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Solve This

  1. Sep 8, 2007 #1
    Solve this

    1 1
    2 1
    1 2 1 1
    1 1 1 2 2 1
    3 1 2 2 1 1
    ? ? ? ? ? ? ? ?​
  2. jcsd
  3. Sep 8, 2007 #2


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  4. Sep 8, 2007 #3


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    Each line describes the previous line (how many of each symbol there were). After 13112221,
    1113213211. I can't see there being any characters but 1,2,3 in any of the later lines, and therefore also any character repeating more than 3 times in a line (ex.1 1 1 1). Also, in the long run, "1" to have most frequent appearance of the 3 numbers.
  5. Sep 8, 2007 #4
    you can't solve something unless there is a problem. you might as well have said:
    solve this
    "bear, racecar, 7,???".
  6. Sep 9, 2007 #5


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    While I agree with matticus, I think both bel and Esd have devined what Dazzlex intended. And, of course, it has nothing to do with mathematics.
  7. Sep 9, 2007 #6
    And yet, this thread is still sitting in the Number Theory section. :rolleyes:
  8. Sep 9, 2007 #7

    matt grime

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    There are some interesting questions about the 'say what you see' sequence. See the work of Conway (its inventor) for more. And it is maths, Halls, though not necessarily number theory.
  9. Sep 9, 2007 #8


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    I don't think you have understood what Conway is saying.
  10. Sep 10, 2007 #9

    matt grime

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    Really? I don't? So the fact that JH Conway (a mathematician in a maths department; this doesn't automatically make it maths, of course - has Conway been quoted as saying 'and of course this is completely unrelated to the entirety of mathematics') has done some interesting mathematicalwork on this isn't sufficient for it to be maths? And Conway's constant isn't an interesting thing? What on earth does something have to be to be considered maths, then Halls? Perhaps be in a tedious book on calculus taught to engineers?

    Exercise: prove that nearly all say what you see sequences (or look and say) have the ratio digits of consecutive numbers converges to a constant, further show it is algebraic, and find its minimal polynomial.
    Last edited: Sep 10, 2007
  11. Sep 10, 2007 #10


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  12. Sep 11, 2007 #11
    so close, it was actually badger.
  13. Oct 15, 2007 #12
    I saw this puzzle on the last page of the book titled "mathematical mysteries:the beauty and mathematics of numbers" by Calvin C Clawson. I broke my head over it for awhile. And eventually couldn't resist the temptation to check the solution. Though the solution was there, plain and bare, there was not a description for how he arrived at the given solution.

    Im glad i saw it on this thread today.
  14. Oct 15, 2007 #13
    could it be??

    1 3 1 1 1 2 1
  15. Oct 15, 2007 #14
    No Bel and Esd got the right answer: "one 3 one 1 two 2's and two 1's" is the description of the previous term. The answer is this description in "shorthand", i.e., "13112221". See the posts by Matt Grime for more background on this type of sequence.
    Last edited: Oct 15, 2007
  16. Oct 15, 2007 #15
    By the way, in general, this is called (decimal) run length encoding. While it may not have direct applications to most of number theory, algebra, or combinatorics (though clearly some) and likely no useful connections to any other field of mathematics, it is a very commonly used compression method (precisely because it is extremely simple) for fairly regular strings of words. Thus, it is both studied and used by computer scientists.

    You have a base set, a set of words, and a subset of the natural numbers (could this be generalized to an ordered groupoid? I'm not sure) that you use to count them with.

    In some cases, the encoding of your string isn't going to be unique, in some cases it is.

    For instance, let's say your words are A, B, and C, and you're using the numbers 1-99
    Then the string AAAAAAABBCCCCCBBBAAAA encodes to 7A2B5C3B4A, which is significantly shorter than the length of the original string. The nice thing about this encoded string is that when you look at each symbol you know immediately whether it's a word or a number by its position, so you can expand it very easily.

    Let's say your words are instead 0:A 1:B 2:AB 3:AAB 4:AAAB
    Then the string BAAABBAABAAAAABAABABBAAA encodes to 111411133023121130, though clearly this is not unique.

    Furthermore, you can allow a lot of really weird things with your words, like letter that means "erase the previous symbol if one exists" or something like that.

    The sequence that this topic was started about of course uses the the numbers 0-9 as words and the numbers 1-9 as counters. The interesting thing is that it's done in such a way so that you can keep applying the code to the result.

    There are a lot of interesting problems with this type of encoding. For instance, given a set, finite number of words, a set bound on your counters, and a set of strings, how can you find the optimal set of words to encode the set, and how given such an encoding, how can you find the optimal encoding of each string?

    It's also fairly easy to show (I have not done this. I have not even done more than heard this topic in passing, but the result is intuitive, and I've heard that it's not too difficult to show) that if the words need to be stored along with the encoding of the sets, that if an optimal set of words can always be found, encoding a second time cannot decrease the total number of symbols needed to store the encoding of the entire set of strings along with the words


    Err... anyway, my point is that, yes, this is real math
  17. Oct 16, 2007 #16
    Cool. I think I understand this sequence now. thanks
  18. Oct 16, 2007 #17
    Well done to everyone who understood the problem. I saw it in a book and it startled me at first but after a while it just looked so simple, so I just had to share it :)
  19. Oct 18, 2007 #18
    mathematical mysteries:the beauty and mathematics of numbers" by Calvin C Clawson
  20. Oct 20, 2007 #19
    This is deductive reasoning right? Is the answer 13112221?
  21. Oct 20, 2007 #20
    This is the general pattern that I see, is that right?:

    x x x x x x x 1
    x x x x x x 1 1
    x x x x x x 2 1
    x x x x 1 2 1 1
    x x 1 1 1 2 2 1
    x x 3 1 2 2 1 1
    1 3 1 1 2 2 2 1
    1 3 3 1 1 2 1 1
    1 3 1 1 1 2 2 1
    3 3 3 1 2 2 1 1
    3 3
    3 3
    1 3
    1 3
    1 3
    Last edited: Oct 20, 2007
  22. Oct 21, 2007 #21
    It is right uptil the 7th line. After that it's wrong, im afraid !!
  23. Oct 21, 2007 #22
    I know, I never continued it past that line, just showed the pattern.
  24. Oct 21, 2007 #23
    After the seventh line,

    1 3 1 1 2 2 2 1

    1 1 1 3 2 1 3 2 1 1

    3 1 1 3 1 2 1 1 1 3 1 2 2 1

    1 3 2 1 1 3 1 1 1 2 3 1 1 3 1 1 2 1 1 1

    ... and goes on i believe
  25. Oct 25, 2007 #24
    Then you are indicating that each column follows a repetitive pattern. That is inductively correct but that is wrong according to the actual rule used to form the sequence.
    Last edited: Oct 25, 2007
  26. Oct 25, 2007 #25
    You can of course deduce that each column will eventually follow a pattern just as the decimal expansions of rational numbers all eventually follow a pattern. But the eventual pattern for column 4 is not 1122 it is 1223, the first 1 is not part of the repeating period.
    Last edited: Oct 25, 2007
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