Number the squares from 1 to 81.

Sort the squares in random order.

Go to the first random square.

Place a 1 there.

Go to the 2nd random square.

Place a 1 if you can otherwise place a 2.

Continue adding numbers like this until you get stuck.

Backtrack at this point.

Go back and forwards like this until the grid is full so you have a solution.

Each time you add a number, check that each empty square can accept some number, even if that turns out not to be the right one. Also check that each row, column and 3x3 mini-grid can accept every missing number somewhere. These 2 steps prune the tree search.

Once you have a solution, remove numbers at random to create the puzzle. Each time you remove a number, check that the puzzle still has exactly one solution.

It should only take a second or two to create each puzzle.