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Four more "oldies"

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R.p has been slow lately.  Here are four Oldies-but-Goodies:

(1) How much of the Earth's surface is North of latitude 30°N, i.e. the
thirtieth parallel north.  Assume Earth's surface is a perfect sphere.

 - - - - - - - - -

For puzzles (2), (3), (4) assume that the team meets during the days before the
actual contest, that they discuss the rules, agree to cooperate, and devise an
optimal strategy. During the actual contest, no communication among them is
allowed except as stated.  The colors or numbers appearing on their hats
are chosen at random.
For puzzles (3) and (4) assume each player has a computer with UNLIMITED
memory and computational power.

(2) The four members of the team sit at a table.  Each can see the other
players' colors (each is red or green), but not his or her own.  On a piece
of paper hidden from view, each player writes 'Red', 'Green' or 'Pass.'
The team wins if and only if
 . a.  at least one player guesses his/her color correctly; and
 . b.  no player guesses the wrong color.
For example, if three players write 'Pass', and the 4th player has a Green hat,
the team wins if and only if that 4th player writes 'Green.'

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.

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(3) Now instead of a color, a numeral 0 to 9 is written on the back of each
player's hat.  Instead of 4 players, there are a hundred.
The players are lined up so that the first-to-speak (the first to guess his
numeral out loud) can see all numerals except his own, and the k'th-to-speak
can hear the (k-1) numerals guessed so far, and see all numerals
except his own and those of the (k-1) who have already guessed.

In turn, each player guesses the numeral on his hat.  The team wins if at least
99 of the 100 players guess their numeral correctly.

 - - - - - - - - -

(4) This is the same contest as (3) EXCEPT
 . a. There are a (countably) infinite number of players.  There is a 1st, 2nd
and 3rd player as before, and a 4th, 5th player, etc., but this line of
players goes on forever.
 . b. Guesses are submitted silently.  The k'th player has no information
except the (infinitely many) numerals he can see in front of him.
 . c. You do assume the Axiom of Choice.

Devise a strategy such that only a finite number of the players will
fail to guess their numeral correctly.

 - - - - - - - - -

Cheers,
James

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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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