sqing wrote: The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases. Consider a $2018\times 2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated). Colour the board as shown below. (Alternating “double” rows of red and green) Note that since the tiles are taken from the same row, # $\text{of green tiles}$ - # $ \text{of red tiles} \equiv \text{2}(\text{mod 4})$. But for a staircase # $\text{of covered green}$ - # $\text{of covered red} \equiv \text{0}(\text{mod 4)}$ which leads to a contradiction. $\blacksquare$

invariant mod(4)