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Long ago,in a planet far far away, there lived three races of intelligent inhabitants: the blues (who always tell the truth), the whites (who always lie), and the pinks (who, when asked a series of questions, start with a lie and then tell the truth and lie alternately). To three creatures, chosen from the planet and seated facing each other at $A$, $B$, and $C$ (see figure), the following three questions are put:

  1. What race is your left-hand neighbour?
  2. What race is your right-hand neighbour?
  3. What race are you?

Here are their answers:

  1. (i) White (ii) Pink (iii) Blue
  2. (i) Pink (ii) Pink (iii) Blue
  3. (i) White (ii) Blue (iii) Blue


What is the actual race of each of the three creatures?

  1. $A$ is Pink, $B$ is White, $C$ is Blue.
  2. $A$ is Blue, $B$ is Pink, $C$ is White.
  3. $A$ is Pink, $B$ is Blue, $C$ is Pink.
  4. $A$ is White, $B$ is Pink, $C$ is Blue.
  5. Cannot be determined from the above data.
in Mathematical Logic by Boss (30.8k points)
edited by | 917 views
0
The figure is not shown in the GO BOOk pdf and for solving this question we need the given figure.
0
Can this question be solved if options are not given?

2 Answers

+25 votes
Best answer

If $A$ is Blue (honest), then

  • Whatever $A$ says about $B$ and $C$ must be True.
  • $A$ says that $B$ is White(liar) and $C$ is Pink(alternating). So, if $A$ is Blue, $B$ must be White and $C$ must be Pink.
  • $B$ says that $C$ is Pink. But $B$ is a liar, and $B$ agrees with $A$ on the race of $C$ (they must not agree). Thus, we reached a contradiction.

So, $A$ can't be Blue.


If $B$ is Blue (honest), then

  • Whatever $B$ says about $A$ and $C$ must be True.
  • $B$ says that $A$ is Pink(alternating) and $C$ is Pink(alternating). So, if $B$ is Blue, $A$ must be Pink and $C$ must be Pink.
  • Since $A$ is pink, it must lie about $B$, say the truth about $C$ and then lie about itself. Which it does.
  • Since $C$ is pink, it must lie about $A$, say the truth about $B$, and then lie about itself. Which it does.

So we see that Blue B, Pink A and Pink C is a possible solution!

Thus, option (C) is correct.


However, there is another option (E), which says Cannot be determined from the above data.

So, what if there are multiple solutions that satisfy these constraints? If that is the case, option e will be correct. Sadly, there is no way of proving that no other solutions work except checking each one of them (using branch and bound to somewhat improve). Sadly, that will be lengthy.

Here is a Python3 program that finds all solutions to this problem: http://ideone.com/7EFXCn

by Boss (22.9k points)
edited by
+1

Superb answer.Thanks, Pragy Agarwal .

0
Actually there is a way of proving that no answer other than C exists

After checking A,B,C only B can be blue ,other cases only exist when B is not blue

This implies,none of the A,B,C cannot be blue.

So, A,B,C are either white or pink.

This further implies first and last statements of the A,B,C are false.

B->  C->Pink A->Pink  B->Blue

A->  B->White C->Pink A->Blue

C->  A->White B->Blue C->Blue

negating first statements of B,A,C we get

C->White

B->Pink

A->Pink

which are to be considered true.

now if A->Pink is a true statement then second statement of A must be true which says C->Pink but from above we can infer C->White which is a contradiction.

Hence ,C) is the answer
+3 votes

Blue always say truth , White always say lie and Pink say lie and truth alternatively starting from lie.

1.If we consider the A as Blue then according to A answers B will be white and C will be Pink. 
now if we match the the C answer it will not match. He says right is blue which is white according to A so its a lie which is supposed to be truth.

2. Now consider B as blue then according to to B answer A will be Pink and C will also be Pink.
Then answer of A and C will match.
So answer is Option C A is pink B is blue and C is pink.

by Boss (16.5k points)
Answer:

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