The Troll’s Paradox Riddle

You and your brother have discovered another realm and set off exploring the new wonderful world. Along the way, you see a troll catching creatures in an enormous net. The troll agrees to release the creatures if you can come up with a statement that is both truth and false. Can you come up with the correct sentence and force the troll to release them? Dan Finkel shows how.

Transcript:

You’ve discovered a doorway to another realm, and now you and your brother are off exploring the wonderful world of Paradoxica. Fantastically paradoxical creatures crawl, run, and fly around you. And then you see the troll.

It’s catching all the creatures in an enormous net. You bravely step forward and demand it let them go. The troll laughs. “If you’re such a fan of paradoxes,” it says, “then I’ll make you an offer. If you say something true, I’ll release all these creatures.” You’re about to say, “You are a troll,” but before you can, the troll grabs your brother. “If you say something false,” he continues, “then I’ll release your brother.” Your statement can only be a single sentence. And as you can see, I hate paradoxes more than anything. If you try to cheat by saying something paradoxical, like, ‘this statement is false,’ then I’ll eat your brother and the creatures.”

What true/false statement can you say to force the troll to free your brother and the paradoxical creatures?

This seems like an impossible situation, but incredibly, you can say something that will force the troll to release all its prisoners. This is an example of coercive logic, invented by the great logician and puzzle creator Raymond Smullyan.

The trick Smullyan came up with involves saying a statement whose truth or falseness depends on what you want the troll to do.

Your statement still has to be carefully crafted. For example, if you were to say, “You are going to free the creatures and my brother,” the troll could respond, “that’s false… I’m only going to free your brother.” Similarly, if you said, “You will free the paradoxes,” the troll could say, “That’s true,” and free the paradoxes.

But watch what happens if you say, “You will free my brother.”

The statement can’t be false, because if it were, the troll, by its own rules, would have to free your brother. That would make the statement paradoxically true and false. But the troll hates paradoxes and would never willingly create one. So his only option is for the statement to be true. If “you will free my brother” is true, then the troll has to release your brother. And by its own rules, the troll has to free the creatures as well, since you said a true statement. By wielding just 5 words like a logical scalpel, you’ve forced the troll to free all its prisoners.

As the troll stomps off in anger, the paradoxes cheer you for winning them their freedom, and promise to lead you to the treasure at the top of the stairs. If you can reach it.

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