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Re: Four more "oldies"

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Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:
> On 26/08/2026 08:08, James Dow Allen wrote:
> > Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:
> >> 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:
> >>>>>
> >>>>> (2)  ...
> >>>>> With optimal strategy, what is the probability that the team wins?
> >>>>> This is tricky. First try it with 3 players....
> >>>
> >>> "Tricky" in an almost perverse way.  Do try it first with only 3 players.
> >>
> >> For 3 players I see ...
> >>
> >> Specifically, players can effectively bet on a 2-1 split which occurs
> >> with probability 3/4, ...
> > 
> > Beautiful!
> > 
> >>
> >> But I don't feel that's helped me with the 4-player game.
> > 
> > I assume you DO have a 3/4 solution for the 4-player game?
> > Gur "gevpx" vf gung gurer vf ab jnl gb vzcebir ba gung.
> 
> At the time I previously posted I only had the 1/2 solution I'd previously described.
> 
> Since then I've written a (pretty naive) program to search for the best strategies, and for the 
> 3-player game it confirms the best we can do gives a 3/4 chance for the team to win.
> 
> For the 4-player game I doubt I will be able to run the program to completion ... :(
> But as luck would have it, it's found a handful of 3/4 
> strategies near the beginning of its search tree
> - these strategies have one of the players always 
> passing regardless of what hats that player sees!
> 
> But these results don't really improve my understanding.

The N=3 problem was presented long ago, I think right here in r.p.

Extending it to N=4  was my own "invention".  Cutesy because the
solution -- (the ONLY way to get 3/4 ??) -- is to nominate one
player ("George") and instruct him to always pass.  A sadistic variation, perhaps
since many will ignore that easy way to get 3/4.

IIRC when the puzzle was presented long ago, some mathematicians pointed out
that 3/4 could be exceeded for N>4 using ideas borrowed from Hamming (error-correcting)
codes.  Perhaps the puzzle had been cross-posted to sci.math or sci.crypt.

Especially good solutions exist for
 - N=3 ... p(Win) = 3/4
 - N=7 ... p(Win) = 7/8
 - N=15 .. p(Win) = 15/16

These are like Hamming error-correction codes.  Except that for N=7
there are 2^4 Correct codes, whereas in this puzzle it is precisely
those 2^4 "correct" codes that result in failure.


> >> I'm probably overlooking some key insight!

You and me both, Mike!  What's embarrassing is that 50 years ago
I worked with Hamming codes but now get easily confused.
 
> > For N=5 or larger the puzzle becomes horrifically tedious and difficult.
> > (I'll supply keywords for Google search if you wish.)

N=5 or 6 may be sadistic.  But BIG KUDOS to anyone who demonstrates
correct strategy for the relatively easy(??) N=7 or N=15.

> > 
> >> Mike.
> >>
> > 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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