full transcript

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

Unscramble the Blue Letters

In the 1920's, the German mathematician dviad Hilbert devised a famous thughot 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 hnwrairdokg night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. A man wlkas 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 romos, there is a new room for each existing guest. This leaves room 1 open for the new customer. The psercos can be repeated for any finite number of new gtseus. If, say, a tour bus unloads 40 new people looking for rooms, then every eistixng 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 gseut 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 binmoog more than ever. Well, actually, it is booming exactly the same amount as ever, banking an infinite number of dollars a night. Word srapdes about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an ifninite line of infinitely lgrae 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 moeny, and he will surely lose his job. Luckily, he remembers that around the year 300 B.C.E., Euclid poervd 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 cnrerut 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 neumbr 2^7, which is room 128. The night manager then tkeas 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 anissegd powers of the next pmrie, 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 fcartos, there are no overlapping room numbers. All the buses' passengers fan out into rooms using unique room-assignment schemes based on uquine prime nrmbues. In this way, the night manager can accommodate every pgneasser 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. llkuicy, 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 cbonatule infinity of the natural numbers, 1, 2, 3, 4, and so on. greog Cantor called this level of infinity aleph-zero. We use nautarl numbers for the room numbers as well as the seat numbers on the bsues. If we were dealing with higher orders of itnfiniy, such as that of the real numbers, these suurecrttd strategies would no lnoegr be possible as we have no way to systematically include every number. The Real Number Infinite heotl 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 ecxept free dsseert. 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 nihgt menaagr serve to rmnied us of just how hard it is for our relatively finite minds to grasp a ccopnet 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.

Open Cloze

In the 1920's, the German mathematician _____ Hilbert devised a famous _______ 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 ___________ night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. A man _____ 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 _____, 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 ______. If, say, a tour bus unloads 40 new people looking for rooms, then every ________ 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 _____ 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 _______ more than ever. Well, actually, it is booming exactly the same amount as ever, banking an infinite number of dollars a night. Word _______ about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an ________ line of infinitely _____ 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 _____, and he will surely lose his job. Luckily, he remembers that around the year 300 B.C.E., Euclid ______ 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 _______ 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 ______ 2^7, which is room 128. The night 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 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 ________ powers of the next _____, 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 _______, there are no overlapping room numbers. All the buses' passengers fan out into rooms using unique room-assignment schemes based on ______ prime _______. In this way, the night manager can accommodate every _________ 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. _______, 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 _________ infinity of the natural numbers, 1, 2, 3, 4, and so on. _____ Cantor called this level of infinity aleph-zero. We use _______ numbers for the room numbers as well as the seat numbers on the _____. If we were dealing with higher orders of ________, such as that of the real numbers, these __________ strategies would no ______ be possible as we have no way to systematically include every number. The Real Number Infinite _____ 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 ______ free _______. 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 _____ _______ serve to ______ us of just how hard it is for our relatively finite minds to grasp a _______ 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.

Solution

  1. current
  2. countable
  3. hotel
  4. expect
  5. david
  6. dessert
  7. concept
  8. number
  9. remind
  10. georg
  11. guest
  12. longer
  13. unique
  14. infinity
  15. process
  16. night
  17. money
  18. structured
  19. assigned
  20. proved
  21. existing
  22. natural
  23. passenger
  24. numbers
  25. salary
  26. large
  27. hardworking
  28. spreads
  29. booming
  30. thought
  31. luckily
  32. factors
  33. infinite
  34. walks
  35. buses
  36. rooms
  37. guests
  38. manager
  39. prime
  40. takes

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