Years ago, logician and riddle master Raymond Smullyan developed a more difficult logic question that, as far as we know, has not been developed. In this article, we will tackle this logic question and examine some of its interesting aspects.
Hardest Logic Question
Let there be 3 gods (A, B, C). The names of these gods are, in any order, Truthful, Liar, and Random. The Righteous god always tells the truth, the Liar god always lies, and the Random god is completely random truth or lie. Your task is to determine which gods A, B, C are Truthful, which Liar, and which are Random, provided you ask only 3 yes-no questions. You can only ask 1 god per question. The gods understand Turkish; but they speak a language of their own. In this language, the words "yes" and "no" correspond to the words "da" and "ja"; but you don't know which is "yes" and which is "no".
Frequently asked Questions
Before we get to the answer, we would like to answer some of the questions that are sometimes asked about this riddle:
- You can ask a god more than one question (so you can choose not to ask either or both of the gods at all).
- What the second question will be and which god to ask may depend on the answer to the first question. The same is true for the third question.
- You can compare whether the random god is telling the truth or not, like a coin tossed in his mind coming up heads or tails. If it comes heads, you can imagine it telling the truth, if it comes heads, you can imagine it lying.
- The random god also responds to any yes-no question with "da" or "ja" answers.
3 Riddles on the Road to Solution
Before we move on to the solution, we'll take a look at 3 similar but much simpler riddles. Then, by combining the answers of these 3 puzzles, we will solve the most difficult logic question ever. You may be familiar with the last two of these; but the first is a new puzzle derived during our reflection on this question.
Riddle 1
I placed 2 aces and 1 jack cards face down on the table, side by side, provided that their positions were recorded in advance. You don't know which card goes where. Your problem is pointing at any of the three cards and then asking me a single yes-no question. From the answer to this question, you can definitively determine that one of the three cards is an ace. If you pointed to one of the aces before asking the question, I will answer your question honestly. But if you pointed it out to the valet, I would answer "yes" or "no" to your yes-no question in a completely random fashion.
Riddle 2
Let's say you're not talking to the Random god, but to one of the Truthful and Liar gods; but you don't know which one you're talking to. And for some reason, the god you spoke to chose to speak to you in Turkish. For some reason you have to find out if Dushanbe is in Kyrgyzstan. If you could ask one question to determine whether Dushanbe is in Kyrgyzstan, what would you ask this god, whose truth or liar you do not know?
Riddle 3
Now, you definitely know that you are talking to the Righteous god; but god refuses to speak English with you. So it only gives "da" and "ja" answers. If you could ask the truthful god just one question, which question would you ask if Dushanbe is in Kyrgyzstan?
3 Puzzle Solutions
Here are the solutions to these riddles:
The Solution to Riddle 1
Point to the middle card and say, "Is the left card an ace?" ask. If I answer "yes", select the card on the left; if I answer "no", select the card on the right. Regardless of whether the middle card is an ace or not, by choosing the left card if I say "yes" or the right card if I say "no", you are guaranteed to find an ace.
The reason is this: If the middle card is an ace and therefore my answer is correct; If I say "yes" the left card should be an ace, if I say "no" the right card should be an ace. But if the middle card is a jack, then both sides (both left and right) must be aces. So if I say "yes" to your question, again the left card must be an ace (in which case it doesn't matter if the right card is an ace), and if I say "no" the right card must be an ace (in which case it doesn't matter if the left card is an ace).
Solutions to Riddles 2 and 3
To solve these riddles, we will use the concept of if and only (abbreviation: "iff"). In logic, "if and only" works like this: If you put "if and only" or "iff" between two propositions that are both true or both false, you get the proposition true. If you put "iff" between two propositions, one true and one false, the proposition you get is false. So, for example:
- "The moon is made of Gorgonzola if and only if Rome is in Russia." is a true proposition; because "The moon is made of Gorgonzola." The proposition is, "Rome is in Russia." proposition is also wrong.
- The moon is made of Gorgonzola; if and only if Rome is in Italy" is a false proposition because the first of the statements is false and the second is true.
- "There is no air on the moon if and only if it is in Rome, Russia." is a false proposition; because the second of the propositions is false and the first is true.
- "There is no air on the moon if and only if it is in Rome, Italy." is a true proposition; because both propositions are true.
The proposition "if and only" has nothing to do with causes, explanations, or laws of nature.
To solve the second riddle, ask the god "Is Dushanbe in Kyrgyzstan?" Instead of asking a simple question like "Are you the God of Truth if and only if Dushanbe is in Kyrgyzstan?" Ask a complex question like In this case (and in the absence of any geographic information), there are 4 possibilities:
- God is the Righteous god and is in Dushanbe, Kyrgyzstan: In that case, your answer would be "yes".
- God is the Righteous god and not in Dushanbe, Kyrgyzstan: the answer you get then is "no".
- God is the Liar god and is in Dushanbe, Kyrgyzstan: In that case, your answer would be "yes"; because only one of the propositions is true and therefore the correct answer to the whole proposition is "no"; but the Liar god says "yes" to lying.
- God is a Liar god and not in Dushanbe, Kyrgyzstan: then your answer would be "no"; because both propositions are false and therefore the correct answer to the whole proposition is "yes"; but the Liar god says "no" to lying.
In this case, if the answer you get is "Yes", Dushanbe is in Kyrgyzstan. If no, it isn't. It doesn't matter which god you talk to. By jotting down the answer to your complex question, you can find out if Dushanbe is in Kyrgyzstan.
The point to be noted here is that you can talk to the Truthful or Liar god and ask, "Are you the Truthful god if and only if X is?" it is asked. The answer comes in Turkish. Under these conditions, you get "yes" if X is true and "no" if it's false. It doesn't matter which god you talk to.
The solution to riddle 3 is very similar. Ask the righteous god "Is Dushanbe in Kyrgyzstan?" instead of the question "If and only if it is in Dushanbe, Kyrgyzstan, does the word 'da' mean yes?" ask. Again, there are 4 possibilities:
- Da means yes and is in Dushanbe, Kyrgyzstan: in this case the Righteous god will say "da".
- Da means yes, and Dushanbe is not in Kyrgyzstan: the Righteous god would then say "ja" (and "ja" would mean "no").
- Da means no and is in Dushanbe, Kyrgyzstan: the Righteous god will then say "da" (and "da" turns out to be "no").
- Da means no and Dushanbe is not in Kyrgyzstan: in this case both propositions are false, so the answer to the whole proposition would be "yes" in English, so the Righteous god would say "ja".
In this case, if Dushanbe is in Kyrgyzstan you will get the answer "da", otherwise you will get the answer "ja". So by asking this question, you can find out if Dushanbe is in Kyrgyzstan without knowing which of the words "da" or "ja" is yes and which is no.
This time he asked the Righteous god, "If and only if Y means yes too?" The rationale for asking is that if Y is true you get the answer "da", if false you get the answer "ja". It doesn't matter which is which.
Combining these two points, knowing that you will only get the answer "da" or "ja", you can ask one of the Truth or Liar gods the very complex question: "If and only if X is you, the Truthful god; does "da" mean yes if and only if? "
When you ask this question, you will get the answer "da" if and only if X is true, and "ja" if it is false. In this case, it doesn't matter whether the god you ask is a Truth or a Liar. Similarly, it is irrelevant which of the "da" and "ja" answers is which.
Now we can solve the Hardest Logic Problem Ever.
Solving The Hardest Logic Problem Ever
Your first move is to find a god that you are sure is not Random. So you have to identify the Truthful or Liar god.
Move 1: Eliminate the Random God!
To do this, turn to god A and ask Question 1: "If and only if god B is a Random god and you're a Righteous god, does 'also' mean yes if and only if you're a Righteous god?"
If A is either a Truthful or a Liar god, and you get the answer "da", then god B must be a Random god, and therefore C is either a Truthful god or a Liar god.
But if god A is a Truthful or a Liar god and you get the answer "ja" then god B is not a Random god and therefore god B is either a Truthful god or a Liar god.
What if god A is a Random god?
If god A is a Random god, neither god B nor god C can be a Random god.
So if god A is a Random god and you get the answer "da", then god C isn't a random god (neither can be B, but that's trivial). And so god C is either a Truthful god or a Liar god.
If god A is a Random god and you get the answer "ja", then god B cannot be a Random god (and god C cannot be a Random god, but that's irrelevant). Therefore, god B is either a Truthful god or a Liar god.
So, regardless of whether god A is Truthful, Liar, or Random, if the answer you get to your 1st question is "ja", god C is either a Truthful god or a Liar god; If the answer you get to your 1st question is "da," then god B is either a Truthful god or a Liar god.
Move 2: Ask Your Partially Identified God!
Now, whichever you have partially unidentified (either B or C has to be a Truth or a Liar), turn to it. Let's say thanks to the answer to question 1 we realize that god B has to be either a Truth or a Liar (if it's C, change the names of god B to god C from now on).
Now, ask him Question 2: "If and only if Rome is in Italy, does "also" mean yes?"
The truthful god will answer "da", the false god "ja". So with the first two questions, you've determined that god B is the Truth (or you've determined that god B is a Liar). Thus, you have definitively identified a god.
Move 3: Finish With the God You've Completely Unidentified!
From this point on, it's simpler: For your third and final question, turn to the god whose identity you've fully decoded and ask Question 3: "If god A is the god of Random, does 'da' mean yes?" The answers include 4 possibilities and they all get the job done:
Let's say B is the Righteous god. In this case:
- If you get the answer "da" then god A is Random god and hence: god A is random god, B god is Truthful god, god C is Liar god. Job done, you solved the riddle.
- If you get the answer "ja" then god A is not Random god and hence: god A is a Liar god, god B is a Truthful god, god C is a Random god. Job done, you solved the riddle.
Let's say B is the Liar god. In this case:
- If you get the answer "da", since god B is a liar, god A is not a Random god and hence: god A is a Truthful god, god B is a Liar god, god C is a random god. Job done, you solved the riddle.
- If you get the answer "ja", since god B is a liar, god A is a god of Random, and hence: god A is god of Random, god B is a Liar god, god C is a Truthful god. Job done, you solved the riddle.
That's it!
We weren't exaggerating when we said "the hardest logic question ever," are we?
Conclusion and Importance of the Riddle
Before we finish our words, let's briefly touch on the importance of this riddle.
In logic, there is a law of logic called the "law of the excluded middle". Accordingly, either X is true or the proposition "X is not" is true. This applies to all X's. On the other hand, the "Law of Contradiction" says that the propositions X and not X cannot both be true at the same time.
Mathematicians and philosophers occasionally attack the "law of the excluded middle", saying that it is not a valid law. It is not possible for us to conclude this discussion in this article; but it's worth noting that in Riddle 1 we inevitably resort to this law by saying, "Ace, with or without the middle card..."
As we can see from the Hardest Logic Question Ever and our simpler Riddle 1, our ability to grapple with alternative possibilities even in everyday life would be completely paralyzed if it weren't for the "law of the excluded middle."
Oh, by the way... Dushanbe is in Tajikistan. Not in Kyrgyzstan.