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        riddles >> medium >> Metalogical Card Game
        
(Message started by: ThudnBlunder on Sep 11^th, 2011, 8:36am)

Title: Metalogical Card Game
Post by ThudnBlunder on Sep 11^th, 2011, 8:36am

Three perfect logicians, A, B, and C, play the following game of cards. Each of them randomly chooses a playing card from a well-shuffled deck and holds it to his/her forehead. Thus they can each see the other two chosen cards but not their own. Aces high, high wins, and (as suits don't matter) draws/ties are possible. Then each of them makes the strongest possible statement they can be certain of from the following:

'I WIN' = I have a higher card then anybody else.
'I LOSE' = Someone else has a higher card than me.
'I DON'T WIN' = I either lose or tie.
'I DON'T LOSE' = I either win or tie.
'I TIE AS A WINNER = I have the highest card, but at least one other person also has it.
'I DON't KNOW'

What can you infer about the dealt cards from the following scenarios?

1)
A says "I don't know."
B says "I lose."
C says "I lose."

2)
A says "I don't know."
B says "I don't win."
C says "I win."

3)
A says "I don't know."
B says "I don't win."
C says "I tie as a winner."

4)
A says "I don't know."
B says "I don't win."
C says "I don't win.

Title: Re: Metalogical Card Game
Post by ThudnBlunder on Sep 11^th, 2011, 8:42am

For example, for 1), [hide]A has an ace, B and C have lower than an ace, but not two 2's.
REASONING:
A said 'I don't know'.
If A saw an ace she would say 'I don't win.' Hence she didn't see an ace.
So B knows he doesn't have an ace.
And the only reason for B to say 'I lose' is that he can see A has an ace.
Similarly for C.

And if A saw {2,2} she would say 'I don't lose'. Hence she didn't see {2,2} either. [/hide]

Title: Re: Metalogical Card Game
Post by aicoped on Sep 11^th, 2011, 5:27pm

on the second scenario by similar logic A does not see an Ace, but he could see a king theoretically. B ses that C has a king and A has less than a king so the only thing B knows is he cant beat C. This means that C sees B's card which must be lower than a king so since C deduces by Bs statement C must have a king and C sees that B and A have Queens or lower so he wins.

Title: Re: Metalogical Card Game
Post by ThudnBlunder on Sep 11^th, 2011, 6:19pm

aicoped, having translated your Vogon prose into Vulcan, I think you are correct.

[hide]C has a king; A and B have lower cards.
REASONING:
As in 1), A sees no ace, nor pair of 2's, and says 'I don't know'.
B sees no ace either and therefore knows that king is high.
And B says 'I don't win' (outright) because he can see a king.
C knows this but cannot see a king on A or B.
Hence C infers he has a king and says 'I win'. [/hide]

Title: Re: Metalogical Card Game
Post by Hippo on Sep 12^th, 2011, 6:59am

[hide]
3)
A says "I don't know."
{B,C}#{2,2}, 'A'\not\in {B,C}
B says "I don't win."
{B,C}#{2,2}, 'K'\in {A,C}, 'A'\not\in {A,B,C}
C says "I tie as a winner."
{B,C}#{2,2}, 'K'\not\in {A}, {B,C}={'K','K'}

4)
A says "I don't know."
{B,C}#{2,2}, 'A'\not\in {B,C}
B says "I don't win."
{B,C}#{2,2},'K'\in {A,C}, 'A'\not\in {A,B,C}
C says "I don't win.
{B,C}#{2,2},'K'\in {A}, 'A'\not\in {B,C}
[/hide]

[edit]wrong reasoning on small cards ..[/edit]

Title: Re: Metalogical Card Game
Post by ThudnBlunder on Sep 12^th, 2011, 10:16am

3)[hide] B and C have kings; A has less than a king.
REASONING:
As in 2), kings are high.
B sees a king on C and says 'I don't win'.
C sees that A has no king, but B does, and declares 'I tie as a winner'.[/hide]

4)[hide] A has a king; B and C have less then a king, but not a pair of 2's.
REASONING:
As in 2) and 3), kings are high.
B sees a king on A and says 'I don't win'.
C sees a king on A and says 'I don't win'.
(If A cannot see a king, she will claim victory in the next round.) [/hide]

5) Deduce their cards and the final comments of B and C if
A says "I don't know."
B says "I don't know."
C says "I don't know."
A says "I lose."
B says ?
C says ?

6) Deduce their cards if
A says "I don't know."
B says "I don't know."
C says "I don't know."
A says "I don't know."
B says "I don't know."
C says "I don't know."
A says "I don't know."
B says "I don't know."
C says "I win."

Title: Re: Metalogical Card Game
Post by Hippo on Sep 12^th, 2011, 11:27am

5)[hide] Deduce their cards and the final comments of B and C if
A says "I don't know." 'A'>B>2,'A'>C>2
B says "I don't know." 'K'>C>3,'K'>A>3,'A'>B>2
C says "I don't know." 'Q'>A>4,'Q'>B>4,'K'>C>3
A says "I lose." C='Q'
B says ....... "I lose."
C says ....... "I win."
[/hide]
6)[hide] Deduce their cards if
A says "I don't know." 'A'>B>2,'A'>C>2
B says "I don't know." 'K'>C>3,'K'>A>3,'A'>B>2
C says "I don't know." 'Q'>A>4,'Q'>B>4,'K'>C>3
A says "I don't know." 'J'>B>5,'J'>C>5,'Q'>A>4
B says "I don't know." 'T'>C>6,'T'>A>6,'J'>B>5
C says "I don't know." 9>A>7,9>B>7,'T'>C>6
A says "I don't know." A=8,B=8,C visible ???
B says "I don't know."
C says "I win."
[/hide]

Title: Re: Metalogical Card Game
Post by ThudnBlunder on Sep 12^th, 2011, 5:17pm

5) [hide]C has a queen; A,B have less than a queen and do not both have less than 4.
REASONING:
As before,
A cannot see an ace, otherwise she would say 'I don't win'.
B cannot see an ace or king, otherwise he would say 'I don't win'.
C cannot see an ace or king or queen, otherwise he would say 'I don't win'.
A says 'I lose' because she can see a queen.
It must be on C because C cannot see it.
B realizes this and says 'I lose'.
C knows he has a queen and says 'I win'.

As for the lower cards,
A saying 'Don't know' means she cannot see {2,2}.
B saying 'Don't know' means he cannot see {2,2} or {3,2}. (If he could see {2,3} he would say 'I don't lose'.)
C saying 'Don't know' means he cannot see {2,2}, {3,2}, or {3,3}.[/hide]

6) [hide] A has a 5; B has a 4 or 5; C has a 6.
REASONING:
As before,
A cannot see an ace, otherwise she would say 'I don't win'.
B cannot see an ace or king - otherwise he would say 'I don't win'.
C cannot see an ace, king, or queen - otherwise he would say 'I don't win'.
A cannot see an ace, king, queen, or jack - otherwise she would say 'I don't win'.
B cannot see an ace, king, queen, jack, or 10 - otherwise he would say 'I don't win'.
C cannot see an ace, king, queen, jack, 10, or 9 - otherwise he would say 'I don't win'.
A cannot see an ace, king, queen, jack, 10, 9, or 8 - otherwise she would say 'I don't win'.
B cannot see an ace, king, queen, jack, 10, 9, 8, or 7 - otherwise he would say 'I don't win'.
Hence
A and C both have a 6 or lower.
B has a 7 or lower. [/hide]

For the lower cards I managed to find this (http://www.tomas.rokicki.com/bluffhead/) file.

Title: Re: Metalogical Card Game
Post by Hippo on Sep 13^th, 2011, 5:00am

on 09/12/11 at 17:17:17, ThudnBlunder wrote:

For the lower cards I managed to find this (http://www.tomas.rokicki.com/bluffhead/) file.

Have they answer I don't lose?

Title: Re: Metalogical Card Game
Post by ThudnBlunder on Sep 13^th, 2011, 6:10am

on 09/13/11 at 05:00:58, Hippo wrote:

Have they answer I don't lose?

Yeah, I don't understand what 'Threes case' means.
The page seems to be part of a larger file.

However, the 'Fifth problem' is the same as our 6), which comes from Scientific American.
Looks like he wrote a program dealing with metalogic.
I think it is becoming a new field for research.
See Reasoning about Knowledge (http://www.amazon.co.uk/Reasoning-about-Knowledge-Ronald-Fagin/dp/0262562006/ref=sr_1_1?ie=UTF8&qid=1315919014&sr=8-1).