The puzzle of the honest pirates
in [Math]
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From: Herman Jurjus on 13 Nov 2009 04:20 Years ago I read a puzzle (I think it was in a book by Halmos, but I'm not sure) about five honest pirates who had stolen a number of coconuts. Before going to sleep, they made the arrangement to meet in the morning and divide the coconuts evenly among them. But during the night, one pirate woke up, took 1/5 of the coconuts, and left; then another woke up, took 1/5 of whatever he found present (not knowing about what the first had done), and so did the other three. Puzzle: assuming that no pirate had to chop pieces off any coconut, what's the least possible number of coconuts left over at the end, and with (at least) how many did they start? The puzzle itself is not very exciting; but it was accompanied by the remark that this puzzle can be elegantly solved using /eigenvalues/. My question: what could have been meant with that remark? -- Cheers, Herman Jurjus
From: Frederick Williams on 13 Nov 2009 07:46 Herman Jurjus wrote: > > Years ago I read a puzzle (I think it was in a book by Halmos, but I'm > not sure) about five honest pirates who had stolen a number of coconuts. > Before going to sleep, they made the arrangement to meet in the morning > and divide the coconuts evenly among them. > But during the night, one pirate woke up, took 1/5 of the coconuts, and > left; then another woke up, took 1/5 of whatever he found present (not > knowing about what the first had done), and so did the other three. > Puzzle: assuming that no pirate had to chop pieces off any coconut, > what's the least possible number of coconuts left over at the end, and > with (at least) how many did they start? In the version I know, each time after taking one fifth there's one left over which is given to a monkey. The initial number may be minus four. > The puzzle itself is not very exciting; but it was accompanied by the > remark that this puzzle can be elegantly solved using /eigenvalues/. > > My question: what could have been meant with that remark? -- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
From: dan73 on 12 Nov 2009 21:52 > Years ago I read a puzzle (I think it was in a book > by Halmos, but I'm > not sure) about five honest pirates who had stolen a > number of coconuts. > Before going to sleep, they made the arrangement to > meet in the morning > and divide the coconuts evenly among them. > But during the night, one pirate woke up, took 1/5 of > the coconuts, and > left; then another woke up, took 1/5 of whatever he > found present (not > knowing about what the first had done), and so did > the other three. > Puzzle: assuming that no pirate had to chop pieces > off any coconut, > what's the least possible number of coconuts left > over at the end, and > with (at least) how many did they start? > > The puzzle itself is not very exciting; but it was > accompanied by the > remark that this puzzle can be elegantly solved using > /eigenvalues/. > > My question: what could have been meant with that > remark? > > -- > Cheers, > Herman Jurjus > I have no idea but is the total number of coconuts stolen = 5^5 ? Leaving the least amount of 1024 coconuts after each pirate took his 1/5 of the remainder. Dan
From: Bart Goddard on 13 Nov 2009 12:43 Herman Jurjus <hjmotz(a)hetnet.nl> wrote in news:hdj8cq$37l$1(a)news.eternal- september.org: > The puzzle itself is not very exciting; but it was accompanied by the > remark that this puzzle can be elegantly solved using /eigenvalues/. > > My question: what could have been meant with that remark? Assuming that the problem has one left over coconut at each stage: Eigenvalues are fixed points. If there are x coconuts in the pile before a pirate messes with it, then there are f(x) = 4/5(x-1) coconuts after he messes with it. The easiest solution is the _fixed point_ of this function, x=-4. Then argue that all integer solutions must be congruent modulo 5^<something>. So I guess his point was "invariance." Bart -- Cheerfully resisting change since 1959.
From: Mensanator on 13 Nov 2009 13:06
On Nov 13, 3:20 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Years ago I read a puzzle (I think it was in a book by Halmos, but I'm > not sure) about five honest pirates who had stolen a number of coconuts. > Before going to sleep, they made the arrangement to meet in the morning > and divide the coconuts evenly among them. > But during the night, one pirate woke up, took 1/5 of the coconuts, and > left; then another woke up, took 1/5 of whatever he found present (not > knowing about what the first had done), and so did the other three. > Puzzle: assuming that no pirate had to chop pieces off any coconut, > what's the least possible number of coconuts left over at the end, and > with (at least) how many did they start? The real question is: why are they called "honest"? > > The puzzle itself is not very exciting; but it was accompanied by the > remark that this puzzle can be elegantly solved using /eigenvalues/. > > My question: what could have been meant with that remark? > > -- > Cheers, > Herman Jurjus |