Initially, all the squares of an 8 \times 8 grid are white. You start by choosing one of the squares and coloring it gray. After that, you may color additional squares gray one at a time, but you may only color a square gray if it has exactly 1 or 3 gray neighbors at that moment (where a neighbor is a square sharing an edge). For example, the configuration below (of a smaller 3 \times 4 grid) shows a situation where six squares have been colored gray so far. The squares that can be colored at the next step are marked with a dot.
Is it possible to color all the squares gray? Justify your answer.
