"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.

(In reply to Solution by Dej Mar)

Dej Mar,

You have some ideas which play an important role in the solution, but I believe you've also made some mistakes along the way. I'll try to accompany your line of thought to expose them:

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.

I agree with your statement above. I've found it convenient to developed a notation to record and build upon this assessment:

(Y r = Y KU) <-> Y U?

That is, by monitoring yellow's answer to Ud, we can establish whether yellow's knighthood status (at the time of answering Ud) is like or unlike to yellow's ability to speak red.

If he [yellow] 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?.

This is questionable: yellow's answer to Ud we does not allow us to ascertain his language. Is it your intention to propose yellow's red-languagity as a free assumption? Even so, this does not mean that green's must give the same answer to Ud as to Spoergsmaal (although you would then be able to conclude that green would have consistent truth type between the two questions).

What can we say? You've established that green's truth type can be ascertained by his answer to Ud, i.e. we do know

G KU = not G U? 

We've also seen that  Y r <-> (G KU = G KS)  so. combining:  Y r <-> (G KS = not G U?)

In other words, if we were to know yellow's language then we could ascertain the reliability of greens answer to spoergsmaal, which would allow us to determine the truthfulness of yellow's answer to spoergsmaal as well:

Y r  and  G S?  -> Y KS

Y r  and  not G S?  ->  not Y KS 

If the two answers are not the same, we know that the yellow-dressed native of day one is a g-speaking Liar.

I believe you mean to say that green must be a liar if he gives unlike responses to Ud and Spoergsmaal. This is wrong though,  as you could see by sanity checking against the 3 knight case. With three knights green answers no to Ud and yes to spoergsmaal; yet g is not a liar.

I broke off examining your comment at this point. (If there were any misunderstandings, please point it out). Like I pointed out at the start, you've had some profitable insights, so I hope you will continue your efforts. I'd also like to encourage you to use my notation (or invent your own it you like). The notation proves to be a big help in accounting and recombining the diverse information in the problem and essential in writing the syllogism.

Posted by FrankM on 2008-04-21 20:54:55