people,” but cons tend to respond with details about gun varieties.

      So it may be less about one side or the other, and more about trying to win an argument through confusion.

      But if we’re generous, maybe people aren’t *trying* to win via confusion; maybe being really interested in a subject leads people to know lots about it and knowing lots about it can cause them to overrate the importance of specific details. Perhaps people struggle with probability, and so overrate the likelihood of rare events. For example, I people consistently overestimate the size of groups like trans people, gay people, black people, and immigrants. In order to remember things, our brains probably have to devote about equal bits of brain power to each item we’re remembering, so different categories of things with wildly different sizes end up with roughly equal mental markers, (speaking a little metaphorically about brains.) Remembering how big each category is requires attaching a second piece of data to it, which is just more to remember so that’s harder. So Ironically, as people become more informed about a subject, they may lose a certain sense of perspective.

      Since liberalism and academia are well-correlated, libs might fall pretty reliably into this category of “knows a lot about a subject but doesn’t have good size tags attached to it.”

      Certainly if a real human were involved in the conversation about eye colors, I would object that perfectly reasonable exceptions (such as adoption) exist that could cause a child to have unusual eye colors. But I wouldn’t call someone scientifically illiterate for a Twitter comment about the most likely cause being infidelity. (Disclaimer: I think my parents once got in a fight because supposedly my eyes don’t match their eyes and my mom still claims my eyes don’t match but the weird thing is my eyes do match… somehow my parents just don’t know my eye color. IDK it’s weird.)

      I’m not convinced that internet debates are really a productive way to disseminate the idea that “people like this exist.” Often on the internet the most click-baitey arguments get rewarded, leading to a screaming match between two of the worst representatives of both sides.

      What even is “a community that needs to be reminded general tendencies exist”? How can you recognize it? What problems does it have?
      Communists ;)
      The one you’ve already identified is probably the biggest, but distributed around different issues. Feminists are opposed to the idea that women could have any natural distribution of strengths and aptitudes that could result in doing worse than men in any high-status occupations (no one cares about ladies’ skills as garbage collectors,) eg.

      Policies about everything from education (folks are convinced “the schools are failing” because data about black test scores isn’t disaggregated from general scores) to housing/loans, from job hiring to criminal justice are effected by this conviction that outcomes ought to be evenly distributed.

      I remember a conversation with some libs who swore blacks don’t make worse SAT scores, an average, than whites. (I’d like to know why they think affirmative action exists if not to make up for the test score gap.) The conversation quickly degenerated as they became extremely angry.

      So Affirmative Actin exists because blacks have worse test scores than whites, but we’re not allowed to say so because saying blacks have worse test scores than whites is racist.

      And then there’s the class of people who know X, but politely don’t say X, and so you get caught in this whole trap of “are you clever enough to know the latest thing you’re supposed to claim to believe but also clever enough not to actually believe it?” Take whites who claim to be anti-racist but also live in majority white neighborhoods because “they’re safe” or “for the schools” or some other factor that correlates strongly with whiteness.

      I agree, though, that there are a LOT of bravery debates.

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      • I think a lot of people actually don’t realize that, say, living a place “for the schools” has such a big racial component. If you’ve grown up in a lot of places, it’s quite possible the only blacks you’ve encountered are either on TV or middle class (native or immigrant) and the closest you see to poor people are white trash or proles, because those are the neighborhoods you’ve been allowed to go to. I think there is a strong enough taboo, then, that it can be easy to keep up the illusion. Plus, so many people are terrible at statistical risks, anyway. Car crashes vs kidnapping as a risk for kids is probably an obvious example. (I’ll try not to get on a rant about people who are against strict border controls and then fret about sex trafficking…)

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      • >Just to make sure I’ve gotten them all, is this your last major comment recently that I haven’t responded to, yet?

        Right after you asked that, I left one here. But please don’t feel obligated to answer every comment.

        >Like, when people say, “We need to ban assault rifles,” what they really mean is, “Let’s prevent people from getting their hands on big scary guns that can easily kill lots of people,” but cons tend to respond with details about gun varieties.

        I thought they did that because they wanted to get clear exactly how the proposed ban would work. I don’t know a lot about guns, but I know enough to know that if you tried to literally write a law saying, “We ban assault rifles,” it…wouldn’t make any sense and the implementation would be a giant mess. I thought pointing that out, and trying to hash out what a likely law would actually do, was the reason for the focus on details. Trying to *clear up* confusion, not spread it.

        >Perhaps people struggle with probability, and so overrate the likelihood of rare events.

        Some people have a principle about not crushing minorities just because they’re minorities, even extreme minorities. “The Ones Who Walk Away From Omelas.” J. S. Mill. Etc. Anyway, people like this wouldn’t need to overrate the likelihood of rare events to care about them.

        More generally, your proposed mechanism seems logical, but I’m not sure it’s necessary. I think this is an example of the cognitive bias in favor of cognitively available events, such as those that happened to them or that they hear a lot about on the news. I think the reason people evolved to focus on cognitively available events is so they don’t have to estimate probabilities at all. The focus doesn’t cause them to overestimate probabilities and then decide the issue is more important than it is, the focus just causes them to…decide it’s important since it’s cognitively available. The “gut feeling” that it’s important since it’s cognitively available, then also puts its oar in when they are asked to estimate probabilities, leading to…overestimates of the size of groups like trans people, gay people, black people, and immigrants.

        I don’t think this is necessarily a bad thing. If everyone’s a crusader on an issue they care about because it affects them personally, the issues that affect most people will be addressed. In a democracy they will, anyway.

        >I remember a conversation with some libs who swore blacks don’t make worse SAT scores, an average, than whites. (I’d like to know why they think affirmative action exists if not to make up for the test score gap.) The conversation quickly degenerated as they became extremely angry.

        My boomer teachers knew about the test score gap, therefore they couldn’t accept that the tests meant anything. Millennials I meet tend to accept that the test scores mean something, therefore they can’t accept the test score gap.

        >I’d like to know why they think affirmative action exists if not to make up for the test score gap.

        The people who don’t know about the test score gap think affirmative action exists to make up for other (“real”) areas of bias. Like the same “biased” teachers who discipline black kids more often and severely.

        :sigh: Remember when affirmative action existed to create diversity in the ruling class so that its culture would evolve to be more welcoming to minorities?

        >Take whites who claim to be anti-racist but also live in majority white neighborhoods because “they’re safe” or “for the schools” or some other factor that correlates strongly with whiteness.

        I’m pretty sure the things they say they care about are the things they actually care about, and they wish those things didn’t correlate so much with whiteness so they could have more diversity too. If they even allow themselves conscious awareness of the correlation at all.

        It’s a mess.

        Like

      • BTW, your link showed up fine on my end.
        A couple of further thoughts related to our other continuing conversations: 1. Your insight into the Puritan mindset was recently useful to me in understanding a conversation with a Puritan relative (although, unfortunately, only in retrospect.) We were discussing school shootings, in which I argued that perhaps we would have fewer of them if we allowed students who don’t like each other more options to avoid each other, and he argued that this would only cause the problem to replicate in new environments (conversation summarized and translated). If there is a certain bullying dynamic that produces shooters, then trying to separate future shooters from bullies will fail, they argued, because the students will just repeat the dynamic in their new environments, producing the same results.

        By contrast, my experience in school was that different schools had radically different students. Some I got along with, despite my myriad of flaws. Some I didn’t get along with. And some were atrociously bad. This wasn’t just “I have to learn how to get along with people and socialize;” this was “I can’t do this.” I really don’t think he understood my point that different groups of kids actually interacted very differently.

        It was a frustrating conversation. I have opinions about people who downplay the effects of bullying or give bullshit excuses for it like “oh they just have low self-esteem.”

        Back to home management as a “job”. I do think there is some level on which I object to characterizing it as a “job” but not “work.” In my experience, most people (especially most married people) acknowledge that housework and the like is actual work. I don’t think most people think that Ma Ingalls didn’t work extremely hard, for example.
        I have occasionally encountered people who think “housework isn’t a real job.” IME, they’re nasty people. From an economics perspective, it’s work and we can calculate its economic value by looking at what it would cost to hire someone to do the same job. People fall into this weird mental trap where they think that if they hire someone to wash the dishes, it’s “economic work” whereas if they do it themselves it’s not, but the end product (Washed dishes) is obviously the same and of equal value either way.

        But by the same token, there’s something about how I think about “jobs” and “housework” that’s different, and I think it’s that I like my house better than my job. Lots of people work jobs they hate just because they have to; we need money to buy food, after all. Jobs are often a bit adversarial: the employer wants to get as much labor out of the employees for the least pay, while the employees want the opposite. The employee doesn’t directly benefit from a job well-done except inasmuch as they stay employed. By contrast, as a person who lives in a house, I benefit directly from housework well-done. Anything I do around the house is useful and makes my life better (whether I do it or not.)

        My parents, I confess, often fought about housework, and as a result it didn’t get done, and as a result we were all kind of unhappy about the state of the house. Which doesn’t mean that “therefore trying to share the housework is bad; housework is the woman’s job,” it means that I think my mother’s attitude/way of handling things (she was really the dominant person in the household) wasn’t the best. “There has to be a better way,” says I.

        As I see it, almost every animal has a home or shelter of some sort. Even a coral polyp builds a house. A bear has its den and a spider spins a web. Birds and chimps build nests. Having a home and keeping it nice is part of being human. It’s work–even a spider does work to keep its web nice. By contrast, a “job” is just something you’re paid to do. If you don’t get paid, you stop working. If you don’t like it, you can (hopefully) search for a new job. By contrast, a spider can’t just say “no more web; I’m going to go hunt prey in a different way.” A job is external; a creature maintaining its habitat is intrinsic.

        This is largely a matter of attitude–whether you see your activities as basically benefiting yourself or whether you see them as benefiting someone else.

        If two people each have their own household, then obviously they each do their own housework; men naturally take care of their own laundry and schedules when they live alone. Living together reduces each person’s workload, though it can lead to mooching. (Of course having kids greatly increases the workload, though not as much as raising kids on your own.) So I guess my overall attitude is “taking care of the home is a natural part of being human and people should find an arrangement that suits them,” rather than “Your lack of emptying the dishwasher proves that you default to thinking about women in oppressive, patriarchal terms and don’t see me as a human being.”

        Back on topic: Guns
        I agree that if one side is making vague and inaccurate statements about what kind of gun control they want in order to convey the notion “we want more gun control” then it makes sense for the other side to point out “if we passed the law you are advocating for it wouldn’t actually work.” However, It seems like both sides have this kind of rehearsed script where they get bogged down yelling “ban this” and the other side goes “that’s not even the kind of gun you actually want to ban” and no one addresses the real point, “Let’s keep dangerous weapons out of the hands of dangerous people.”

        “I think the reason people evolved to focus on cognitively available events is so they don’t have to estimate probabilities at all. “
        Certainly; and in small communities, cognitively available events were probably things happening in your community and so things you really should worry about, whereas now we can watch the news and hear about things happening on the other side of the planet.

        See: the spread of newspapers and the spread of the witchhunt hysteria.

        “Remember when affirmative action existed to create diversity in the ruling class so that its culture would evolve to be more welcoming to minorities?”
        Yes.

        “I’m pretty sure the things they say they care about are the things they actually care about, and they wish those things didn’t correlate so much with whiteness so they could have more diversity too. “
        Yeah, I get that. I’d like that, too. It’d be much more convenient and peaceful. But there’s a kind of willful not living in reality to it combined with some nasty classism. Hypocrisy. You know what I mean.

        Yeah, it is.
        Take care.

        Like

      • …links to this blog don’t show up.

        Anyway, I left a comment on “Do Sufficiently Large Organizations Start Acting Like Malevolent AIs? (pt 1)”

        Like

  2. Here’s a math problem I thought up recently. You want to skate on a frozen lake covered with a uniform layer of snow. You stake out a circle 20 feet in diameter and hire John Henry, whose only tool is a snow shovel, to clear it for you. He finishes the job in one hour.

    Now I stake out a circle 40 feet in diameter, 100 feet away. If John Henry never gets tired, so his power output is a constant, how long will it take him to clear my circle?

    Hint: If you’ve actually shoveled snow by hand, you’re more likely to give the correct answer.

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      • No, I just want to be clear that he’s starting a new circle, not expanding the circle he shoveled for you.

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      • John Henry may clear the snow any way he chooses, but let’s presume he’s smart, and always uses the most energy-efficient method. Your question shows you’re on the right track.

        Throw a ball 20 feet, then throw it 40 feet. The second throw requires a sqrt(2) increase in velocity, or double the energy. If air resistance is significant, the second throw requires *more* than double the energy, so you’re better off with two 20-foot throws.

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      • Assuming he deposits each shovelful of snow immediately to his right and works in concentric circles, each circle contains each of the previous circles, but the number of circles depends on the width of his shovel. If I assume his shovel is two feet wide, I get 7 hours, but if I assume it is only 1 foot wide I get 7.45 (repeating) hours.

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      • You’re over-thinking it. He has to move four times as much snow, and move it twice as far, so he must do eight times as much work. Thus the correct answer is eight hours.

        My dad answered “four hours” because he uses a snowblower, which makes throwing distance irrelevant. I shovel my driveway by hand, so I know exactly where its “pole of inaccessibility” is and why it’s the hardest place to clear.

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      • But only the 25% of the snow in the center has to go twice as far. The 75% of the snow within 10 feet of the edge travels the same distance as before.

        You specified a shovel, so I know the snowblower solution is wrong.

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      • You’re treating all distances less than ten feet as being the same, which makes sense only if you have a machine that throws snow exactly ten feet.

        The average distance between a point inside a circle and the outer edge is one-third of the radius of the circle. Thus doubling the radius from 10 feet to 20 feet doubles the average distance that snow must be moved, from 3 feet 4 inches to 6 feet 8 inches.

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      • Let’s assume he has a shovel that’s two feet wide and he dumps the snow immediately to his right as he works (you said nothing about his snow-flinging abilities and you said his power output is constant, so I assume he only makes one action: shoveling. No carrying or flinging.) Whether he starts from the center or the edge doesn’t matter, but the math is easier from the center. At the center, he carves out a circle of area pi2^2. This goes immediately onto the next circle, so even though this circle looks like a ring, it still has pi4^2 area. This goes on the next circle, for pi6^2, and so on, until we reach the final ring, pi10^2. Adding these together, we get: pi(2^2+4^2+6^2+8^2+10^2) = 220pi. This is the total amount of snow he moved in an hour.

        Our next circle is bigger, but we already know the area of the central part, so we can speed things up a bit:
        pi(220+12^2+14^2+16^2+18^2+20^2)= 1540.

        1540/220 (the amount he can do in an hour) gives us 7.

        If he worked from the other direction, the first ring of snow would go right out of the circle–it only needs to move immediately to his right, just like the first ring of the smaller circle. The second ring moves two feet to his right, then out of the circle, just like the second ring on the first circle. The five outer rings on the big circle each have to be shoveled the same distance to the right as the five inner rings on the original circle. Only the 5 inner rings have to move twice as far.

        But this depends on his shovel; if he has a smaller shovel, it’ll take longer.

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      • As your theoretical shovel gets smaller, your answer converges on eight hours. That’s because the time needed to move a snowbank of radius r is proportional to r^2, and integrating this yields r^3. All constants including pi can be disregarded because they are the same in both input cases.

        With a wide shovel, you over-estimate the time needed for the inner-most circles, because your shoveling technique is very inefficient when the radius of the circle is similar to the width of the shovel.

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      • I see now — a thinner shovel, by your method, results in a thinner outer snowbank, which means the entire mass of the circle moves one foot less, a significant energy savings if it were possible. In fact, we must always move snow some extra distance to create a snowbank of non-zero thickness, whose height and width grow with the square root of the radius. Thus the workload grows as something less than the cube of the radius (how much less depends on the thickness and compressibility of the snowpack), converging on cubic for large radii.

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