Thursday, March 4, 2010

The Warden's Game

As in "The prison doors (2)", The prison doors (2) , the warden has set up several fiendish situations for several prisoners. Here are are the door and sign situations:

Prisoner 1:
(one sign is true, and the other is false)
Sign 1: EITHER A TIGER IS IN THIS ROOM OR A LADY IS IN THE OTHER ROOM
Sign 2: A LADY IS IN THE OTHER ROOM

Prisoner 2:
(If a lady is in room 1, then the sign on the door is true ... if it contains a tiger, then the sign is false. In room 2, a lady in the room means the sign is false, and a tiger means the sign is true. It is still possible that both rooms have a lady, or both rooms have a tiger, or that one room has a lady and the other one a tiger.)
Sign 1: BOTH ROOMS CONTAIN LADIES
Sign 2: BOTH ROOMS CONTAIN TIGERS

Prisoner 3:
(same rules apply as for prisoner 2)
Sign 1: AT LEAST ONE ROOM CONTAINS A LADY
Sign 2: THE OTHER ROOM CONTAINS A LADY

Prisoner 4:
(same rules apply as for prisoner 2)
Sign 1: IT MAKES NO DIFFERENCE WHICH ROOM YOU PICK
Sign 2: THERE IS A LADY IN THE OTHER ROOM

Prisoner 5:
(same rules apply as for prisoner 2)
Sign 1: IT DOES MAKE A DIFFERENCE WHICH ROOM YOU PICK
Sign 2: YOU ARE BETTER OFF CHOOSING THE OTHER ROOM

Prisoner 6:
(same rules apply as for prisoner 2 ... the signs have not yet been posted on the doors. The prisoner asked how he was suppose to choose a room when the signs aren't mounted. The warden informed him that a solution was possible without knowing which sign should be on which door.)
Sign (not posted): THIS ROOM CONTAINS A TIGER
Sign (not posted): BOTH ROOMS CONTAIN TIGERS


Will you help these prisoners choose the correct doors, and escape this madman warden?

6 Comments:

Blogger Eketahuna said...

Prisoner 1 - Door with sign 1.
Sign 1 is false (can't be either/or as they are the same) so Sign 2 is True.

March 4, 2010 3:17 PM  
Blogger Anonymous said...

Warden's Game

Optimal doors are given. Assume Warden is twisted and both doors having Tigers is allowed. If they are not allowed the choice will remain the same.

Prisoner 1
Assume door must contain either Tiger or Lady
Given: One True one False sign
Interpret (Either A OR B) as exclusive or (XOR) where XOR evaluates as:
XOR(False,False)=False
XOR(False,True)= True
XOR(True,False)=False
XOR(True,True)=False

Door 1, Door2, Sign1,Sign2
Tiger,Tiger, True, False
Tiger,Lady, False(XOR),False
Lady, Tiger, False, True
Lady, Lady, True,True

Eliminate False,False and True, True. Door 1 has P=1/2 Lady, Door 2 has P=0/2 Lady.
Door 1 is optimal.

Prisoner 2
Door 1, Door2, Sign1,Sign2
Tiger,Tiger,False,True
Tiger,Lady,False,False
Lady,Tiger,False,False
Lady,Lady,True, False

By rules:
- Lady,Tiger must be removed since lady in room 1 and sign is false.
2 Tigers possible, 2 Ladies possible, Tiger in room 1 and Lady in Room 2 possible. If forced to pick door pick Door 2. Door 1 is P=1/3 Lady, Door2 is P=2/3 Lady.
Door 2 is optimal.

Prisoner 3:
Door 1, Door2, Sign1,Sign2
Tiger,Tiger,False,False
Tiger,Lady,True,False
Lady,Tiger,True,True
Lady,Lady,True, True

By rules:
- Tiger, Lady must be removed since Tiger in room 1 and sign is True.
-Tiger,Tiger must be removed since Tiger in room 2 and sign is False.
-Lady, Lady must be removed since Lady is in room 2 and sign is True

2 Tigers not possible, 2 Ladies not possible, Lady in room 1 and Tiger in Room 2 possible. If forced to pick door pick Door 1. Door 1 is P=1/1 Lady, Door2 is P=0/1 Lady.
Door 1 is optimal.

Prisoner 4:
Door 1, Door2, Sign1,Sign2
Tiger,Tiger,True,False
Tiger,Lady,False,False
Lady,Tiger,False,True
Lady,Lady,True, True

By rules:
- Tiger, Tiger must be removed since Tiger in room 1 and sign is True.
-Lady,Tiger must be removed since Lady in room 1 and sign is False.
-Lady, Lady must be removed since Lady is in room 2 and sign is True

2 Tigers not possible, 2 Ladies not possible, Tiger in room 1 and Lady in Room 2 possible. If forced to pick door pick Door 2. Door 1 is P=0/1 Lady, Door2 is P=1/1 Lady.
Door 2 is optimal.

Prisoner 5:
Door 1, Door2, Sign1,Sign2
Tiger,Tiger,False,False
Tiger,Lady,True,False
Lady,Tiger,True,True
Lady,Lady,False,False

By rules:
- Tiger, Tiger must be removed since Tiger in room 1 and sign is True.
-Tiger,Lady must be removed since Tiger in room 1 and sign is True.
-Lady, Lady must be removed since Lady is in room 1 and sign is False
2 Tigers not possible, 2 Ladies not possible, Lady in room 1 and Tiger in room 2 possible. If forced to pick door, pick Door 1. Door 1 is P=1/1 Lady, Door2 P= 0/1 Lady.
Door 1 is optimal.

Prisoner 6:
Door 1, Door2, Sign1,Sign2
Tiger,Tiger,True,True
Tiger,Lady, True,False OR False, False
Lady,Tiger, False,False OR False, True
Lady,Lady,False,False

By rules:
- Tiger, Tiger must be removed since Tiger in room 1 and sign is True.
-Lady, Lady must be removed since Lady is in room 1 and sign is False
-Lady,Tiger must be removed since LAdy in Room 1 and sign is False
Tiger,Lady by rules must be False,False thus signs are arrange as:
Both room contains Tigers on Door 1.
This room contains a Tiger on Door 2.

2 Tigers not possible, 2 Ladies not possible, Tiger in room 1 and Lady in Room 2 possible. If forced to pick door pick Door 2. Door 1 is P=0/1 Lady, Door2 is P=1/1 Lady.
Door 2 is optimal.


Cam

March 4, 2010 4:50 PM  
Blogger Eketahuna said...

Prisoner 2

Door 2 is best...

If a lady behind door 1 sign is tru so sign 1 on door 1, 2 on 2.

So must be a tiger in room 1 to make both signs false, and if both are false then there must be a lady in room 2.

March 4, 2010 5:08 PM  
Blogger Zaux said...

Cam ....
2,3,4,5,and 6 all correct

#1 - the published solution varies from yours ... can you take another look?

March 4, 2010 5:54 PM  
Blogger Anonymous said...

Zaux

for #1 I suspect it's an issue with the interpretation of Either A OR B. statement. If the solution assumes it is a simple OR rather than an XOR then:

Door 1, Door2, Sign1,Sign2
Tiger,Tiger, True, False
Tiger,Lady, True,False
Lady, Tiger, False, True
Lady, Lady, True,True

Then the choice is 50/50. 1/2 times a Lady for Door 1 and 1/2 Times a Lady for Door 2. No choice is optimal.

Cam

March 4, 2010 11:11 PM  
Blogger Zaux said...

Cam ...
exactly right ... there is lady behind doors 1 and 2

March 5, 2010 8:30 PM  

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