Dominoes on chessboard

Question

A $2016\times 2016$ chessboard is tiled with $2 \times 1$ dominoes.

I can prove that there is a grid line that pass through at least $505$ dominoes.

But how to prove or disprove that there is a grid line that pass through at least $506$ dominoes?

Thanks for helping.

Answer 1

:-) Each grid line intersects an even number of dominoes.