# An extension of a question asked in Problem Solving Strategies

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

1. "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 $$a$$"

2. "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)"

3. "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?