"Good afternoon, Mr. Phelps. The Society of Logicians have recently discovered a plot to overthrow the friendly government of Uwalahooloo. Your mission, should you decide to accept it, is to return to Uwalahooloo and appropriate the alabaster crown, a totem in the keeping of the hostile chieftain. Bereft of this artefact, the chieftain will lose his standing with local warriors, and his plot will fail.

You will find the crown, unguarded, somewhere along one of the island's two paths, one coloured red, the other green. The false path is to avoided at all costs, as it leads through deadly quicksand.

Intelligence reports the island to be inhabited by three natives, each one a liar or a knight. The natives are identical in appearance, but may be distinguished by their differing garb. The native in green is a monolingual speaker of language g, while the native in red a monolingual speaker of language r. The native in yellow is also monolingual, but we have not been able to determine which of the two languages, either g or r, he speaks.

You may assemble the natives and pose one question per day. (The same question to each native). As is usual on Uwalahooloo, the questions have different meanings in the two languages. After responding, the natives will retire out of sight until the next day.

Be warned: the natives delight in tricking logicians: when out of sight the yellow native may change clothes with the native with whom he can communicate.

Finally, you need to be alerted to the fact that the natives are only conditionally friendly. They will patiently respond to two queries, but if you try to ask them a third question, they will their lose patience and have you over for dinner (typically covered in barbecue sauce). We therefore urge to you consider carefully which questions to pose.

Here, now, are the candidate questions with their alternative interpretations in languages g and r:

Hvilket?

g: Is the road with the crown the same colour as one of the other two natives' costumes?

r: Are the other two natives able to communicate?

Spoergsmaaler?

g: Is the yellow native the same truth type as the native with whom he can't communicate?

r: Are the other two natives of the same truth type?

Bliver?

g: Are the other two natives able to communicate?

r: Has there been a costume change?

Ud?

g: Has there been a costume change?

r: Is the road with the crown the same colour as one of the other two natives' costumes?

Should you be discovered in Uwalahooloo, the SL will deny any knowledge of your mission. Good luck, Jim. This tape will self destruct in 10 seconds."

Derive a syllogism, based on native responses, for the road containing the crown.

Assuming the natives’ answers are only relevant to the present – i.e., no costume changes could be reported truthfully as having occured during the first day of questioning, the question to ask is Ud?, followed the next day with asking the question Spoergsmaaler?

If the green-dressed native answers Yes to Ud?, we know that he is a Knight; if he answers No to the question, we know he is a Liar.
If the yellow-dressed native answers Yes to Ud?, we can deduce that he is either an r-speaking Knight or a g-speaking Liar; if he answers No to the question, we know he is the opposite -- a g-speaking Knight or an r-speaking Liar. If he is an r-speaking Knight, he could not change costume with the green-dressed native; thus the green-dressed native’s answer to Spoergsmaaler? should be the same as his answer to Ud?.
If the two answers are not the same, we know that the yellow-dressed native of day one is a g-speaking Liar.
Therefore, if the two answers given by the green-dressed native are the same, we know the yellow-dressed native of day one is a Knight.
Since the yellow-dressed native of day one speaks language r, we know both or either the red-dressed and/or yellow-dressed native of day two will answer Spoergsmaaler? truthfully. Therefore, following the given conditions, if both the red-dressed and yellow-dressed natives answer Spoergsmaaler? the same, we know that the red-dressed native is also a Knight, otherwise the red-dressed native of day one must be a Liar.
If the two answers given by the green-dressed native are different, we know the yellow-dressed native of day one is a g-speaking Liar. As we can readily deduce whether the green-dressed native of day one is a Liar or Knight by his answer of Yes or No to Ud?, we can then determine if the red-dressed native is a Knight or Liar by his answer to the question Spoergsmaaler?.
If the red-dressed native is a Knight and his answer to Ud? is Yes, the crown is on the green road else it is on the red road.
If the red-dressed native is a Liar and his answer to Ud? is Yes, the crown is on the red road, else it is on the green road.

Posted by Dej Mar on 2008-04-21 11:28:07