The Blue-Eyed Prisoners Puzzle: A Mind-Bending Logic Challenge
Discover the captivating logic puzzle where blue-eyed prisoners on an island must deduce their own eye color to escape. Explore how common knowledge and inductive reasoning lead to an astonishing outcome.
A ruthless dictator holds 100 prisoners on a remote island. Escape is impossible, but there’s a peculiar rule: each night, any prisoner may request release. If the prisoner has blue eyes, they are freed; if not, they face a deadly fate.
Unbeknownst to the prisoners, all 100 have blue eyes. Raised on the island with no mirrors or reflective surfaces, none can see their own eye color. Communication is strictly forbidden—no talking, gestures, or messages—except for a daily roll call where all prisoners are present.
Each prisoner is perfectly logical and will only risk asking for freedom if absolutely certain of success.
One day, the dictator falls for a truthful young woman who convinces him to let her visit the island and speak to the prisoners. However, she may make only one statement and cannot reveal any new information.
Aware of the situation, she boldly announces to the gathered prisoners: “At least one of you has blue eyes.” Then she leaves.
The dictator believes this statement harmless and expects no change. But after 100 days, the island is empty—all prisoners have requested release and departed. How did this happen? Remember, every prisoner is a flawless logician.
To understand, imagine only two prisoners: Andrew and Maria. Each sees the other has blue eyes but is unsure if they themselves do. On the first night, neither leaves. When morning comes and both are still present, each deduces: if my eyes weren’t blue, the other would have left that night. Since they didn’t, both realize their eyes must be blue and leave the next morning.
Extend this to three prisoners: Andrew, Maria, and Boris. Each sees two with blue eyes but doesn’t know if the others see one or two. They wait the first night, but no one leaves. Boris reasons: if my eyes aren’t blue, Andrew and Maria would leave on the second night. When they don’t, Boris concludes his eyes are blue. Andrew and Maria think similarly, so on the third night, all three leave.
This reasoning continues inductively—if there were four prisoners, they’d leave on the fourth night; five on the fifth; and 100 on the hundredth.
The key concept is common knowledge: everyone knows the statement, everyone knows that everyone knows it, and so on infinitely. The woman’s announcement doesn’t add new facts but establishes common knowledge, enabling the prisoners to deduce their own eye color through collective reasoning.
Thus, the prisoners’ logical minds lead them to freedom after 100 days. This puzzle beautifully illustrates the power of shared information and inductive logic.
Based on a TED-Ed video.
Did you solve the puzzle on your own? Share your thoughts in the comments!
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