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Too much missing information.
If some of the codes above are wrong (ingame, not mathematically) then please let me know and i will correct them.
It is possible that devs themselves made wrong solution, iam just posting codes which should work in-game and made it faster for players to skip Math passages during their play.
If it's not a rotation, then you can start decrypting it by figuring out the differences in between each.
case in point: 1-2-6-24-120- ??? (720, but ignored for this example)
2 - 1 = 1.
6 - 2 = 4
24 - 6 = 18
120 - 24 = 96.
1,4,18,96 isn't really a significant number sequence, so there's a clue it's not an addition/subtraction. If it was a sequence like 1,2,3,5,7,11, 13, 17... (prime numbers), that would be different.
So, if it's not addition/subtraction, lets try division.
2/1 = 2
6/2 = 3
24/6 = 4
120/24 = 5..
AHA! so if we solve for x where x/120 = 6? 6 x 120 = 1440.
These will in a grand majority of cases, but not all.
So, author... 3-5-8-13-21-34, none of these techniques work on. How did you get 34?
3 (+2) 5 (+3) 8 (+5) 21 (+13) 34. If it was a prime number sequence, it skipped 7 and 11.
An easy example of this is 7771-1777-7177-7717. The 1 is moving to the right and rolls over to the start when it drops off the end.
Some like 5040-720-120-24-6 are in fact divisions.
5040 / 7 = 720,
720 / 6 = 120
120 / 5 = 24
24 / 4 = 6
Fun fact, in the 1's, the 1-2-6-24-120-720 chain is in fact the reverse of this
1 * 2 = 2
2 * 3 = 6
6 * 4 = 24
24 * 5 = 720
All of these are math operations of one sort or an another, so for the rotations, ANY code sequence can be represented as abcd -> bcda -> cdab -> dabc or the reverse.