Sunday, December 19, 2010

A solution to the lawn mowing puzzle

This one's got references and a slightly different style, because it's a paper. Given the length and the subject matter, I thought it'd be more at home on a blog than in a journal, so here it is:

Jeremy Gwiazda (2010) presents a puzzle: can someone mow eight lawns with one mower and never have exactly seven left? Clearly something strange would have to happen but if supertasks are possible then it can be done.

Supertasks are not new. Russell (1936, p.144) gives one example, Hawthorne and Weatherson (2004) give another and there are many more. The term appears to come from J. F. Thomson (1954). Supertasks involve doing an infinite number of tasks in a finite stretch of time. For example, if you want to clap your hands an infinite number of times in an hour then you can clap your hands at 2pm, 2.30, 2.45, 2.52 and thirty seconds and so on, halving the gap between claps each time. You will have clapped your hands infinitely many times by 3pm, but at any time before 3pm you will have only clapped a finite number of times. Of course, no human can clap their hands that quickly. Supertasks are strange, but not logically incoherent.

To solve the lawn mowing puzzle you must schedule the mowing of the first two lawns as two supertasks which finish at the same time, although no part of one supertask is simultaneous with a part of the other. They must finish at the same time because if one finished first then there would be exactly seven lawns left until the second one finished. They cannot have parts at the same time because you only have one mower. This is possible because a supertask need not have a last part. Here is how to do it.

Start at 2pm. Mow half of lawn one in fifteen minutes, and half of lawn two in fifteen minutes, then half of what remains of lawn one in seven and a half minutes, then half of what remains of lawn two in seven and a half minutes, then half of what remains of lawn one in three minutes and forty-five seconds, and so on. This will get both lawns mown by three o’clock, except for the very edges. To solve this problem you can start at 1.58 instead and mow the very edges before the supertasks begin. Then at three o’clock you have a well-earned rest before mowing the other six.

References

-Gwiazda, Jeremy 2010: ‘The lawn mowing puzzle’, Philosophia, vol. 38, no. 3, p.629

-Hawthorne, John and Weatherson, Brian 2004: ‘Chopping up gunk’, The Monist 87, pp.339-50

-Russell, Bertrand 1936: ‘The limits of empiricism’, Proceedings of the Aristotelian Society, New Series, vol. 36 (1935-1936), pp.131-150

-Thomson, J. F. 1954: ‘Tasks and super-tasks’, Analysis vol. 15, no. 1 (Oct. 1954), pp.1-13

11 comments:

  1. I like this, but what if I say that the lawn mowing puzzle is (obviously) intended to apply to humans who are (obviously) not able to perform supertasks? Your solution works, albeit not for humans, but the puzzle was intended for humans, and so, in a sense, it also doesn't work. Would you be happy with that?

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  2. Can you mow eight square meters of lawn without ever having exactly 7 square meters of lawn left to mow.

    You can't mow any area instananeously, no matter how small, but you can mow a sq meter in an arbitrarily small amount of time.

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  3. If time's gunky I think you can do it without much trouble, even if you don't want to have a continuous bout of mowing which includes parts of both the first and second square metres. You'd do that by supertasking the first square metre and backwards-supertasking the second, with the same limit.

    I suspect that if time's not gunky you'd have to have 7 square metres left for at least an instant, but I'll have to think about it.

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  4. Interesting. I was thinking of the following backwards supertask: for each n, between 1/(n+1) and 1/n hours past noon mow the first two sq meters if they haven't already been mown, and do nothing otherwise.

    None of those instructions require you to be able to mow instananeously, but if you have to mow at any point there'll have been a point where there were 7 sq meters left. But if you've followed the rule at every point you never do have to mow; the first two sq meters will be mown at any point strictly after noon.

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  5. Actually I'm not sure how your solution works anymore. Are you doing the lawns in succession? After you've done the first lawn what's the area left?

    If you stipulate that all the mowing has to be done by you (so ruling out the scenario I gave) I don't think you can do it. That no area is mowed instantaneously means that the function from time past noon to area mowed must be continuous (whether or not time is gunky) so at some point there will be exactly 1 sq meter mowed (by the intermediate value theorem.)

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  6. I thought the intermediate value theorem would probably figure in a proof that you couldn't do it without doing something more peculiar than supertasks.

    The role gunkiness was playing was to mean that there are no temporal points, so there's no noon, and at any time before noon you won't have finished the supertask of mowing the first square metre, and at any time after noon you'll have already started the backwards-supertask of mowing the second.

    I added the requirement that the end of the first square metre and the start of the second aren't parts of the same bout of mowing because if they can be and there are no temporal points then it's too easy.

    I had two ideas for exotic ways of doing it if time's not gunky but I don't know enough maths to know if they'd work. What you need is to have the area mown increase non-continuously even though your mowing is continuous.

    One idea was to mow the kind of indeterminately-sized regions exploited in the Tarski-Banach theorem, and then hope their sum jumps when you join them up. The other idea was to use a point-wide mower and mow along a space-filling curve, hoping that the area you've mown is zero until you've done the whole curve. But I don't know whether either of these can work.

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  7. Right, so I guess one way for there not to be a time at which there's exactly 7 meters left is for there to be no times at all! But you can still pose similar questions that don't have a trivial answer - e.g. whether there's ever an interval during which exactly the first meter was mowed, and I was taking it that the function that takes lengths of intervals starting at "noon" to area must be continuous so you could run the same argument (I'm assuming intervals of time can still have their lengths measured by real numbers, even if they aren't isomorphic to the reals themselves.)

    The thing with space filling curves is that the area of f([0, a]), for a in [0,1], generally increases continuously with a. (For example, the one on the wikipedia page looks like the range of the function on [0,1/4] will fill the first quadrant, etcetera.)

    Wouldn't mowing two non-measurable sets, which add up to 2 sq meters involve a discontinuous jump in area. Also, what counts as mowing continuously in that context?

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  8. One way to mess with the structure of time that might work would be if time were like the rationals instead of the reals. You could then be mowing at at every time strictly greater than root 2 minutes past noon, mowing at one sq meter per hour, and there would be no time at which there were exactly 7 meters left. The mowing would be continuous in the epsilon delta sense, and the argument I gave fails, since the IVT doesn't hold for the rationals.

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  9. I like the last one, though I suppose that would still leave an interval during which exactly the first square metre was mowed, between root 2 minutes past noon and root 2 minutes past one.

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  10. I guess I could add compositional nihilism into the mix and deny there are any intervals during which the first meter was mowed. :-p

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