In two words, Im-possible: The problem with counterintuition

Nearly 40 years ago, we were asked this question at school:

Imagine a string tied around the middle of an orange, in effect forming a circumference. Now imagine another string, this time tied around the middle of the earth, at the equator.Okay? Now increase the length of each of these strings by a foot. Imagine each string now suspended around its sphere as an annulus. Tell me, which string will be further away from the sphere it contains?

And we all answered “the one around the orange, of course”. Or words to that effect. And we wondered why someone would ask such a silly question.

And then we did the math.

  • C=2pi r
  • C+1=2pi R
  • R-r=(C+1)/2pi -C/2pi
  • Or (cancelling out the Cs), R-r=1/2pi

What?!?! How can this be? How can the change in radius be independent of the circumference of the sphere (or for that matter the radius)? You mean that both strings will be the same distance away from “their” sphere? Im-possible.

It didn’t matter how many times we invoked Sam Goldwyn (he was still alive at the time), the answer did not change. No hidden tricks. No small print. No scams involving oranges and geoids. Just the facts. When you increase circumference by X, the radius increases by X/2pi. Regardless of what the original radius was. Regardless of what the original sphere was. One string round a table tennis ball, the other round the sun, same answer.

I tell you, it kept me up nights as a boy, it just didn’t make any sense to me. I had to drill the answer into my head, drag it there kicking and screaming. It took time, but the pain subsided in the end.

And then.

And then I bought two fascinating books by Julian Havil: Nonplussed and Impossible. Books that were tailormade to fit in to that odd space in my library, between Martin Gardner and John Allen Paulos.

And went through all that pain again. From “does not compute” to “im-possible” to “I don’t believe it”. So if you’ve got a similar penchant for mind-masochism, go out and buy the books. Both of them. You won’t regret it.

I need to keep challenging my biases and prejudices, the anchors and frames I cannot see. And books like these help me exercise my mind, they ensure that I don’t reject ideas just because they’re counterintuitive.

12 thoughts on “In two words, Im-possible: The problem with counterintuition”

  1. Hey JP, good to know another fan of the counter-intuitive. I’m guessing you share my view that the Read/Write Web brings a whole host of tipping points that lead us into counter-intuitive design parameters and that understanding them can bring considerable advantage.

    Your little math exercise looks similar to the scribble in my notebook after reading p25 of Shirky’s ‘here comes everybody’ in which he talks about the birthday ‘paradox’. When you have n people in a room there are n*(n-1)/2 pairs – a number that increases more quickly than the size of the group. One of those counter-intuitive factors that is handy in considering mesh vs hub/spoke designs for things.

    Thanks for the pointer to nonplussed, time for some masochistic brain massage ;-)

    Fang – Mike Seyfang

  2. Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”
    “I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.

    Who was right, Alice or the Red Queen? Alice of course! But then do the thinking: it turns out both were equally correct.

  3. I don’t understand why would you think the orange is further away? I tried to logically think it out and it still came out same…no maths involved.

    and yes it was independent of the circumference

  4. Mike, the 23 person birthday thing gets covered as well in one of the books, but (like Bev says for this one) I didn’t find it counterintuitive.

    That’s why I think of intuition as influenced by personal prejudices and biases, anchors and frames. We don’t all find the same things intuitive… or counterintuitive… which makes for some interesting arguments.

  5. Intuition is the output of the unconscious mind. The process of really learning something means transferring the knowledge about that thing from the conscious mind to the unconscious mind – we are only really expert in something (be it speaking a language, playing chess, doing integral calculus, playing a musical instrument, doing ballet, writing software, making management decisions…) when we can do it with the unconscious mind. Getting to expert level in something takes about 2-3 years. Getting really good at something (becoming a master) takes at least 5 years. When the unconscious mind knows something it is smarter and faster than the conscious mind, but when it doesn’t know something it is far stupider. So part of being smart is knowing whether to trust your intuition in something (that is knowing whether your unconscious mind is an expert or better in that area). One of the problems of intuitive decisions is that we don’t immediately know why we reached a particular conclusion, but often it is important to know why (to convince other people, to explain to others so they can learn, or to double-check the decision), so another part of being smart is understanding why the unconscious mind made a particular decision and bringing that knowledge into the conscious mind.

    It takes a long time to bring knowledge into the unconscious mind, but it also takes a long time to remove incorrect knowledge. So it’s very important to guard what goes in. At a simple level this means avoiding bad habits when we learn a new musical instrument or a new sport. But it also means actively challenging the prejudices, perceived wisdom and practices of those around us. In particular we need to guard against insidious cultural norms – if the norm is of hopelessness or powerlessness or uncaring or disrespectful of particular groups of people then there is risk of those norms entering our minds, unless we actively keep them out.

    Our prejudices live in the unconscious mind. We want to increase the number of correct prejudices and decrease the number of incorrect ones. That is we want to hone our intuition. So we need to keep challenging the unconscious. We can only do this in the area where the conscious interacts with the subconscious. We need to challenge both what is obviously right and what is obviously wrong. We shouldn’t reject ideas just because they are counterintuitive, but equally we shouldn’t accept ideas because they are obviously correct.

  6. Hi JP!

    We did this one at school too. I still have a stack of ~30 year old Martin Gardner books…

    It helps to look at this one the other way round. If you increase the radius r of a circle by a constant c, how much does the circumference grow?

    It grows by 2 pi (r+c) – 2 pi r = 2 pi c which is a constant independent of r.

    In particular, to, uh, close the loop back to your example, if you increase the radius by c = 1/2pi, the circumference always grows by 1.


  7. Thanks a lot ! I am quite a fan of the classic probability counter-intuitive examples (there are so many) but this one is so simple … and so effective. Thanks for the book pointers, I just bought them. I assume that you are familiar with “Fooled by randomness”, by N. Taleb, but if you aren’t you would probably like it :)
    — Yves
    PS: two random bits of thoughts about this:
    (a) one reason it seems counter-intuitive is that if you move from a math to a physics problem adding weight, tension and friction into account, the answser is different
    (b) my own math training makes me think at it this way, the additional effect of adding a foot of rope, * for a given length of a circular rope section * is inversely proportional to the curvatue (it costs nothing with a flat rope). Hence the “two effects compensate themselves” …

  8. I think one thing is being missed here. The reason we are suprised is that there seems to be a conflict between our initial intuition and the analytical answer. In fact, there isnt any such conflict. Our intuition says that if we add a foot to the circumference of the earth and add a foot to the circumferance of an orange, the piece of string that was round the orange MUST be further away from the surface of the orange than the string that was round the earth is from the surface of the earth. Of course they arent – the distances are the same – obviously. BUT the relative distances are not the same! The string that was round the earth is now at a relative distance of (1/2pi)/R where R is the radius of the earth. The string that was round the orange is now at a relative distance of (1/2pi)/r, r being the radius of the orange. It is clear that string that was round the orange is now many times further away (in terms of ‘r’) than the string that was around the earth.

  9. slight correction – I meant to say that (1/2pi)/R is the change relative to the radius of the earth.

  10. hi
    just to add what i think
    C = 2pi r
    since 2pi is a constant, then the graph C vs r will be linear
    in other words, constant slope, rate of change, dr/dC is a constant, or however you want to put it….

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