full transcript

From the Ted Talk by Jeff Dekofsky: The Infinite Hotel Paradox

Unscramble the Blue Letters


In the 1920's, the gmaren mathematician David Hilbert devised a famous thought eminxrepet to show us just how hard it is to wrap our minds around the concept of infinity. Imagine a hotel with an infinite number of rooms and a very hraoknrwidg night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. A man walks into the heotl and asks for a room. Rather than turn him down, the night manager ddeices to make room for him. How? Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every guset moves from room number "n" to room number "n+1". Since there are an infinite number of rooms, there is a new room for each existing guest. This leaves room 1 open for the new customer. The pseocrs can be repeated for any finite number of new guests. If, say, a tour bus ulodnas 40 new people looking for rooms, then every existing guest just moves from room number "n" to room number "n+40", thus, opening up the first 40 rooms. But now an itefinlniy lagre bus with a countably infinite number of passengers pllus up to rent rooms. cltnabouy infinite is the key. Now, the infinite bus of infinite paegernsss perplexes the night manager at first, but he realizes there's a way to place each new person. He asks the guest in room 1 to move to room 2. He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on. Each current guest moves from room number "n" to room number "2n" — filling up only the infinite even-numbered rooms. By doing this, he has now emptied all of the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus. Everyone's happy and the hotel's business is booming more than ever. Well, actually, it is booming exactly the same amount as ever, bnaikng an infinite number of dollars a night. Word spreads about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an infinite line of infinitely large buess, each with a countably infinite number of passengers. What can he do? If he cannot find rooms for them, the hotel will lose out on an infinite amount of mneoy, and he will serluy lose his job. Luckily, he remembers that around the year 300 B.C.E., Euclid proved that there is an infinite quantity of prime numbers. So, to accomplish this seemingly impossible task of finding infinite beds for infinite buses of infinite weary taveerrls, the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number. So, the current ocpnucat of room number 7 goes to room nbmeur 2^7, which is room 128. The ngiht manager then tkaes the people on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the peowr of their seat number on the bus. So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers. All the buses' passengers fan out into rooms using uinque room-assignment schemes based on unique prime numbers. In this way, the night manager can aoctoamdmce every passenger on every bus. Although, there will be many rooms that go unfilled, like room 6, since 6 is not a power of any prmie number. lcikluy, his bsesos weren't very good in math, so his job is safe. The night manager's strategies are only possible because while the initfnie Hotel is certainly a logistical nightmare, it only deals with the lowest level of infinity, mainly, the catounlbe infinity of the natural numbers, 1, 2, 3, 4, and so on. Georg Cantor called this level of infinity aleph-zero. We use nraatul numbers for the room numbers as well as the seat nrbmeus on the buses. If we were dealing with hhiger orders of iitnifny, such as that of the real numbers, these structured strategies would no longer be possible as we have no way to sctleimltasyay include every number. The Real Number Infinite Hotel has nvgaiete number romos in the basement, fiacrontal rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1. Square root rooms, like room radical 2, and room pi, where the guests epecxt free dessert. What self-respecting night manager would ever want to work there even for an infinite saarly? But over at Hilbert's Infinite Hotel, where there's never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night meanagr srvee to remind us of just how hard it is for our relatively finite minds to gasrp a concept as large as infinity. Maybe you can help tackle these problems after a good night's seelp. But honestly, we might need you to change rooms at 2 a.m.

Open Cloze


In the 1920's, the ______ mathematician David Hilbert devised a famous thought __________ to show us just how hard it is to wrap our minds around the concept of infinity. Imagine a hotel with an infinite number of rooms and a very ___________ night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. A man walks into the _____ and asks for a room. Rather than turn him down, the night manager _______ to make room for him. How? Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every _____ moves from room number "n" to room number "n+1". Since there are an infinite number of rooms, there is a new room for each existing guest. This leaves room 1 open for the new customer. The _______ can be repeated for any finite number of new guests. If, say, a tour bus _______ 40 new people looking for rooms, then every existing guest just moves from room number "n" to room number "n+40", thus, opening up the first 40 rooms. But now an __________ _____ bus with a countably infinite number of passengers _____ up to rent rooms. _________ infinite is the key. Now, the infinite bus of infinite __________ perplexes the night manager at first, but he realizes there's a way to place each new person. He asks the guest in room 1 to move to room 2. He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on. Each current guest moves from room number "n" to room number "2n" — filling up only the infinite even-numbered rooms. By doing this, he has now emptied all of the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus. Everyone's happy and the hotel's business is booming more than ever. Well, actually, it is booming exactly the same amount as ever, _______ an infinite number of dollars a night. Word spreads about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an infinite line of infinitely large _____, each with a countably infinite number of passengers. What can he do? If he cannot find rooms for them, the hotel will lose out on an infinite amount of _____, and he will ______ lose his job. Luckily, he remembers that around the year 300 B.C.E., Euclid proved that there is an infinite quantity of prime numbers. So, to accomplish this seemingly impossible task of finding infinite beds for infinite buses of infinite weary _________, the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number. So, the current ________ of room number 7 goes to room ______ 2^7, which is room 128. The _____ manager then _____ the people on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the _____ of their seat number on the bus. So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers. All the buses' passengers fan out into rooms using ______ room-assignment schemes based on unique prime numbers. In this way, the night manager can ___________ every passenger on every bus. Although, there will be many rooms that go unfilled, like room 6, since 6 is not a power of any _____ number. _______, his ______ weren't very good in math, so his job is safe. The night manager's strategies are only possible because while the ________ Hotel is certainly a logistical nightmare, it only deals with the lowest level of infinity, mainly, the _________ infinity of the natural numbers, 1, 2, 3, 4, and so on. Georg Cantor called this level of infinity aleph-zero. We use _______ numbers for the room numbers as well as the seat _______ on the buses. If we were dealing with ______ orders of ________, such as that of the real numbers, these structured strategies would no longer be possible as we have no way to ______________ include every number. The Real Number Infinite Hotel has ________ number _____ in the basement, __________ rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1. Square root rooms, like room radical 2, and room pi, where the guests ______ free dessert. What self-respecting night manager would ever want to work there even for an infinite ______? But over at Hilbert's Infinite Hotel, where there's never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night _______ _____ to remind us of just how hard it is for our relatively finite minds to _____ a concept as large as infinity. Maybe you can help tackle these problems after a good night's _____. But honestly, we might need you to change rooms at 2 a.m.

Solution


  1. surely
  2. luckily
  3. countable
  4. accommodate
  5. infinite
  6. process
  7. numbers
  8. number
  9. systematically
  10. takes
  11. higher
  12. buses
  13. sleep
  14. decides
  15. power
  16. grasp
  17. infinity
  18. expect
  19. unloads
  20. passengers
  21. hotel
  22. countably
  23. natural
  24. large
  25. negative
  26. banking
  27. occupant
  28. guest
  29. german
  30. night
  31. rooms
  32. manager
  33. infinitely
  34. prime
  35. unique
  36. hardworking
  37. experiment
  38. salary
  39. serve
  40. fractional
  41. bosses
  42. pulls
  43. money
  44. travelers

Original Text


In the 1920's, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds around the concept of infinity. Imagine a hotel with an infinite number of rooms and a very hardworking night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. A man walks into the hotel and asks for a room. Rather than turn him down, the night manager decides to make room for him. How? Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every guest moves from room number "n" to room number "n+1". Since there are an infinite number of rooms, there is a new room for each existing guest. This leaves room 1 open for the new customer. The process can be repeated for any finite number of new guests. If, say, a tour bus unloads 40 new people looking for rooms, then every existing guest just moves from room number "n" to room number "n+40", thus, opening up the first 40 rooms. But now an infinitely large bus with a countably infinite number of passengers pulls up to rent rooms. countably infinite is the key. Now, the infinite bus of infinite passengers perplexes the night manager at first, but he realizes there's a way to place each new person. He asks the guest in room 1 to move to room 2. He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on. Each current guest moves from room number "n" to room number "2n" — filling up only the infinite even-numbered rooms. By doing this, he has now emptied all of the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus. Everyone's happy and the hotel's business is booming more than ever. Well, actually, it is booming exactly the same amount as ever, banking an infinite number of dollars a night. Word spreads about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an infinite line of infinitely large buses, each with a countably infinite number of passengers. What can he do? If he cannot find rooms for them, the hotel will lose out on an infinite amount of money, and he will surely lose his job. Luckily, he remembers that around the year 300 B.C.E., Euclid proved that there is an infinite quantity of prime numbers. So, to accomplish this seemingly impossible task of finding infinite beds for infinite buses of infinite weary travelers, the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number. So, the current occupant of room number 7 goes to room number 2^7, which is room 128. The night manager then takes the people on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seat number on the bus. So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers. All the buses' passengers fan out into rooms using unique room-assignment schemes based on unique prime numbers. In this way, the night manager can accommodate every passenger on every bus. Although, there will be many rooms that go unfilled, like room 6, since 6 is not a power of any prime number. Luckily, his bosses weren't very good in math, so his job is safe. The night manager's strategies are only possible because while the Infinite Hotel is certainly a logistical nightmare, it only deals with the lowest level of infinity, mainly, the countable infinity of the natural numbers, 1, 2, 3, 4, and so on. Georg Cantor called this level of infinity aleph-zero. We use natural numbers for the room numbers as well as the seat numbers on the buses. If we were dealing with higher orders of infinity, such as that of the real numbers, these structured strategies would no longer be possible as we have no way to systematically include every number. The Real Number Infinite Hotel has negative number rooms in the basement, fractional rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1. Square root rooms, like room radical 2, and room pi, where the guests expect free dessert. What self-respecting night manager would ever want to work there even for an infinite salary? But over at Hilbert's Infinite Hotel, where there's never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night manager serve to remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity. Maybe you can help tackle these problems after a good night's sleep. But honestly, we might need you to change rooms at 2 a.m.

Frequently Occurring Word Combinations


ngrams of length 2

collocation frequency
room number 13
night manager 9
infinite number 6
infinite hotel 3
countably infinite 3
guest moves 2
existing guest 2
infinitely large 2
infinite bus 2
current guest 2
prime numbers 2
infinite buses 2
seat number 2
prime number 2
room numbers 2

ngrams of length 3

collocation frequency
countably infinite number 2

Important Words


  1. accommodate
  2. accomplish
  3. amount
  4. asks
  5. assigned
  6. assigns
  7. banking
  8. base
  9. based
  10. basement
  11. beds
  12. booked
  13. booming
  14. bosses
  15. bus
  16. buses
  17. business
  18. called
  19. cantor
  20. change
  21. completely
  22. concept
  23. continues
  24. countable
  25. countably
  26. current
  27. customer
  28. david
  29. dealing
  30. deals
  31. decides
  32. dessert
  33. devised
  34. dollars
  35. easy
  36. emptied
  37. euclid
  38. existing
  39. expect
  40. experiment
  41. faced
  42. factors
  43. famous
  44. fan
  45. filing
  46. filling
  47. find
  48. finding
  49. finite
  50. fractional
  51. free
  52. full
  53. georg
  54. german
  55. good
  56. grasp
  57. guest
  58. guests
  59. guy
  60. happy
  61. hard
  62. hardworking
  63. higher
  64. hilbert
  65. honestly
  66. hospitable
  67. hotel
  68. imagine
  69. impossible
  70. include
  71. incredible
  72. infinite
  73. infinitely
  74. infinity
  75. job
  76. key
  77. large
  78. leaves
  79. level
  80. line
  81. logistical
  82. longer
  83. lose
  84. lowest
  85. luckily
  86. man
  87. manager
  88. math
  89. mathematician
  90. minds
  91. money
  92. move
  93. moves
  94. natural
  95. negative
  96. night
  97. nightmare
  98. number
  99. numbers
  100. occupant
  101. open
  102. opening
  103. orders
  104. overlapping
  105. passenger
  106. passengers
  107. people
  108. perplexes
  109. person
  110. pi
  111. place
  112. pour
  113. power
  114. powers
  115. prime
  116. problems
  117. process
  118. proved
  119. pulls
  120. quantity
  121. radical
  122. raised
  123. real
  124. realizes
  125. remembers
  126. remind
  127. rent
  128. repeated
  129. room
  130. rooms
  131. root
  132. safe
  133. salary
  134. scenarios
  135. schemes
  136. seat
  137. seemingly
  138. sees
  139. serve
  140. show
  141. sleep
  142. spreads
  143. square
  144. strategies
  145. structured
  146. surely
  147. suspects
  148. systematically
  149. tackle
  150. takes
  151. task
  152. thought
  153. totally
  154. tour
  155. travelers
  156. turn
  157. unfilled
  158. unique
  159. unloads
  160. unthinkable
  161. vacancy
  162. walks
  163. weary
  164. wide
  165. word
  166. work
  167. wrap
  168. year