full transcript

From the Ted Talk by Alex Gendler: Can you solve the famously difficult green-eyed logic puzzle?

Unscramble the Blue Letters

Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator. There's no escape, except for one strange rule. Any peoisnrr can approach the guards at night and ask to leave. If they have green eyes, they'll be reeelasd. If not, they'll be tesosd into the vonclao. As it happens, all 100 prisoners have green eyes, but they've lived there since birth, and the dictator has ensured they can't learn their own eye color. There are no reflective seuarcfs, all water is in opaque containers, and most importantly, they're not allowed to ctocnumaime among themselves. Though they do see each other during each morning's head count. Nevertheless, they all know no one would ever risk trying to leave without atloubse certainty of success. After much preursse from human rights groups, the dtaocitr reluctantly agrees to let you visit the island and speak to the prisoners under the following conditions: you may only make one statement, and you cannot tell them any new information. What can you say to help free the prisoners without incurring the dictator's wrath? After thinking long and hard, you tell the crowd, "At least one of you has green eyes." The dictator is suspicious but reassures himself that your steentamt couldn't have changed anything. You leave, and life on the iansld seems to go on as before. But on the hundredth moninrg after your visit, all the prisoners are gone, each having asked to leave the previous night. So how did you outsmart the dictator? It might help to realize that the amount of prisoners is arbitrary. Let's simplify things by imagining just two, Adria and Bill. Each sees one prseon with geern eyes, and for all they know, that could be the only one. For the first night, each styas put. But when they see each other still there in the morning, they gain new information. Adria realizes that if Bill had seen a non-green-eyed person next to him, he would have left the first ngiht after concluding the statement could only reefr to himself. Bill simultaneously realizes the same thing about Adria. The fact that the other person waited tells each prisoner his or her own eyes must be green. And on the second morning, they're both gone. Now imagine a third prisoner. Adria, Bill and Carl each see two green-eyed poelpe, but aren't sure if each of the others is also seeing two green-eyed people, or just one. They wait out the first night as before, but the next morning, they still can't be sure. Carl thinks, "If I have non-green eyes, Adria and Bill were just watching each other, and will now both leave on the second night." But when he sees both of them the third morning, he realizes they must have been witahcng him, too. Adria and Bill have each been going through the same process, and they all leave on the third night. Using this sort of ivucnidte reasoning, we can see that the pattern will repeat no matter how many pnorreiss you add. The key is the concept of common knowledge, coined by philosopher divad leiws. The new information was not contained in your statement itself, but in telling it to everyone simultaneously. Now, besides knowing at least one of them has green eyes, each prisoner also knows that everyone else is kienpeg track of all the green-eyed people they can see, and that each of them also knows this, and so on. What any given prisoner doesn't know is whether they themselves are one of the green-eyed people the others are keeping track of until as many nights have passed as the number of prisoners on the island. Of course, you could have spared the prisoners 98 days on the island by telling them at least 99 of you have green eyes, but when mad dictators are involved, you're best off with a good headstart.

Open Cloze

Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator. There's no escape, except for one strange rule. Any ________ can approach the guards at night and ask to leave. If they have green eyes, they'll be ________. If not, they'll be ______ into the _______. As it happens, all 100 prisoners have green eyes, but they've lived there since birth, and the dictator has ensured they can't learn their own eye color. There are no reflective ________, all water is in opaque containers, and most importantly, they're not allowed to ___________ among themselves. Though they do see each other during each morning's head count. Nevertheless, they all know no one would ever risk trying to leave without ________ certainty of success. After much ________ from human rights groups, the ________ reluctantly agrees to let you visit the island and speak to the prisoners under the following conditions: you may only make one statement, and you cannot tell them any new information. What can you say to help free the prisoners without incurring the dictator's wrath? After thinking long and hard, you tell the crowd, "At least one of you has green eyes." The dictator is suspicious but reassures himself that your _________ couldn't have changed anything. You leave, and life on the ______ seems to go on as before. But on the hundredth _______ after your visit, all the prisoners are gone, each having asked to leave the previous night. So how did you outsmart the dictator? It might help to realize that the amount of prisoners is arbitrary. Let's simplify things by imagining just two, Adria and Bill. Each sees one ______ with _____ eyes, and for all they know, that could be the only one. For the first night, each _____ put. But when they see each other still there in the morning, they gain new information. Adria realizes that if Bill had seen a non-green-eyed person next to him, he would have left the first _____ after concluding the statement could only _____ to himself. Bill simultaneously realizes the same thing about Adria. The fact that the other person waited tells each prisoner his or her own eyes must be green. And on the second morning, they're both gone. Now imagine a third prisoner. Adria, Bill and Carl each see two green-eyed ______, but aren't sure if each of the others is also seeing two green-eyed people, or just one. They wait out the first night as before, but the next morning, they still can't be sure. Carl thinks, "If I have non-green eyes, Adria and Bill were just watching each other, and will now both leave on the second night." But when he sees both of them the third morning, he realizes they must have been ________ him, too. Adria and Bill have each been going through the same process, and they all leave on the third night. Using this sort of _________ reasoning, we can see that the pattern will repeat no matter how many _________ you add. The key is the concept of common knowledge, coined by philosopher _____ _____. The new information was not contained in your statement itself, but in telling it to everyone simultaneously. Now, besides knowing at least one of them has green eyes, each prisoner also knows that everyone else is _______ track of all the green-eyed people they can see, and that each of them also knows this, and so on. What any given prisoner doesn't know is whether they themselves are one of the green-eyed people the others are keeping track of until as many nights have passed as the number of prisoners on the island. Of course, you could have spared the prisoners 98 days on the island by telling them at least 99 of you have green eyes, but when mad dictators are involved, you're best off with a good headstart.

Solution

  1. volcano
  2. watching
  3. green
  4. prisoners
  5. released
  6. lewis
  7. keeping
  8. dictator
  9. stays
  10. pressure
  11. prisoner
  12. morning
  13. refer
  14. statement
  15. communicate
  16. island
  17. absolute
  18. person
  19. people
  20. inductive
  21. surfaces
  22. tossed
  23. david
  24. night

Original Text

Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator. There's no escape, except for one strange rule. Any prisoner can approach the guards at night and ask to leave. If they have green eyes, they'll be released. If not, they'll be tossed into the volcano. As it happens, all 100 prisoners have green eyes, but they've lived there since birth, and the dictator has ensured they can't learn their own eye color. There are no reflective surfaces, all water is in opaque containers, and most importantly, they're not allowed to communicate among themselves. Though they do see each other during each morning's head count. Nevertheless, they all know no one would ever risk trying to leave without absolute certainty of success. After much pressure from human rights groups, the dictator reluctantly agrees to let you visit the island and speak to the prisoners under the following conditions: you may only make one statement, and you cannot tell them any new information. What can you say to help free the prisoners without incurring the dictator's wrath? After thinking long and hard, you tell the crowd, "At least one of you has green eyes." The dictator is suspicious but reassures himself that your statement couldn't have changed anything. You leave, and life on the island seems to go on as before. But on the hundredth morning after your visit, all the prisoners are gone, each having asked to leave the previous night. So how did you outsmart the dictator? It might help to realize that the amount of prisoners is arbitrary. Let's simplify things by imagining just two, Adria and Bill. Each sees one person with green eyes, and for all they know, that could be the only one. For the first night, each stays put. But when they see each other still there in the morning, they gain new information. Adria realizes that if Bill had seen a non-green-eyed person next to him, he would have left the first night after concluding the statement could only refer to himself. Bill simultaneously realizes the same thing about Adria. The fact that the other person waited tells each prisoner his or her own eyes must be green. And on the second morning, they're both gone. Now imagine a third prisoner. Adria, Bill and Carl each see two green-eyed people, but aren't sure if each of the others is also seeing two green-eyed people, or just one. They wait out the first night as before, but the next morning, they still can't be sure. Carl thinks, "If I have non-green eyes, Adria and Bill were just watching each other, and will now both leave on the second night." But when he sees both of them the third morning, he realizes they must have been watching him, too. Adria and Bill have each been going through the same process, and they all leave on the third night. Using this sort of inductive reasoning, we can see that the pattern will repeat no matter how many prisoners you add. The key is the concept of common knowledge, coined by philosopher David Lewis. The new information was not contained in your statement itself, but in telling it to everyone simultaneously. Now, besides knowing at least one of them has green eyes, each prisoner also knows that everyone else is keeping track of all the green-eyed people they can see, and that each of them also knows this, and so on. What any given prisoner doesn't know is whether they themselves are one of the green-eyed people the others are keeping track of until as many nights have passed as the number of prisoners on the island. Of course, you could have spared the prisoners 98 days on the island by telling them at least 99 of you have green eyes, but when mad dictators are involved, you're best off with a good headstart.

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
keeping track 2

Important Words

  1. absolute
  2. add
  3. adria
  4. agrees
  5. allowed
  6. amount
  7. approach
  8. arbitrary
  9. asked
  10. bill
  11. birth
  12. carl
  13. certainty
  14. changed
  15. coined
  16. color
  17. common
  18. communicate
  19. concept
  20. concluding
  21. contained
  22. containers
  23. count
  24. crowd
  25. david
  26. days
  27. dictator
  28. dictators
  29. ensured
  30. escape
  31. eye
  32. eyes
  33. fact
  34. free
  35. gain
  36. good
  37. green
  38. groups
  39. guards
  40. hard
  41. head
  42. headstart
  43. human
  44. hundredth
  45. imagine
  46. imagining
  47. importantly
  48. imprisoned
  49. incurring
  50. inductive
  51. information
  52. involved
  53. island
  54. keeping
  55. key
  56. knowing
  57. knowledge
  58. learn
  59. leave
  60. left
  61. lewis
  62. life
  63. lived
  64. logicians
  65. long
  66. mad
  67. matter
  68. morning
  69. night
  70. nights
  71. number
  72. opaque
  73. outsmart
  74. passed
  75. pattern
  76. people
  77. perfect
  78. person
  79. philosopher
  80. pressure
  81. previous
  82. prisoner
  83. prisoners
  84. process
  85. put
  86. realize
  87. realizes
  88. reasoning
  89. reassures
  90. refer
  91. reflective
  92. released
  93. reluctantly
  94. repeat
  95. rights
  96. risk
  97. rule
  98. sees
  99. simplify
  100. simultaneously
  101. sort
  102. spared
  103. speak
  104. statement
  105. stays
  106. strange
  107. success
  108. surfaces
  109. suspicious
  110. telling
  111. tells
  112. thinking
  113. thinks
  114. tossed
  115. track
  116. visit
  117. volcano
  118. wait
  119. waited
  120. watching
  121. water
  122. wrath