Sudoku Solver

I am writing a Sudoku solving program in Atari Basic (I may convert it to a C++ console program later but it will not have a graphical UI).

The joystick driven UI is finished so I am now moving on to the logic. I thought there would be any number of example algorithms online but they all seem to rely on brute force back tracking. If anyone knows of a better method then please let me know but if not I am going to write my own code based on forced moves just like I do with pen and paper.

The code will first determine the numbers that are still needed in a row by subtracting the ones that are already entered and then try to fit them into empty squares from left to right. If a number will fit because it is still required in the row and it is not in the column  or 3×3 grid of the empty square then the position will be tagged but not filled in and the program will try to fit the same number in the next empty square along. If a second possible fit is found in the row then the program will abandon that number and move on to the next required number. A number will only be filled in if it has only one possible location in the row. Once all the required numbers have been tried the program will do the same with the next row and then the columns. The process will repeat until the grid is complete or there are no more forced moves in which case it will then look for forced moves in the 3×3 grids before running through the rows and columns again and so on.