# How many adjacent edges in an $n \times n$ grid of squares

arithmeticcombinatorics

I'm trying to find a general expression for the number of adjacent edges in a $$n \times n$$ grid of uniform squares.

A square can have adjacent edges above, below, to the left, or to the right, if there is another square there. No diagonals.

So for example, a

a) $$2\times2$$ grid has $$4$$ squares and $$4$$ adjacent edges,

b) $$3\times3$$ grid has $$9$$ squares and $$12$$ adjacent edges,

c) $$4\times4$$ grid has $$16$$ squares and $$24$$ adjacent edges,

and so on.

It seems really simple but I can't seem to find a correct generalization.

There are $$n$$ adjacent edges in each horizontal line segment and there are $$n-1$$ such line segments. This leads to a total of $$n(n-1)$$ $$\color{red}{\text{horizontal adjacent edges}}$$. And since a square is symmetric, there are an equal number of $$\color{blue}{\text{vertical adjacent edges}}$$ too. Therefore, the total number of adjacent edges in a $$n\times n$$ grid is given by: $$2n(n-1)$$