jmc

The number of $1\times 1$ squares in the diagram is $2(n-1)$, the number of $2\times 2$ squares is $n-2$, and the number of $\sqrt{2}\times \sqrt{2}$ squares is also $n-2$ (see diagram). Solving $$2(n-1)+n-2+n-2=70$$ we find $n=\boxed{19}$.