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← Can you find the next number in this sequence? - Alex Gendler

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Showing Revision 2 created 06/29/2017 by Jessica Ruby.

  1. These are the first five elements
    of a number sequence.
  2. Can you figure out what comes next?
  3. Pause here if you want
    to figure it out for yourself.
  4. Answer in: 3
  5. Answer in: 2
  6. Answer in: 1
  7. There is a pattern here,
  8. but it may not be the kind
    of pattern you think it is.
  9. Look at the sequence again
    and try reading it aloud.
  10. Now, look at the next number
    in the sequence.
  11. 3, 1, 2, 2, 1, 1.
  12. Pause again if you'd like to think
    about it some more.
  13. Answer in: 3
  14. Answer in: 2
  15. Answer in: 1
  16. This is what's known as
    a look and say sequence.
  17. Unlike many number sequences,
  18. this relies not on some mathematical
    property of the numbers themselves,
  19. but on their notation.
  20. Start with the left-most digit
    of the initial number.
  21. Now, read out how many times
    it repeats in succession
  22. followed by the name of the digit itself.
  23. Then move on to the next distinct digit
    and repeat until you reach the end.
  24. So the number 1 is read as "one one"
  25. written down the same way
    we write eleven.
  26. Of course, as part of this sequence,
    it's not actually the number eleven,
  27. but 2 ones,
  28. which we then write as 2 1.
  29. That number is then read out
    as 1 2 1 1,
  30. which written out we'd read as
    one one, one two, two ones, and so on.
  31. These kinds of sequences were first
    analyzed by mathematician John Conway,
  32. who noted they have
    some interesting properties.
  33. For instance, starting with the number 22,
    yields an infinite loop of two twos.
  34. But when seeded with any other number,
  35. the sequence grows in some
    very specific ways.
  36. Notice that although the number
    of digits keeps increasing,
  37. the increase doesn't seem
    to be either linear or random.
  38. In fact, if you extend the sequence
    infinitely, a pattern emerges.
  39. The ratio between the amount of digits
    in two consecutive terms
  40. gradually converges to a single number
    known as Conway's Constant.
  41. This is equal to a little over 1.3,
  42. meaning that the amount of digits
    increases by about 30%
  43. with every step in the sequence.
  44. What about the numbers themselves?
  45. That gets even more interesting.
  46. Except for the repeating sequence of 22,
  47. every possible sequence eventually breaks
    down into distinct strings of digits.
  48. No matter what order these strings
    show up in,
  49. each appears unbroken in its entirety
    every time it occurs.
  50. Conway identified 92 of these elements,
  51. all composed only of digits 1, 2, and 3,
  52. as well as two additional elements
  53. whose variations
    can end with any digit of 4 or greater.
  54. No matter what number the sequence
    is seeded with,
  55. eventually, it'll just consist
    of these combinations,
  56. with digits 4 or higher only appearing
    at the end of the two extra elements,
  57. if at all.
  58. Beyond being a neat puzzle,
  59. the look and say sequence
    has some practical applications.
  60. For example, run-length encoding,
  61. a data compression that was once used for
    television signals and digital graphics,
  62. is based on a similar concept.
  63. The amount of times a data value repeats
    within the code
  64. is recorded as a data value itself.
  65. Sequences like this are a good example
    of how numbers and other symbols
  66. can convey meaning on multiple levels.