"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 re: Comments on the Bliver/Ud connection by Dej Mar)

.. (my 3-year-old daughter, who unbeknownst to me, while I had stepped away to heed the call of her mother, had filled various random cells with miscellaneous typed data. When I had returned and sorted the data, I had then arrived at my incorrect assumption).

Ah! the impish hands of a 3 year old - well known blight in the career of many a logician..

The two questions to ask are "Hvilket? & "Spoergsmaaler?".
The following is a table of the day-one and day-two responses by the green-, red-, and yellow-dressed natives (Y = "Yes.", N = "No.", x = Y or N):

Congratulations. Your persistence has paid off. That is indeed the solution.

Did you notice there is a brief way to encapsulate the solution, namely:

"Take the red road if and only if there is a even number of positive answers to Hvilket? plus red's response to Spoersmaaler?"

I worked hard to create a problem that would be nearly impervious to computer solution, but a small computer type cross section remained, and you found it.

See the published for a surprisingly brief argument in support of the solution. Two points to getting there were key:  1) recognising that parity in the number of knights (mysteriously coded in red's answer to Spoergsmaal?) is a significant quantity, and  2) recognising that the question of whether or not natives change costumes is completely irrelevant.

FYI Hvilket spoergesmaal bliver ud? is Danish for "Which question drops out?".

I hope you enjoyed working through the problem as much as I did creating it!

Posted by FrankM on 2008-04-27 05:48:49