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Posers and Puzzles

Posers and Puzzles

  1. 20 Sep '06 16:13
    Got too much work on at the moment, so am having a break from RHP for at least the next few months.

    Since this is the only forum I have really posted in I thought I'd say bye and leave you with a couple of (hard) puzzles!!

    1) A knight starts at the corner of a chessboard, and at each time t = 1, 2, 3, 4, ... it moves once, with all possible moves equally likely. Let T be the first time the knight returns to its starting corner. What is the expected value of T?

    2) A monkey is typing on a typewriter, with each letter A, B, C, ... , Z equally likely. How long before you expect to see the sequence ABRACADABRA?

    Enjoy!
  2. 20 Sep '06 18:16
    An ugly incomplete solution for question 1

    I can solve it with a system of 64 equations in 64 variables, in some mathematical software. Basically we need to write a number in each square of the chessboard, such that at a1 we have 0, and at each other square the number is 1 more than the average of the numbers which are at a knight's-move distance. Then the solution is the number at b3 (or c2) plus 1. (I think)
  3. 20 Sep '06 18:18
    I managed to solve problem 2 in the binary case - the monkey types 0 or 1, and we wait for a specific string of 0's and 1's. But I didn't solve yet the original problem. Maybe you should try the binary case first.
  4. Standard member PBE6
    Bananarama
    20 Sep '06 19:32
    Originally posted by SPMars
    Got too much work on at the moment, so am having a break from RHP for at least the next few months.

    Since this is the only forum I have really posted in I thought I'd say bye and leave you with a couple of (hard) puzzles!!

    1) A knight starts at the corner of a chessboard, and at each time t = 1, 2, 3, 4, ... it moves once, with all possible moves equall ...[text shortened]... ... , Z equally likely. How long before you expect to see the sequence ABRACADABRA?

    Enjoy!
    Good luck SPMars! I always enjoyed your posts and analyses. Take care! We'll work on these in the meantime.
  5. 22 Sep '06 03:06
    Originally posted by David113
    I managed to solve problem 2 in the binary case - the monkey types 0 or 1, and we wait for a specific string of 0's and 1's. But I didn't solve yet the original problem. Maybe you should try the binary case first.
    it's just 26^11 is it not?
  6. 22 Sep '06 10:31 / 1 edit
    No. The waiting time depends on which string of 0's and 1's you are waiting for.

    Example:

    If you wait for 01 or 10 then the expected waiting time is 4; but if you wait for 00 or 11 then the expected waiting time is 6.
  7. Subscriber AThousandYoung
    It's about respect
    23 Sep '06 02:38
    Originally posted by SPMars
    Got too much work on at the moment, so am having a break from RHP for at least the next few months.

    Since this is the only forum I have really posted in I thought I'd say bye and leave you with a couple of (hard) puzzles!!

    1) A knight starts at the corner of a chessboard, and at each time t = 1, 2, 3, 4, ... it moves once, with all possible moves equall ...[text shortened]... ... , Z equally likely. How long before you expect to see the sequence ABRACADABRA?

    Enjoy!
    2) 26^11 letters is how many I'd expect before that sequence occurred.
  8. Subscriber AThousandYoung
    It's about respect
    23 Sep '06 02:39
    Originally posted by David113
    No. The waiting time depends on which string of 0's and 1's you are waiting for.

    Example:

    If you wait for 01 or 10 then the expected waiting time is 4; but if you wait for 00 or 11 then the expected waiting time is 6.
    What? Why?
  9. 23 Sep '06 18:41
    See proof here:

    http://www.qbyte.org/puzzles/p082s.html