KöMaL 2023 A865

A crossword is a grid of black and white cells such that every white cell belongs to some $2 \times 2$ square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid. Show that the total number of words in an $n\times n$ crossword cannot exceed $\frac{{(n+1)}^2}{2}$.