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| From | James Dow Allen <user4353@newsgrouper.org.invalid> |
|---|---|
| Newsgroups | rec.puzzles |
| Subject | Re: Four more "oldies" |
| References | <1787507044-4353@newsgrouper.org> |
| Date | 2026-08-29 00:04 +0000 |
| Message-ID | <1787961881-4353@newsgrouper.org> (permalink) |
| Organization | jamesdowallen@gmail.com |
James Dow Allen <user4353@newsgrouper.org.invalid> posted:
>
> (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?
SPOILERS for N=3 and N=7 follow.
22 Stanza's from Hunting the Snark as spoiler warning space
============================================================
"Just the place for a Snark!" the Bellman cried,
As he landed his crew with care;
Supporting each man on the top of the tide
By a finger entwined in his hair.
"Just the place for a Snark! I have said it twice:
That alone should encourage the crew.
Just the place for a Snark! I have said it thrice:
What I tell you three times is true."
The crew was complete: it included a Boots--
A maker of Bonnets and Hoods--
A Barrister, brought to arrange their disputes--
And a Broker, to value their goods.
A Billiard-marker, whose skill was immense,
Might perhaps have won more than his share--
But a Banker, engaged at enormous expense,
Had the whole of their cash in his care.
There was also a Beaver, that paced on the deck,
Or would sit making lace in the bow:
And had often (the Bellman said) saved them from wreck,
Though none of the sailors knew how.
There was one who was famed for the number of things
He forgot when he entered the ship:
His umbrella, his watch, all his jewels and rings,
And the clothes he had bought for the trip.
He had forty-two boxes, all carefully packed,
With his name painted clearly on each:
But, since he omitted to mention the fact,
They were all left behind on the beach.
The loss of his clothes hardly mattered, because
He had seven coats on when he came,
With three pairs of boots--but the worst of it was,
He had wholly forgotten his name.
He would answer to "Hi!" or to any loud cry,
Such as "Fry me!" or "Fritter my wig!"
To "What-you-may-call-um!" or "What-was-his-name!"
But especially "Thing-um-a-jig!"
While, for those who preferred a more forcible word,
He had different names from these:
His intimate friends called him "Candle-ends,"
And his enemies "Toasted-cheese."
"His form is ungainly--his intellect small--"
(So the Bellman would often remark)
"But his courage is perfect! And that, after all,
Is the thing that one needs with a Snark."
He would joke with hyenas, returning their stare
With an impudent wag of the head:
And he once went a walk, paw-in-paw, with a bear,
"Just to keep up its spirits," he said.
He came as a Baker: but owned, when too late--
And it drove the poor Bellman half-mad--
He could only bake Bridecake--for which, I may state,
No materials were to be had.
The last of the crew needs especial remark,
Though he looked an incredible dunce:
He had just one idea--but, that one being "Snark,"
The good Bellman engaged him at once.
He came as a Butcher: but gravely declared,
When the ship had been sailing a week,
He could only kill Beavers. The Bellman looked scared,
And was almost too frightened to speak:
But at length he explained, in a tremulous tone,
There was only one Beaver on board;
And that was a tame one he had of his own,
Whose death would be deeply deplored.
The Beaver, who happened to hear the remark,
Protested, with tears in its eyes,
That not even the rapture of hunting the Snark
Could atone for that dismal surprise!
It strongly advised that the Butcher should be
Conveyed in a separate ship:
But the Bellman declared that would never agree
With the plans he had made for the trip:
Navigation was always a difficult art,
Though with only one ship and one bell:
And he feared he must really decline, for his part,
Undertaking another as well.
The Beaver's best course was, no doubt, to procure
A second-hand dagger-proof coat--
So the Baker advised it--and next, to insure
Its life in some Office of note:
This the Banker suggested, and offered for hire
(On moderate terms), or for sale,
Two excellent Policies, one Against Fire,
And one Against Damage From Hail.
Yet still, ever after that sorrowful day,
Whenever the Butcher was by,
The Beaver kept looking the opposite way,
And appeared unaccountably shy
============================================================
Just this N=3 may seem amazing. ANY guess has precisely a 50% chance
of being wrong. Assume w.l.o.g. that George guesses. (Someone has to guess or
the team loses by default.) George fails 50% of the time. Whatever the other
teammates do, they must fail at least 50% of the time!
Or so it would seem.
The trick is to combine failures. If you see two of the same color, guess the
other color. Otherwise pass. Here are the results. W means player makes a winning
guess; L, a losing guess.
- RRR - LLL
- RRG - ..W
- RGR - .W.
- RGG - W..
- GRR - W..
- GRG - .W.
- GGR - ..W
- GGG - LLL
See? Half the guesses are indeed wrong, but they're bunched together
and the team wins 6 out of 8 cases. The table of cases can be abbreviated
- 2 x LLL
- 6 x W
To win 7/8 of the time when N=7 we want a table of cases
- 16 x LLLLLLL
- 112 x W
Assign the seven 3-bit labels (001, 010, ..., 111) to the seven players.
(All bit triplets except 000.) Each player sums (via XOR) the labels
on the players he sees wearing Green. 1/8 of the time the total sum will
be 000 -- in those cases we LOSE. But otherwise we win.
Suppose the total seen (all but self) is 000. Self writes "Green." if
green is correct, All others pass (see next paragraph). (If Red,
it is one of the losing configs. In fact EVERY player will announce Green
-- and lose -- in that losing config case.)
Suppose players 001 and 010 are Green and all others are Red. Those players
compute XOR sums 010 and 010 respectively and all others see 011.
Everyone Passes, except the
player labeled 011 -- the same as the xor total he sees. That player writes
"Red." (Green would give a losing config.)
This is the same mechanic as a single-bit-error correcting Hamming code,
except that the goal is reversed.
Cheers, James
Back to rec.puzzles | Previous | Next — Previous in thread | Find similar | Unroll 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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