Imagine a game where you have 4 rows of dots. Each row has 1,3,5,7 dots.

A legal move is a circling or capturing of some kind of any number of consecutive dots on a single row.

If you take the last dot, you lose.

The question: If you can only take from the ends of the rows, the winning moves are pretty simple to figure out. However, if you can take from the middles (which you can according to the game), it's not quite.

However, after studying it for a while, I decided that taking from the center makes no difference in the winning moves list, that is, you cannot take dots from the middle of a (end-only) winning position to form a different (end-only) winning position.

Is this true, and how would you prove it in a way you could probably generalize?

(by the way if it helps, here are some positions that if you leave your opponent in the end-only game, you win:

1, 22, 33, 44, 55, 111, 123, 145, 246, 257, 347, 356, 1122, 1133, 1144, 1155, 1246, 1347, 2222, 2233, 2244, 2345, 3333, 11111, 11123, etc.)