"Consider a sequence of integers $x_1, x_2, ..., x_8$. From this sequence, create a new sequence whose elements are $|x_2-x_1|, |x_3-x_2|, ... |x_8-x_7|, |x_1-x_8|$. Find all initial sequences such that after finitely many applications of the transformation, the elements will become the same."
After creating a program to facilitate my solution, I noticed that all of the initial sequences I've put in satisfy the condition in the problem. Also, I noticed that the last sequence is always composed of an even integer.

Question

"Consider a sequence of integers $x_1, x_2, ..., x_8$. From this sequence, create a new sequence whose elements are $|x_2-x_1|, |x_3-x_2|, ... |x_8-x_7|, |x_1-x_8|$. Find all initial sequences such that after finitely many applications of the transformation, the elements will become the same."
After creating a program to facilitate my solution, I noticed that all of the initial sequences I've put in satisfy the condition in the problem. Also, I noticed that the last sequence is always composed of an even integer.

blockcolder · Answer

So I thought that maybe all possible initial sequences will terminate. How do I prove this rigorously, though?

kinggeorge · Answer

My first thought, is that you might be able to show that the new sequence will have a total value less than or equal to the original sequence. I.e., $$|x_1+x_2+...+x_8|\geq||x_2-x_1|+|x_3-x_2|+...+|x_1-x_8||$$

kinggeorge · Answer

And from there, possibly show that you only get equality when the elements are the same.

kinggeorge · Answer

However, on second thought, there are counterexamples to that idea, so that's not correct.

kinggeorge · Answer

I can show that if you have 7 of the same number, and one different, you end up at a single number after 7 more transformations.

blockcolder · Answer

At some point, the integers became all even. I wonder if this is relevant?

anonymous · Answer

I wonder if this is related to the triangle inequality.