Sum of Neighbouring Cells

There are $16$ simultaneous linear equations to solve for the grid entries. This set of equations is consistent, and the associated $16\times16$ matrix has rank $14$ , so there is a $2$ -parameter family of solutions as follows: $\begin{array}{|c|c|c|c|} \hline a & b & b+2 & a+2 \\ \hline a-b & 4-a & 1-a & a-b \\ \hline 2+a-b & 3-a & 2-a & 1+a-b \\ \hline 1+a & 2+b & 3+b & a-1 \\ \hline \end{array}$ The facts that $1-a$ , $a-1$ and $3+b$ are integers between $0$ and $3$ tell us that $a=1$ and $b=0$ , making the grid $\begin{array}{|c|c|c|c|} \hline 1 & 0 & 2 & 3 \\ \hline 1 & 3 & 0 & 1 \\ \hline 3 & 2 & 1 & 2 \\ \hline 2 & 2 & 3 & 0 \\ \hline \end{array}$ and hence there are $\boxed{5}$ $2$ s in the grid.