Can you solve the prisoner hat riddle? - Alex Gendler

TED-Ed

TED-Ed

4 min, 35 sec

Ten captives must pass an alien test of logic to avoid being eaten by determining the colors of their hats.

Summary

  • The alien overlords have challenged ten humans with a test to determine if they are logical and cooperative enough to avoid being eaten.
  • The captives are lined up in size order and each given a black or white hat; they must guess the color of their own hat without any direct clues.
  • The first person in the line communicates the parity of black hats he sees to help others deduce their hat colors.
  • By using the parity information, each subsequent captive can deduce the color of their hat, with the group only allowed one incorrect guess.

Chapter 1

The Alien Challenge

0:10 - 19 sec

Captives are presented with a logic test by aliens who cannot eat highly logical beings.

Captives are presented with a logic test by aliens who cannot eat highly logical beings.

  • Ten individuals are captured by aliens who find humans tasty but cannot consume logical beings.
  • The aliens are uncertain about humans' logical abilities and offer a test to prove their capacity for logic and cooperation.

Chapter 2

Test Instructions

0:32 - 59 sec

The aliens explain the rules of the test that will determine the captives' fate.

The aliens explain the rules of the test that will determine the captives' fate.

  • The captives will be lined up in size order, unable to see behind them or leave the line.
  • They will each have a randomly assigned black or white hat and must guess its color when prompted, without using signals or words other than 'black' or 'white'.
  • If at least nine captives guess correctly, they will be spared; otherwise, they will be eaten.

Chapter 3

Planning the Strategy

1:31 - 25 sec

The captives must devise a strategy that uses limited communication to save themselves.

The captives must devise a strategy that uses limited communication to save themselves.

  • The captives have five minutes to discuss a plan using the limited communication allowed by the aliens.
  • The key to the strategy lies in the first person's ability to see all other hats and communicate coded information.

Chapter 4

Decoding the Solution

1:56 - 31 sec

The solution centers on using the parity of visible black hats to inform the others.

The solution centers on using the parity of visible black hats to inform the others.

  • The solution is to use the first person's statement to indicate the parity of the number of black hats he sees.
  • By saying 'black' or 'white', the first person conveys whether the number of black hats ahead is odd or even, allowing others to deduce their hat color.

Chapter 5

Practical Application of the Strategy

2:27 - 57 sec

The captives apply the agreed-upon strategy to determine their hat colors.

The captives apply the agreed-upon strategy to determine their hat colors.

  • The tallest captive starts by indicating the parity he observes, sacrificing his chance to guess correctly.
  • Each captive uses the parity information and the previous guesses to deduce the color of their own hat.

Chapter 6

Outcome and Implications

3:25 - 48 sec

The strategy ensures the captives' survival, demonstrating their logical prowess.

The strategy ensures the captives' survival, demonstrating their logical prowess.

  • The strategy proves successful as it allows each captive, except the first, to guess their hat color with certainty.
  • The parity rule ensures that the aliens cannot eat the captives due to their logical and cooperative behavior.

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