full transcript

From the Ted Talk by Alex Gendler: Can you solve the three gods riddle?

Unscramble the Blue Letters

Created by logician Raymond Smullyan and popularized by his colleague George boloos, this riddle has been called the hardest logic puzzle ever. You and your team have crash-landed on an aecnint planet. The only way off is to appease its three alien overlords, Tee, Eff, and Arr, by giving them the correct artifacts. Unfortunately, you don't know who is who. From an inscription, you laern that you may ask three yes or no qitenosus, each addressed to any one lord. Tee's answers are always true, Eff's are always flsae, and Arr's answer is random each time. But there's a problem. You've deciphered the language enough to ask any question, but you don't know which of the two wrods 'ozo' and 'ulu' means yes and which means no. How can you still figure out which alien is which? Pause here if you want to figure it out for yourself! Answer in: 3 2 1 At first, this puzzle seems not just hard, but downright ilsbspoime. What good is asking a question if you can neither understand the answer nor know if it's true? But it can be done. The key is to carefully formulate our questions so that any answer yields useful information. First of all, we can get around to not knowing what 'ozo' and 'ulu' mean by including the words themselves in the questions, and secondly, if we load each question with a hypothetical cntooidin, whether an alien is lying or not won't actually matter. To see how that could work, iginame our question is whether two plus two is four. Instead of posing it directly, we say, "If I akesd you whether two plus two is four, would you answer 'ozo'?" If 'ozo' means yes and the overlord is Tee, it truthfully replies, "ozo." But what if we ask Eff? Well, it would answer "ulu," or no to the embedded question, so it lies and replies 'ozo' instead. And if 'ozo' actually means no, then the aeswnr to our ebeedmdd question is 'ulu,' and both Tee and Eff still reply 'ozo,' each for their own reasons. If you're confused about why this works, the reason iveonvls logical srturtuce. A double positive and a double negative both result in a ptviiose. Now, we can be sure that asking either Tee or Eff a question put this way will yield 'ozo' if the hoattpyceihl question is true and 'ulu' if it's false regardless of what each word actually means. Unfortunately, this doesn't help us with Arr. But don't worry, we can use our first question to idnitfey one alien lord that definitely isn't Arr. Then we can use the second to find out whether its Tee or Eff. And once we know that, we can ask it to identify one of the others. So let's begin. Ask the alien in the middle, "If I asked you whether the overlord on my left is Arr, would you answer 'ozo'?" If the reply is 'ozo,' there are two possibilities. You could already be talking to Arr, in which case the answer is meaningless. But otherwise, you're talking to either Tee or Eff, and as we know, getting 'ozo' from either one means your hypothetical qsieoutn was croerct, and the left overlord is indeed Arr. Either way, you can be sure the aieln on the right is not Arr. slimarliy, if the answer is 'ulu,' then you know the alien on the left can't be Arr. Now go to the overlord you've determined isn't Arr and ask, "If I asked 'are you Eff?' would you answer 'ozo'?" Since you don't have to worry about the random possibility, either answer will establish its identity. Now that you know whether its answers are true or false, ask the same alien whether the center overlord is Arr. The process of elimination will identify the remaining one. The satisfied overlords help you repair your ship and you prepare for takeoff. Allowed one final question, you ask Tee if it's a long way to Earth, and he answers "ozo." Too bad you still don't know what that means.

Open Cloze

Created by logician Raymond Smullyan and popularized by his colleague George ______, this riddle has been called the hardest logic puzzle ever. You and your team have crash-landed on an _______ planet. The only way off is to appease its three alien overlords, Tee, Eff, and Arr, by giving them the correct artifacts. Unfortunately, you don't know who is who. From an inscription, you _____ that you may ask three yes or no _________, each addressed to any one lord. Tee's answers are always true, Eff's are always _____, and Arr's answer is random each time. But there's a problem. You've deciphered the language enough to ask any question, but you don't know which of the two _____ 'ozo' and 'ulu' means yes and which means no. How can you still figure out which alien is which? Pause here if you want to figure it out for yourself! Answer in: 3 2 1 At first, this puzzle seems not just hard, but downright __________. What good is asking a question if you can neither understand the answer nor know if it's true? But it can be done. The key is to carefully formulate our questions so that any answer yields useful information. First of all, we can get around to not knowing what 'ozo' and 'ulu' mean by including the words themselves in the questions, and secondly, if we load each question with a hypothetical _________, whether an alien is lying or not won't actually matter. To see how that could work, _______ our question is whether two plus two is four. Instead of posing it directly, we say, "If I _____ you whether two plus two is four, would you answer 'ozo'?" If 'ozo' means yes and the overlord is Tee, it truthfully replies, "ozo." But what if we ask Eff? Well, it would answer "ulu," or no to the embedded question, so it lies and replies 'ozo' instead. And if 'ozo' actually means no, then the ______ to our ________ question is 'ulu,' and both Tee and Eff still reply 'ozo,' each for their own reasons. If you're confused about why this works, the reason ________ logical _________. A double positive and a double negative both result in a ________. Now, we can be sure that asking either Tee or Eff a question put this way will yield 'ozo' if the ____________ question is true and 'ulu' if it's false regardless of what each word actually means. Unfortunately, this doesn't help us with Arr. But don't worry, we can use our first question to ________ one alien lord that definitely isn't Arr. Then we can use the second to find out whether its Tee or Eff. And once we know that, we can ask it to identify one of the others. So let's begin. Ask the alien in the middle, "If I asked you whether the overlord on my left is Arr, would you answer 'ozo'?" If the reply is 'ozo,' there are two possibilities. You could already be talking to Arr, in which case the answer is meaningless. But otherwise, you're talking to either Tee or Eff, and as we know, getting 'ozo' from either one means your hypothetical ________ was _______, and the left overlord is indeed Arr. Either way, you can be sure the _____ on the right is not Arr. _________, if the answer is 'ulu,' then you know the alien on the left can't be Arr. Now go to the overlord you've determined isn't Arr and ask, "If I asked 'are you Eff?' would you answer 'ozo'?" Since you don't have to worry about the random possibility, either answer will establish its identity. Now that you know whether its answers are true or false, ask the same alien whether the center overlord is Arr. The process of elimination will identify the remaining one. The satisfied overlords help you repair your ship and you prepare for takeoff. Allowed one final question, you ask Tee if it's a long way to Earth, and he answers "ozo." Too bad you still don't know what that means.

Solution

  1. involves
  2. question
  3. questions
  4. hypothetical
  5. correct
  6. boolos
  7. asked
  8. impossible
  9. answer
  10. condition
  11. embedded
  12. alien
  13. positive
  14. learn
  15. words
  16. imagine
  17. false
  18. ancient
  19. similarly
  20. identify
  21. structure

Original Text

Created by logician Raymond Smullyan and popularized by his colleague George Boolos, this riddle has been called the hardest logic puzzle ever. You and your team have crash-landed on an ancient planet. The only way off is to appease its three alien overlords, Tee, Eff, and Arr, by giving them the correct artifacts. Unfortunately, you don't know who is who. From an inscription, you learn that you may ask three yes or no questions, each addressed to any one lord. Tee's answers are always true, Eff's are always false, and Arr's answer is random each time. But there's a problem. You've deciphered the language enough to ask any question, but you don't know which of the two words 'ozo' and 'ulu' means yes and which means no. How can you still figure out which alien is which? Pause here if you want to figure it out for yourself! Answer in: 3 2 1 At first, this puzzle seems not just hard, but downright impossible. What good is asking a question if you can neither understand the answer nor know if it's true? But it can be done. The key is to carefully formulate our questions so that any answer yields useful information. First of all, we can get around to not knowing what 'ozo' and 'ulu' mean by including the words themselves in the questions, and secondly, if we load each question with a hypothetical condition, whether an alien is lying or not won't actually matter. To see how that could work, imagine our question is whether two plus two is four. Instead of posing it directly, we say, "If I asked you whether two plus two is four, would you answer 'ozo'?" If 'ozo' means yes and the overlord is Tee, it truthfully replies, "ozo." But what if we ask Eff? Well, it would answer "ulu," or no to the embedded question, so it lies and replies 'ozo' instead. And if 'ozo' actually means no, then the answer to our embedded question is 'ulu,' and both Tee and Eff still reply 'ozo,' each for their own reasons. If you're confused about why this works, the reason involves logical structure. A double positive and a double negative both result in a positive. Now, we can be sure that asking either Tee or Eff a question put this way will yield 'ozo' if the hypothetical question is true and 'ulu' if it's false regardless of what each word actually means. Unfortunately, this doesn't help us with Arr. But don't worry, we can use our first question to identify one alien lord that definitely isn't Arr. Then we can use the second to find out whether its Tee or Eff. And once we know that, we can ask it to identify one of the others. So let's begin. Ask the alien in the middle, "If I asked you whether the overlord on my left is Arr, would you answer 'ozo'?" If the reply is 'ozo,' there are two possibilities. You could already be talking to Arr, in which case the answer is meaningless. But otherwise, you're talking to either Tee or Eff, and as we know, getting 'ozo' from either one means your hypothetical question was correct, and the left overlord is indeed Arr. Either way, you can be sure the alien on the right is not Arr. Similarly, if the answer is 'ulu,' then you know the alien on the left can't be Arr. Now go to the overlord you've determined isn't Arr and ask, "If I asked 'are you Eff?' would you answer 'ozo'?" Since you don't have to worry about the random possibility, either answer will establish its identity. Now that you know whether its answers are true or false, ask the same alien whether the center overlord is Arr. The process of elimination will identify the remaining one. The satisfied overlords help you repair your ship and you prepare for takeoff. Allowed one final question, you ask Tee if it's a long way to Earth, and he answers "ozo." Too bad you still don't know what that means.

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
hypothetical question 2

Important Words

  1. addressed
  2. alien
  3. allowed
  4. ancient
  5. answer
  6. answers
  7. appease
  8. arr
  9. artifacts
  10. asked
  11. bad
  12. boolos
  13. called
  14. carefully
  15. case
  16. center
  17. colleague
  18. condition
  19. confused
  20. correct
  21. created
  22. deciphered
  23. determined
  24. double
  25. downright
  26. earth
  27. eff
  28. elimination
  29. embedded
  30. establish
  31. false
  32. figure
  33. final
  34. find
  35. formulate
  36. george
  37. giving
  38. good
  39. hard
  40. hardest
  41. hypothetical
  42. identify
  43. identity
  44. imagine
  45. impossible
  46. including
  47. information
  48. inscription
  49. involves
  50. key
  51. knowing
  52. language
  53. learn
  54. left
  55. lies
  56. load
  57. logic
  58. logical
  59. logician
  60. long
  61. lord
  62. lying
  63. matter
  64. meaningless
  65. means
  66. middle
  67. negative
  68. overlord
  69. overlords
  70. pause
  71. planet
  72. popularized
  73. posing
  74. positive
  75. possibilities
  76. possibility
  77. prepare
  78. problem
  79. process
  80. put
  81. puzzle
  82. question
  83. questions
  84. random
  85. raymond
  86. reason
  87. reasons
  88. remaining
  89. repair
  90. replies
  91. reply
  92. result
  93. riddle
  94. satisfied
  95. ship
  96. similarly
  97. smullyan
  98. structure
  99. takeoff
  100. talking
  101. team
  102. tee
  103. time
  104. true
  105. truthfully
  106. understand
  107. word
  108. words
  109. work
  110. works
  111. worry
  112. yield
  113. yields