In the planet Realmivory, all the inhabitants are either Knights, who always speak truthfully; Liars, who always speak falsely; Knaves, who make statements that are alternately true and false, but in which order is unknown; or those few Revoltosos who do not abide by the planet's traditions.

A Revoltoso is an eccentric person who can choose to speak truthfully like Knights, or speak falsely like Liars or, alternate between true and false statements (not necessarily in that order) like Knaves.

Four inhabitants: Atys, Euryalus, Lausus and Turnus were busy in a conversation. In response to a query posited by an inquisitive visitor from a neighboring planet regarding their identity, they say:

Atys:
1. I am a Knight. 
2. Euryalus is not a Knave.

Euryalus: 
1. I am a Knave. 
2. Lausus  is not a Liar.

Lausus:  
1. I am a Liar. 
2. Turnus is not a Revoltoso.

Turnus: 
1. I am a Revoltoso. 
2. Atys is not a Knight.

(i) Determine the probability that:
(Atys, Euryalus, Lausus, Turnus) = (Knight, Revoltoso, Knave, Liar) given that the types of the four inhabitants are distinct.

(ii) What is the answer to (i) if the types of the four inhabitants may or may not be distinct?

A knight cannot claim to be a knave,  liar, or a revoltoso.  Therefore, E, L, and T are not knights.  So A is the knight by process of elimination, and by his second statement, E is not a knave.  Which means either L or T is the knave.

T cannot be the knave since his statements would both be lies.  Therefore, by process of elimination, L is the knave.

Since L is not a liar, E's second statement is true, so I is not a liar.  So T is the liar, and E is the revoltoso.

Atys: knight
Euryalus: revoltoso
Lausus: knave
Turnus: liar

So the probability of this is 1.

Posted by Dustin on 2012-07-16 05:40:32