How to Solve the Hardest Logic Puzzle Ever

While a doctoral student at Princeton University, American mathematician, Raymond Smullyan would occasionally visit New York City. On one such visit, he met a “very charming lady musician.” On their first date, Smullyan, proceeded to flirt via a logic puzzle.

“Would you please do me a favor?” he asked her. “I am to make a statement. If the statement is true, would you give me your autograph?”

She replied, “I don’t see why not.”

“If the statement is false,” he continued, “you don’t give me your autograph.”

“Alright …”

His statement: “You’ll give me neither your autograph nor a kiss.”

Clever. A truthful statement gets him an autograph, but if the statement is true, it leads to a contradiction. The giving of an autograph is therefore ruled out, making the statement false. And if the statement is false, then the charming lady musician must give him either an autograph or a kiss. But there’s the trap, she’s already agreed not to reward a false statement with an autograph. With logic, Smullyan turned a false statement into a kiss. The two would eventually marry.

It is this sort of logical playfulness that Smullyan is well-known for. His books on math and logic, with titles like What Is the Name of This Book? and To Mock a Mockingbird, are not only entertaining, they also changed how math and logic are taught.

Smullyan’s legacy is vast, but one of his most storied contributions is his most difficult puzzle, known as the Hardest Logic Puzzle Ever.

The Hardest Logic Puzzle Ever goes like this:

Three gods A, B, and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. 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.

I tried my hand at the puzzle for nearly two hours to no avail. Clearly, my questions weren’t compelling the gods to answer.

In search of enlightenment, I found the solution by George Boolos of MIT, who solved the puzzle in 1996. How he did it turns out to be a fantastic lesson in logic and truth. You can read his solution in The Harvard Review of Philosophy.

If you’re a more visual and auditory person, Alex Gendler shows how to solve it in a TedEd talk below:

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