# 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 / 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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