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Re: Four more "oldies"

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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.

 - - - - - - - - - - -

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?

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!"

> 
> (3)  ... Everybody else can see the mod-10 sum of all 98 of those hats apart from 
> their own,
"can see OR has heard"

> 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?

> [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!!

> 
> Regards,
> Mike.
> 

Cheers,
James.

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Thread

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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