THE’Riddle of the troll who hates paradoxes It’s a logic game that requires a good deal of cunning. You are in this situation: you are walking in a forest with your brother and sisterwhen suddenly a troll appears, who he captures them both. The troll tells you that he will keep them prisoner forever, but makes you a proposal: you can try to save them by speaking just one sentence.
However, it gives you some very specific rules:
- if you say something about True, will free yours sister;
- if you say something about false, will free your Brother;
- if you say a sentence paradoxical, will keep them both prisoners. If, for example, I said “I’m lying”, his brothers would be doomed. This, in fact, is a paradoxical sentence: if it is true then it is false, if it is false then it is true. If the person who says it is not lying, then his sentence should be true, but he himself tells us that he is lying! If the person who says it is lying, then the sentence “I am lying” should be false and therefore should be telling the truth. If you gave yourself a headache trying to understand it, it’s because this is a logical paradox: this sentence can be neither true nor false.
The troll who captured our brothers he hates paradoxes more than anything else in the world and would do anything to avoid them. What can you say to convince him to release both of them? Let’s see the solution together.
The solution to the riddle
At first glance, the riddle seems impossible. As you can save both saying a sentence that is both true and false, without however falling into a paradox?
For example, if I said:
You will free both my sister and my brother
the troll might reply: “It’s false, I will only free your brother”. Your sentence would therefore be false and according to the rules he gave you earlier he should only free your brother and could keep your sister prisoner.
If instead I said:
You will free my sister
he would say it’s true, based on the rules he gave you there would liberate, but would keep your brother prisoner.
There seems to be no way out. And yet, incredibly, there is one phrase that forces the troll to free both prisoners.
The sentence is:
You will free my brother.
Let’s see why it works.
Suppose this sentence is false. According to the troll’s rules, if you say something false he has to release your brother. But in doing so, the phrase “You will free my brother” would immediately become true (since your brother was freed), even though it was considered false. This would create a paradox, but the troll hates paradoxes!
So, to avoid creating a paradox himself, the troll is forced to consider the sentence as true. If the sentence is true, however, “You will free my brother” must actually come true: the troll must free your brother. Plus, by its own rules, if the sentence you said is true, it must free your sister too.
By doing this, using just three words, you have forced the troll to free both your brother and sister.
This strategy is an example of coercive logica concept popularized by the great logician and puzzle creator Raymond Smullyan (author, among others, of this riddle and the famous “hardest puzzle in the world”). In coercive logic, a person is forced to perform an unwanted action (in this case the troll is forced to free both our sister and our brother, even though he wants to keep them both prisoner) because not doing so would lead him to violate the rules of the game that he has agreed to respect. In our case, the final paradox is that not only did the troll accept the rules of the game, but he even created them himself, screwing himself.
