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.
-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