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Mission: Impossible? (Posted on 2008-04-20) Difficulty: 5 of 5

"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.

See The Solution Submitted by FrankM    
Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips BBQ reprieve - Still Not There. (Walking together, you recount discoveries. I ask questions)7 | Comment 7 of 15 |
(In reply to Solution -- after the BBQ reprieve by Dej Mar)

Dej Mar -

The two questions to ask are "Bliver?" followed by "Ud?".

Let's run thru it together..

From the first day of questioning, .. we know whether the red-dressed native is a Knight or Liar by his response to the question "Bliver?". A "Yes." to the question would indicate he is a Liar and a "No." would indicate if he is a Knight. The same is true of a yellow-dressed native r speaking native. If the yellow-dressed native spoke g, then, identically, the same responses would apply to identifying his truth type..

Absolutely true. That gives:   G KB = -G B?    and   Y KB = -Y B?

Therefore a yellow-dressed native (of day one) answering the two questions...

The bulletted conclusions following this statement are wrong.

In case you're still in doubt, here's a counterexample which disproves one of your conclusions, namely:

..."Yes. Yes." would indicate that he is Liar, speaks r, switched costumes with the red-dressed native, and the road with the crown is red. With the switch of clothing the red-dressed native of day one could only follow a "Yes." with a "No." and a "No." with a "Yes.".

Assume we were to start with GRY = KKL (i.e., two knight and a liar), yellow speaks g, costume change (so that GRY goes to LKK when answering Ud?) with the crown on the green path. Then native responses would be:

G B? = No    R B? = No    Y B? = Yes   

G U? = No    R U? = Yes   Y U? = Yes 

________________________________

Following your line of thought, we could indeed draw conclusions about the correct path, but only for a certain subcase. Worse, we lack the information to detect subcase occurrence. Let's take a close look. Firstly:

Yr -> (Y U? = Y KU)

that is, if yellow is an r speaker, then we can identify his truth type from his responses to Bliver? Conversely, if yellow speaks g, we aren't able to conclude anything about his truth type from his response.

For the fortuitous subcase that yellow speaks r and gives the same answer to Ud? and Bliver?, we could be sure that a costume change has taken place, so that red's truth type responding to Bliver? matches yellows' truth type responding to Ud? (because they are the same person). So, we could say that

[ Y r ^ (Y B? = Y U?) ] -> [ (R U? = Y B?) = R ]

where R means that the crown is on the Red path.

In not-so-plain english, this works out to the following: If we knew that yellow was a red speaker, and if he answered both questions identically, then the crown would be on the red/green path according as to whether or not red's response to Ud? matches with yellow's response to Bliver?

___________________________

I've included some hints especially for you in the comment Hint 3. You have some good insights, I hope you are not discouraged.


  Posted by FrankM on 2008-04-24 06:51:51
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