Quote:
Originally Posted by FlectiNonFrangi
OK SL,
Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random manner. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which. |
FNF I had to break out a bottle of JW Blue Label to figure this one out! But here goes....
The "first move is to find a god that you can be certain is not Random, and hence is either True or False".[2] There are many different questions that will achieve this result. One strategy is to use complicated logical connectives in your questions (either biconditionals or some equivalent construction).
Boolos' question was:
Does 'da' mean yes if and only if you are True if and only if B is Random?[3]
Equivalently:
Are an odd number of the following statements true: you are False, 'ja' means yes, B is Random?
The puzzle's solution can be simplified by using counterfactuals.[4][5] The key to this solution is that, for any yes/no question Q, asking either True or False the question
If I asked you Q, would you say 'ja'?
results in the answer 'ja' if the truthful answer to Q is yes, and the answer 'da' if the truthful answer to Q is no. The reason this works can be seen by looking at the eight possible cases.
Assume that 'ja' means yes and 'da' means no.
(i) True is asked and responds with 'ja'. Since he is telling the truth the truthful answer to Q is 'ja', which means yes.
(ii) True is asked and responds with 'da'. Since he is telling the truth the truthful answer to Q is 'da', which means no.
(iii) False is asked and responds with 'ja'. Since he is lying it follows that if you asked him Q he would instead answer 'da'. He would be lying, so the truthful answer to Q is 'ja', which means yes.
(iv) False is asked and responds with 'da'. Since he is lying it follows that if you asked him Q he would in fact answer 'ja'. He would be lying, so the truthful answer to Q is 'da', which means no.
Assume 'ja' means no and 'da' means yes.
(v) True is asked and responds with 'ja'. Since he is telling the truth the truthful answer to Q is 'da', which means yes.
(vi) True is asked and responds with 'da'. Since he is telling the truth the truthful answer to Q is 'ja', which means no.
(vii) False is asked and responds with 'ja'. Since he is lying it follows that if you asked him Q he would in fact answer 'ja'. He would be lying, so the truthful answer to Q 'da', which means yes.
(viii) False is asked and responds with 'da'. Since he is lying it follows that if you asked him Q he would instead answer 'da'. He would be lying, so the truthful answer to Q is 'ja', which means no.
Using this fact, one may proceed as follows.[6]
Ask god B, "If I asked you 'Is A Random?', would you say 'ja'?". If B answers 'ja', then the answer indicates that A is indeed Random and B and C are not Random. If B answers 'da', then either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is not Random. Either way, A is not Random.
Go to the god who was identified as not being Random by the previous question (either A or C), and ask him: "If I asked you 'Are you True?', would you say 'ja'?". Since he is not Random, an answer of 'ja' indicates that he is True and an answer of 'da' indicates that he is False.
Ask the same god the question: "If I asked you 'Is B Random?', would you say 'ja'?". If the answer is 'ja' then B is Random; if the answer is 'da' then the god you have not yet spoken to is Random. The remaining god can be identified by elimination.
[edit] Random's behaviour
Most readers of the puzzle assume that Random will provide completely random answers to any question asked of him; however, the puzzle does not actually state this. In fact, Boolos' third clarifying remark explicitly refutes this assumption.
Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.
This says that Random randomly acts as a liar or a truth-teller, not that Random answers randomly.
A small change to the question above yields a question which will always elicit a meaningful answer from Random. The change is as follows:
If I asked you Q in your current mental state, would you say 'ja'?[6]
We have effectively extracted the truth-teller and liar personalities from Random and forced him to be only one of them. This completely trivializes the puzzle since we can now get truthful answers to any questions we please.
1. Ask god A, "If I asked you 'Are you Random?' in your current mental state, would you say 'ja'?"
If A answers 'ja', then A is Random:
2a. Ask god B, "If I asked you 'Are you True?', would you say 'ja'?"
If B answers 'ja', then B is True and C is False.
If B answers 'da', then B is False and C is True. In both cases, the puzzle is solved.
If A answers 'da', then A is not Random:
2b. Ask god A, "If I asked you 'Are you True?', would you say 'ja'?"
If A answers 'ja', then A is True.
If A answers 'da', then A is False.
3. Ask god A, "If I asked you 'Is B Random?', would you say 'ja'?"
If A answers 'ja', then B is Random, and C is the opposite of A.
If A answers 'da', then C is Random, and B is the opposite of A.
We can modify Boolos' puzzle so that Random is actually random by replacing Boolos' third clarifying remark with the following.
Whether Random says 'ja' or 'da' should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he says 'ja'; if tails, he says 'da'.
With this modification, the puzzle's solution demands the more careful god-interrogation given at the end of the The Solution section
The key here is to figure out which god is which. That’s the key to making quick work of this puzzle. I pretended the "three" Gods were hookers and each had show cases. This made everything so much easier. Here is why!
First I spotted an ad where the prices were really high. The age was 29, the pictures were obviously photo shopped, well it doesn't take genius to figure out this hooker was a 'FLASE" hooker.
Second I spotted a show case where the pictures had many tattoos, some stretch marks, a few blemishes, with crap all over the floor of the motel room and a gross of condoms on the night stand. I figured out right away this hooker was a 'TRUE" hooker.
Third I spotted a show case where the lady is obviously close to 200-250 lbs. but listed her weight as "curvy". She states she is "open minded" and her pictures had both blonde, brunette and red head pictures all taken in a bath room. Most having a duck face. Well I'm not the brightest hobbyist in the world but it was quite clear that this was a "Random" hooker.
Doing it this way I cut my time by 85%.
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