The Hardest Logic Puzzle Ever facts for kids

The Hardest Logic Puzzle Ever is a famous logic puzzle first shared by an American philosopher named George Boolos in 1996. It was published in The Harvard Review of Philosophy. Boolos also showed different ways to solve this tricky puzzle. An Italian version of the puzzle appeared earlier in a newspaper called La Repubblica.

Here is how the puzzle is set up:

Three gods, A, B, and C, are named True, False, and Random. You don't know which god has which name.

• True always tells the truth.
• False always lies.
• Random might tell the truth or lie, it's completely up to chance.

Your goal is to figure out which god is A, which is B, and which is C. You can ask only three yes-no questions. Each question must be asked to just one god. The gods understand English, but they will answer in their own language. Their words for yes and no are da and ja. You do not know which word means yes and which means no.

Boolos added some important rules:

• You can ask the same god more than one question.
• Your next question can depend on the answer you just got.
• Random's answers are like flipping a fair coin in their mind. If it's "heads," Random tells the truth. If it's "tails," Random lies.

Solving the Puzzle

Boolos shared his solution in the same article where he introduced the puzzle. He said the first step is to find a god you know for sure is not Random. This god must be either True or False. There are many ways to do this. One way is to use very complex logical questions.

Boolos's original question to God A was:

Does da mean yes if and only if you are True, if and only if B is Random?

This question is quite complicated!

Later, in 2001 and 2008, other thinkers found a simpler way to solve the puzzle. They used a special type of question called a "counterfactual." This means asking a "what if" question.

The key idea is this: If you ask either True or False a question like this:

If I asked you Q, would you say ja?

You will get ja if the real answer to Q is yes. You will get da if the real answer to Q is no. This works no matter if the god is True or False, and no matter if ja means yes or no. This makes it much easier to get a straight answer!

Here's how this "what if" trick works in simple terms:

• Imagine ja means yes and da means no.

* If True is asked and says ja, it means the answer to Q is really yes. * If True is asked and says da, it means the answer to Q is really no. * If False is asked and says ja, it means they are lying. So, they would actually say da if asked Q directly. Since they lie, the real answer to Q is yes. * If False is asked and says da, it means they are lying. So, they would actually say ja if asked Q directly. Since they lie, the real answer to Q is no.

• Now imagine ja means no and da means yes.

* If True is asked and says ja, it means the answer to Q is really da, which means yes. * If True is asked and says da, it means the answer to Q is really ja, which means no. * If False is asked and says ja, it means they are lying. So, they would actually say ja if asked Q directly. Since they lie, the real answer to Q is da, which means yes. * If False is asked and says da, it means they are lying. So, they would actually say da if asked Q directly. Since they lie, the real answer to Q is ja, which means no.

As you can see, no matter what, the answer you get (ja or da) tells you the true answer to Q.

Here are the three questions to solve the puzzle using this trick:

• Question 1: Ask god B, "If I asked you 'Is A Random?', would you say ja?"

* If B answers ja: This means A is Random. So, C must be either True or False (not Random). * If B answers da: This means A is not Random. So, A must be either True or False. * After this question, you will always know one god who is definitely not Random. Let's call this god "X".

• Question 2: Go to god X (the one you know is not Random). Ask them: "If I asked you 'Are you False?', would you say ja?"

* Since X is not Random, they are either True or False. * If X answers da: This means X is True. * If X answers ja: This means X is False. * Now you know exactly who god X is!

• Question 3: Ask the same god X (whose identity you now know): "If I asked you 'Is B Random?', would you say ja?"

* If X answers ja: This means B is Random. * If X answers da: This means C is Random (because A and B are not Random). * Once you know who Random is, you can figure out the last god by simply eliminating the others.

This method helps you find out who each god is in just three questions!

Random's Behavior

Boolos explained how Random acts: Random flips a coin in their mind. Heads means they tell the truth, tails means they lie.

Some people wondered if Random flips the coin for each question, or just once for the whole puzzle. If Random only flips the coin once at the start, the puzzle could be solved in just two questions. This is because the "what if" questions would still work to get a clear answer from Random, whether they decided to be a truth-teller or a liar for the whole session.

Another idea about Random is that they decide to lie or tell the truth before they figure out the answer to the question. If this is true, you could change the "what if" question slightly to:

If I asked you Q in your current mental state, would you say ja?

This change would force Random to act like either True or False, making the puzzle much easier. However, this assumes Random makes their decision about lying or telling the truth *before* thinking about the question, which wasn't clearly stated in the original puzzle rules.

A different way to get clear answers, without making extra assumptions about Random, is to ask:

If I asked you Q, and if you were answering as truthfully as you are answering this question, would you say ja?

This question works because it only relies on Boolos's original rule: Random either speaks truthfully or falsely when answering. This makes the "Hardest Logic Puzzle Ever" have a surprisingly simple solution.

Some experts, like Rabern and Rabern, suggested changing the puzzle slightly to make Random truly random. They said Random's coin flip should decide if they say ja or da, not if they tell the truth or lie. If this change is made, then the more careful three-question solution described above is needed.

Unanswerable Questions and Exploding God-Heads

In a different version of the puzzle, if a god is asked a question that creates a paradox (a statement that cannot be true or false), they won't answer at all. Instead, their "head explodes" (meaning they just can't respond). For example, if you ask True, "Are you going to answer this question with the word that means no in your language?", True cannot answer truthfully.

This "exploding head" idea can help solve the puzzle in just two questions instead of three.

Here's a simpler puzzle that shows this idea:

Three gods A, B, and C are named Zephyr, Eurus, and Aeolus. You don't know which god is which. These gods always tell the truth. Your task is to find out their identities by asking yes-no questions. Each question must go to only one god. The gods understand English and answer in English.

This simpler puzzle can easily be solved in three questions. But with the "exploding head" rule, you can solve it in two. For example, if you ask A: "Is it true that {[(you will answer 'no' to this question) AND (B is Zephyr)] OR (B is Eurus)}?", then:

• A "yes" answer means B is Eurus.
• A "no" answer means B is Aeolus.
• An "exploding head" means B is Zephyr.

This way, you find out B's identity in just one question!

Similar tricks can solve the original "Hardest Logic Puzzle Ever" in two questions. True and False gods cannot answer certain questions because they are forced to either tell the truth or lie. For example, they cannot answer:

Would you answer the same as Random would to the question 'Is Dushanbe in Kirghizia?'?

Since True and False don't know how Random will answer (because Random is truly random), they can't truthfully say if they would answer the same. So, they remain silent. Random, however, will just randomly say ja or da. This difference helps you identify Random.

Another tricky question that True and False cannot answer is:

Would you answer ja to the question of whether you would answer da to this question?

True and False cannot answer this because it creates a paradox for them. They would be forced to say ja if they were supposed to say da, and vice versa. This makes their "heads explode." Random, however, will just randomly blurt out ja or da. This difference can also be used to solve the puzzle in two questions.

See also

In Spanish: El acertijo lógico más difícil para niños