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| From | Mike Terry <news.dead.person.stones@darjeeling.plus.com> |
|---|---|
| Newsgroups | rec.puzzles |
| Subject | Re: Four more "oldies" |
| Date | 2026-08-25 00:41 +0100 |
| Organization | A noiseless patient Spider |
| Message-ID | <116ikr4$39v80$1@dont-email.me> (permalink) |
| References | <1787507044-4353@newsgrouper.org> <116hnp2$2vnov$1@dont-email.me> <1787591053-4353@newsgrouper.org> |
On 24/08/2026 18:04, James Dow Allen wrote: > > Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted: > >> On 23/08/2026 18:44, James Dow Allen wrote: >>> >>> R.p has been slow lately. Here are four Oldies-but-Goodies: >>> ... >>> For puzzle[] ... (4) assume each player has a computer with UNLIMITED >>> memory and computational power. > > A normal computer has 2^N words of memory, each word M bits, where > N and M are finite and smallish. The preposterously UNLIMITED computer > has 2^F words of F bits each where F is infinite; along with appropriate > peripheral hardware. Never mind what hyper-reality must be postulated > to implement such a machine. > >>> >>> (2) ... >>> With optimal strategy, what is the probability that the team wins? >>> This is tricky. First try it with 3 players. >>> Or two with 'pass' not an option. > > "Tricky" in an almost perverse way. Do try it first with only 3 players. ok, thinking about 3 players... (will post later) > > - - - - - - - - - - - > > I will offer brief comments on Mike's excellent answers, > preserving his spoiler warning. > >>> >> .s...... >> ..p..... >> ..o..... >> .i...... >> ..l..... >> ...e.... >> ....r... >> .....s.. >> .....p.. >> ....o... >> .....i.. >> ......l. >> ......r. >> .....r.. >> ....s... >> ...p.... >> ..o..... >> ..i..... >> ...l.... >> ..e..... >> .r...... >> .s...... >> ..p..... >> ..o..... >> .i...... >> ..l..... >> ...e.... >> ....r... >> .....s.. >> .....p.. >> ....o... >> .....i.. >> ......l. >> ......r. >> .....r.. >> ....s... >> ...p.... >> ..o..... >> ..i..... >> ...l.... >> ..e..... >> .r...... >> >> (1) I'll say 1/4, just because I remember some famous result telling us that the area of such a >> band around a sphere is the same as the area of the corresponding band of a cylinder of the same >> radius. And 30N is "1/2 the way" between the equator and N pole, measured by projecting >> perpendicularly onto the earth's axis. (If I hadn't remembered that result I'd have been into >> calculating integrals etc.) > > Did you need to take a sine? I just needed to know about equilateral triangles... > > The famous result is of course by the famous man Richard Tobin mentioned, who may > or may not have famously run through the streets of Syracuse shouting "Eureka!" > Ah yes, I'm not surprised by that. >> >> (3) ... Everybody else can see the mod-10 sum of all 98 of those hats apart from >> their own, > "can see OR has heard" Right. I had misunderstood the problem, thinking player k could see all players hats apart from his own. Now I realise he can only see the hats for players in front of him, but like you say, he has /heard/ the answers given by players 1,2,3...k-1 which were all correct (if they're following the strategy) so that's good enough for the strategy (modified slightly) to work. > >> and so can work out their own hat, given they heard the first playser give the total for >> the 99 hats... >> > >> (4) I'm pretty sure you are aiming here for the 'mathematical' strategy based on the axiom of >> choice [AC]. There is such a 'strategy' going broadly like this: >> - form the set of equivalence classes of sequences of hats which are "eventually the same" >> (or equivalently which differ at only a finite number of places) >> - choose a representative for each class [this is where AC is used] >> The above steps are considered part of the players "agreeing their strategy", so all players >> somehow know all those (uncountably many) representative choices! >> - when in the infinite queue, players look at the hats ahead and "identify" which >> equivalence class of hat-sequences they are in. >> - Then they guess their hat colour based on their position in the queue and >> the corresponding hat colour in the representative hat sequence >> that they "agreed" for that class. >> This ensures that only finitely many guesses will be wrong. > > Yes. > >> >> BUT this isn't really a workable strategy ... >> ... I'm looking forward to seeing whether you can /define/ such a >> computer and how exactly the players use it when they're standing in the queue! :) > > Does the hypothetical infinite (Aleph-1) computer I mention above, in a > hypothetical hyper-reality count? Even as a thought experiment? Well, how do you intend the players to use it, exactly? :) How does "calculation" work with such a computer? Etc.. I think you might as well just leave it as a mathematical argument rather than trying to imagine it being real. > >> [Bottom line is I don't consider (4) a proper "puzzle", although (for mathematicians) the maths >> involved might be considered interesting...] > > I think it's interesting because it seems to imply that the Axiom of Choice > must obviously be false!! I don't see it that way - how does the implication go? Perhaps you start from the "fact" that there can't be any such strategy? Where would that come from? It's tricky trying to apply real-world intuitions to unphysical infinite scenarios. After all, each player in the queue is supposed as having access to an infinite amount of information [viz the infinite sequence of hat colours of those in front of the player] to start with, which is beyond our physical abilities. How do we even argue about what is possible/reasonable in such made up scenarios? Well, we can make mathematical arguments, sure, but I don't see anything there saying no such strategy can exist... Mike.
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Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-23 17:44 +0000
Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-24 07:40 +0000
Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-25 07:27 +0000
Re: Four more "oldies" ram@zedat.fu-berlin.de (Stefan Ram) - 2026-08-25 12:33 +0000
Re: Four more "oldies" richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-24 09:45 +0000
Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 10:33 +0000
Re: Four more "oldies" Phil Carmody <pc+usenet@asdf.org> - 2026-08-27 22:45 +0300
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-24 16:25 +0100
Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 17:04 +0000
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 00:41 +0100
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 01:08 +0100
Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-26 07:08 +0000
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-26 23:18 +0100
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-27 00:48 +0100
Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-27 10:42 +0000
Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-28 04:55 +0100
Re: Four more "oldies" Charlie Roberts <croberts@gmail.com> - 2026-08-24 13:27 -0400
Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-29 00:04 +0000
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