The geometry is some what subtle, but I argue it's entirely possible to formalize a solution for incredibly large sequences based on this relationship. As you can see by the tiers of the pyramid (discrete triangular fractal), this graphical solution works for all sequences involving numbers of adjacent 1's in powers of 2 (i.e. 1, 11, 1111, 11111111, etc.). Identifying sequences becomes more difficult for sequences that have other numbers of adjacent 1's, but the principle is the same and the geometry is not impossible to navagate. The full solution will involve similar pyramids for all powers of prime numbers. However, a majority of sequences can be solved with the geometry from the simplest of these (e.g. powers of 2 - above). I encourage you to try making a pyramid of your own. Start with the sequence of all 0's, with a prime number of adjacent 1's in the middle. Carry the numbers downward, look at the pattern, and see just how many random sequences of 1's and 0's can be accounted for.
Problem Solved. (more or less) :o)
This method is like a big look up table. You can look at your bit string and then compare with your table to see how various sections will result. You are going to need a lot of tables though. This one works for 1,2,4,8,16... What about for a disturbance of 3,5,6,7,9...
ReplyDeletemmmm,, this is very much like a computer science assignment to me. :) but I believe you looked at it deeper and found more characteristics :). I am not sure whose solution is right but I believe we were all right at some point. lol
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