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Even more theory.

A project log for Pisano-carry Checksum algorithm

Add X to Y and Y to X, says the song. And carry on.

Yann Guidon / YGDESYann Guidon / YGDES 05/09/2021 at 20:520 Comments

The last log 22. Some more theory examined the orbits of the systems with w=0, w=1 and w=2. Let's look at the orbits of w=3 now:

This system has 2 orbits with 35 steps each and shows more complexity than the previous ones. The ranges are [0..7] now:

Orbit 1
Starting at 1,0,0
  1 - 1,1,0
  2 - 2,1,0
  3 - 3,2,0
  4 - 5,3,0
  5 - 0,5,1
  6 - 6,0,0
  7 - 6,6,0
  8 - 4,6,1
  9 - 3,4,1
 10 - 0,3,1
 11 - 4,0,0
 12 - 4,4,0
 13 - 0,4,1
 14 - 5,0,0
 15 - 5,5,0
 16 - 2,5,1
 17 - 0,2,1
 18 - 3,0,0
 19 - 3,3,0
 20 - 6,3,0
 21 - 1,6,1
 22 - 0,1,1
 23 - 2,0,0
 24 - 2,2,0
 25 - 4,2,0
 26 - 6,4,0
 27 - 2,6,1
 28 - 1,2,1
 29 - 4,1,0
 30 - 5,4,0
 31 - 1,5,1
 32 - 7,1,0
 33 - 0,7,1
 34 - 0,0,1
 35 - 1,0,0
Orbit 2
Starting from 7,0,0
  1 - 7,7,0
  2 - 6,7,1
  3 - 6,6,1
  4 - 5,6,1
  5 - 4,5,1
  6 - 2,4,1
  7 - 7,2,0
  8 - 1,7,1
  9 - 1,1,1
 10 - 3,1,0
 11 - 4,3,0
 12 - 7,4,0
 13 - 3,7,1
 14 - 3,3,1
 15 - 7,3,0
 16 - 2,7,1
 17 - 2,2,1
 18 - 5,2,0
 19 - 7,5,0
 20 - 4,7,1
 21 - 4,4,1
 22 - 1,4,1
 23 - 6,1,0
 24 - 7,6,0
 25 - 5,7,1
 26 - 5,5,1
 27 - 3,5,1
 28 - 1,3,1
 29 - 5,1,0
 30 - 6,5,0
 31 - 3,6,1
 32 - 2,3,1
 33 - 6,2,0
 34 - 0,6,1
 35 - 7,0,0

This time, most of the points on the lower diagonal (n,n,0) belong to orbit 1 and the others to orbit 2, but this feature is rather unique to this configuration.

Furthermore I don't see a clear pattern to identify the middle of the orbit. With an odd number of steps, 35/2=17.5 falls between (0,2,1) and (3,0,0), which looks a lot like 2^(w-1)-1 but this looks like a fluke so far.

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