full transcript

From the Ted Talk by Yossi Elran: Can you solve the prisoner boxes riddle?

Unscramble the Blue Letters

Your ftorivae band is great at playing music, but not so great at being onraiezgd. They keep misplacing their instruments on tour, and it's driving their manager mad. On the day of the big crnocet, the band wakes up to find themselves tied up in a wnoswldeis, soundproof practice room. Their manager enalpixs what's happening. Outside, there are ten large boxes. Each contains one of your instruments, but don't be felood by the pictures - they've been randomly placed. I'm going to let you out one at a time. While you're outside, you can look inside any five beoxs before security takes you back to the tour bus. You can't touch the instruments or in any way communicate what you find to the others. No marking the boxes, shouting, nothing. If each one of you can find your own instrument, then you can play tonight. Otherwise, the laebl is dropping you. You have three mtieuns to think about it before we start. The band is in despair. After all, each musician only has a 50% chance of finding their instrument by picking five random boxes. And the cacnhes that all ten will succeed are even lower - just 1 in 1024. But snddleuy, the drummer comes up with a valid strategy that has a better than 35% chnace of working. Can you figure out what it was? Pause the video on the next screen if you want to figure it out for yourself! Answer in: 3 Answer in: 2 awensr in: 1 Here's what the drummer said: Everyone first open the box with the pirtuce of your instrument. If your instrument is inside, you're done. Otherwise, look at whatever's in there, and then open the box with that picture on it. Keep going that way until you find your instrument. The bandmates are skeptical, but almnaizgy enough, they all find what they need. And a few hours later, they're playing to thousands of adoring fans. So why did the drummer's strategy work? Each musician follows a linked sequence that starts with the box whose outside matches their instrument and ends with the box actually containing it. Note that if they kept going, that would lead them back to the start, so this is a loop. For example, if the boxes are aerarngd like so, the singer would open the first box to find the drums, go to the eighth box to find the bass, and find her mrchiopnoe in the third box, which would point back to the first. This works much better than random ginssueg because by sitrtnag with the box with the picture of their instrument, each musician restricts their search to the loop that contains their instrument, and there are decent odds, about 35%, that all of the loops will be of length five or less. How do we calculate those odds? For the sake of simplicity, we'll derattsonme with a simplified case, four instruments and no more than two gssuees allowed for each musician. Let's start by finding the odds of failure, the chance that someone will need to open three or four boxes before they find their instrument. There are six distinct four-box loops. One fun way to conut them is to make a square, put an instrument at each corner, and draw the diagonals. See how many unique lopos you can find, and keep in mind that these two are considered the same, they just start at different points. These two, however, are different. We can visualize the eight dtinicst three-box loops using triangles. You'll find four possible triangles depending on which instrument you leave out, and two distinct paths on each. So of the 24 possible caiotbnnomis of boxes, there are 14 that lead to faliure, and ten that result in seccsus. That computational strategy wkros for any even number of musicians, but if you want a sctuhrot, it gielezrnaes to a handy equation. Plug in ten musicians, and we get odds of about 35%. What if there were 1,000 musicians? 1,000,000? As n iecnesras, the odds approach about 30%. Not a guarantee, but with a bit of musician's luck, it's far from hsoeleps. Hi everybody, if you liked this riddle, try solving these two.

Open Cloze

Your ________ band is great at playing music, but not so great at being _________. They keep misplacing their instruments on tour, and it's driving their manager mad. On the day of the big _______, the band wakes up to find themselves tied up in a __________, soundproof practice room. Their manager ________ what's happening. Outside, there are ten large boxes. Each contains one of your instruments, but don't be ______ by the pictures - they've been randomly placed. I'm going to let you out one at a time. While you're outside, you can look inside any five _____ before security takes you back to the tour bus. You can't touch the instruments or in any way communicate what you find to the others. No marking the boxes, shouting, nothing. If each one of you can find your own instrument, then you can play tonight. Otherwise, the _____ is dropping you. You have three _______ to think about it before we start. The band is in despair. After all, each musician only has a 50% chance of finding their instrument by picking five random boxes. And the _______ that all ten will succeed are even lower - just 1 in 1024. But ________, the drummer comes up with a valid strategy that has a better than 35% ______ of working. Can you figure out what it was? Pause the video on the next screen if you want to figure it out for yourself! Answer in: 3 Answer in: 2 ______ in: 1 Here's what the drummer said: Everyone first open the box with the _______ of your instrument. If your instrument is inside, you're done. Otherwise, look at whatever's in there, and then open the box with that picture on it. Keep going that way until you find your instrument. The bandmates are skeptical, but _________ enough, they all find what they need. And a few hours later, they're playing to thousands of adoring fans. So why did the drummer's strategy work? Each musician follows a linked sequence that starts with the box whose outside matches their instrument and ends with the box actually containing it. Note that if they kept going, that would lead them back to the start, so this is a loop. For example, if the boxes are ________ like so, the singer would open the first box to find the drums, go to the eighth box to find the bass, and find her __________ in the third box, which would point back to the first. This works much better than random ________ because by ________ with the box with the picture of their instrument, each musician restricts their search to the loop that contains their instrument, and there are decent odds, about 35%, that all of the loops will be of length five or less. How do we calculate those odds? For the sake of simplicity, we'll ___________ with a simplified case, four instruments and no more than two _______ allowed for each musician. Let's start by finding the odds of failure, the chance that someone will need to open three or four boxes before they find their instrument. There are six distinct four-box loops. One fun way to _____ them is to make a square, put an instrument at each corner, and draw the diagonals. See how many unique _____ you can find, and keep in mind that these two are considered the same, they just start at different points. These two, however, are different. We can visualize the eight ________ three-box loops using triangles. You'll find four possible triangles depending on which instrument you leave out, and two distinct paths on each. So of the 24 possible ____________ of boxes, there are 14 that lead to faliure, and ten that result in _______. That computational strategy _____ for any even number of musicians, but if you want a ________, it ___________ to a handy equation. Plug in ten musicians, and we get odds of about 35%. What if there were 1,000 musicians? 1,000,000? As n _________, the odds approach about 30%. Not a guarantee, but with a bit of musician's luck, it's far from ________. Hi everybody, if you liked this riddle, try solving these two.

Solution

  1. count
  2. organized
  3. combinations
  4. loops
  5. microphone
  6. chances
  7. guessing
  8. distinct
  9. label
  10. concert
  11. chance
  12. starting
  13. guesses
  14. explains
  15. favorite
  16. generalizes
  17. hopeless
  18. windowless
  19. shortcut
  20. demonstrate
  21. amazingly
  22. arranged
  23. minutes
  24. picture
  25. boxes
  26. answer
  27. suddenly
  28. fooled
  29. works
  30. success
  31. increases

Original Text

Your favorite band is great at playing music, but not so great at being organized. They keep misplacing their instruments on tour, and it's driving their manager mad. On the day of the big concert, the band wakes up to find themselves tied up in a windowless, soundproof practice room. Their manager explains what's happening. Outside, there are ten large boxes. Each contains one of your instruments, but don't be fooled by the pictures - they've been randomly placed. I'm going to let you out one at a time. While you're outside, you can look inside any five boxes before security takes you back to the tour bus. You can't touch the instruments or in any way communicate what you find to the others. No marking the boxes, shouting, nothing. If each one of you can find your own instrument, then you can play tonight. Otherwise, the label is dropping you. You have three minutes to think about it before we start. The band is in despair. After all, each musician only has a 50% chance of finding their instrument by picking five random boxes. And the chances that all ten will succeed are even lower - just 1 in 1024. But suddenly, the drummer comes up with a valid strategy that has a better than 35% chance of working. Can you figure out what it was? Pause the video on the next screen if you want to figure it out for yourself! Answer in: 3 Answer in: 2 Answer in: 1 Here's what the drummer said: Everyone first open the box with the picture of your instrument. If your instrument is inside, you're done. Otherwise, look at whatever's in there, and then open the box with that picture on it. Keep going that way until you find your instrument. The bandmates are skeptical, but amazingly enough, they all find what they need. And a few hours later, they're playing to thousands of adoring fans. So why did the drummer's strategy work? Each musician follows a linked sequence that starts with the box whose outside matches their instrument and ends with the box actually containing it. Note that if they kept going, that would lead them back to the start, so this is a loop. For example, if the boxes are arranged like so, the singer would open the first box to find the drums, go to the eighth box to find the bass, and find her microphone in the third box, which would point back to the first. This works much better than random guessing because by starting with the box with the picture of their instrument, each musician restricts their search to the loop that contains their instrument, and there are decent odds, about 35%, that all of the loops will be of length five or less. How do we calculate those odds? For the sake of simplicity, we'll demonstrate with a simplified case, four instruments and no more than two guesses allowed for each musician. Let's start by finding the odds of failure, the chance that someone will need to open three or four boxes before they find their instrument. There are six distinct four-box loops. One fun way to count them is to make a square, put an instrument at each corner, and draw the diagonals. See how many unique loops you can find, and keep in mind that these two are considered the same, they just start at different points. These two, however, are different. We can visualize the eight distinct three-box loops using triangles. You'll find four possible triangles depending on which instrument you leave out, and two distinct paths on each. So of the 24 possible combinations of boxes, there are 14 that lead to faliure, and ten that result in success. That computational strategy works for any even number of musicians, but if you want a shortcut, it generalizes to a handy equation. Plug in ten musicians, and we get odds of about 35%. What if there were 1,000 musicians? 1,000,000? As n increases, the odds approach about 30%. Not a guarantee, but with a bit of musician's luck, it's far from hopeless. Hi everybody, if you liked this riddle, try solving these two.

Frequently Occurring Word Combinations

Important Words

  1. adoring
  2. allowed
  3. amazingly
  4. answer
  5. approach
  6. arranged
  7. band
  8. bandmates
  9. bass
  10. big
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  14. bus
  15. calculate
  16. case
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  19. combinations
  20. communicate
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  22. concert
  23. considered
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  25. count
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  27. decent
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  29. depending
  30. despair
  31. diagonals
  32. distinct
  33. draw
  34. driving
  35. dropping
  36. drummer
  37. drums
  38. eighth
  39. ends
  40. equation
  41. explains
  42. failure
  43. faliure
  44. fans
  45. favorite
  46. figure
  47. find
  48. finding
  49. fooled
  50. fun
  51. generalizes
  52. great
  53. guarantee
  54. guesses
  55. guessing
  56. handy
  57. happening
  58. hopeless
  59. hours
  60. increases
  61. instrument
  62. instruments
  63. label
  64. large
  65. lead
  66. leave
  67. length
  68. linked
  69. loop
  70. loops
  71. luck
  72. mad
  73. manager
  74. marking
  75. matches
  76. microphone
  77. mind
  78. minutes
  79. misplacing
  80. music
  81. musician
  82. musicians
  83. note
  84. number
  85. odds
  86. open
  87. organized
  88. paths
  89. pause
  90. picking
  91. picture
  92. pictures
  93. play
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  95. plug
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  98. practice
  99. put
  100. random
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  102. restricts
  103. result
  104. riddle
  105. room
  106. sake
  107. screen
  108. search
  109. security
  110. sequence
  111. shortcut
  112. shouting
  113. simplicity
  114. simplified
  115. singer
  116. skeptical
  117. solving
  118. soundproof
  119. square
  120. start
  121. starting
  122. starts
  123. strategy
  124. succeed
  125. success
  126. suddenly
  127. takes
  128. ten
  129. thousands
  130. tied
  131. time
  132. tonight
  133. touch
  134. tour
  135. triangles
  136. unique
  137. valid
  138. video
  139. visualize
  140. wakes
  141. windowless
  142. work
  143. working
  144. works