There is this problem that I encountered in the book problem Solving strategies on page 7, as an example.
"Start with a sequence $S = (a,b,c,d)$ of positive integers and find the derived sequence $S_1 = T(S) = (|a-b|,|b-c|,|c-d|,|d-a|)$. Does the sequence $S,S_1,S_2=T(S_1),S_3=T(S_2),...$ always end up with $(0,0,0,0)$?"
I understood the solution for this, however following this, there are a few variants that are asked further
"Replace the number of terms by $2^r$ for any integer $r>0$ instead of 4. Prove that we always reach $(0,0,0,....)$ and for number terms not equal to $2^r$ we get (up to some exceptions) a cycle containing just two numbers: 0 and evenly often some number $a$"
"Let number of terms not be equal to $2^r$ and c(n) be the cycle length. Prove that $c(2n)=2c(n)$(upto some exceptions)"
"Prove that for an odd number of terms , $S = (0,0,....,1,1)$ always lies on a cycle"
A cycle means the sequence of operations done till the same number occurs.
For the last one, the sequence S refers to modulo 2 of the number appearing in the actually sequence.
Please help in solving them, I also am not able to understand why are there exceptions, do they crop somewhere while proving it?
Thanks in advance!!