Sudoku puzzle with exactly 3 solutions

You could use this as a model (this works for $4 \times 4$ but needs to be adapted for $9\times 9$):

$\begin{array}{cccc} 1 & 2 & & \\ 3 & 4 & 1 & 2 \\ & 1 & & \\ & 3 & & 1 \end{array} $

With solutions

$\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \\ 2/4 & 1 & 4/2 & 3 \\ 4/2 & 3 & 2/4 & 1 \end{array} $

and

$\begin{array}{cccc} 1 & 2 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 1 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array} $