The more logical and efficient technique to solve a Sudoku puzzle is to proceed by successive eliminations of the possible figures in any empty cells.
At the beginnig every empty cell can contain any figure from 1 to 9. The game begins by eliminating cell by cell all the figures which are not correct in that cell:
- if a figure appears in a line, it will be eliminated in all the empty cells of that line;
- if a figure appears in a column, it will be eliminated in all the empty cells of that column;
- if a figure appears in a 3x3 box, it will be eliminated in all the empty cases of that box.
After that, every cell contains only the figures that can be correct.
The first and more obvious step is to check if in some cells one single figure is left, in which case that cell is resolved. The option "Assisted Game" does all the above job until this point for you.
Obviously, every time that in the course of the game we assign a figure to a cell, we will have to eliminate that figure in all the cells belonging to the same row, column and box as the newly resolved cell.
After this phase, a bit boring, the real game begins.
Number that appears one single time. First of all we check, for each row, if a digit appears among the possible options in one single cell. As an example, if in a line we have [2 3 4] [4 5 6] [1 3 5] [2 4 6], the only figure that does not repeat at least once is 1, therefore 1 could be ONLY in that case. The same analysis must be made for each column and 3x3 box.
Same two figures in two cells. If in a line (the same obviously can be said for each column or box) we can find the following situation: [1 2] [2 3 4] [1 2] [1 3 4], it means that the digits 1 and 2 have to belong to the two cells which have 1 and 2 as their only possible options, consequently they can be eliminated from the other cells. Therefore we will have: [1 2] [3 4] [1 2] [3 4]. The same can be said, but it is a much rarer case, if in three cells we can see, for example, [1 2 3] as the only possible options.
These steps have to be repeated for each row, column and box until it is not possible to move forward, that is we are not able to solve new cells or to eliminate figures in any cell. At that point, but only at that point, it is necessary to move to a "what if" approach, that is to say that we will have to check what would happen if we assigned to a cell one of the possible digits left.
We should start from a cell with a low number of figures left, better only 2; a digit is assigned and the Sudoku puzzle is resolved on the basis of this assumption. The player shall not forget to mark every cell which is gone to be resolved from this point on, or to use a different color. This is necessary because, if the hypothesis turned out wrong, it should be necessary to go back to the point from which we made the assumption ("rewind"). If the cell contained only two possible figures, and one has turned out wrong, we will assign the other. We will realize that the assumption is wrong if, along the way, at a certain point we face an impossible situation, such as the double presence of the same figure on the same line, or similar cases.
In the more complex puzzles, it can be necessary to make use of sub-assumptions: that is to say that after a first hypothesis on a cell, which does not come out neither right nor wrong, to be able to go on and solve the puzzle we will have to make a second hypothesis on another cell. In this case, if the game is not resolved correctly (and luckily) at the first attempt, it is necessary to move back not only to the second assumption but also to the first one.
