Short Answer

To fill missing numbers in Sudoku logically, we check each empty cell and see which numbers are already present in its row, column, and box. The missing number is the one that does not repeat in any of these areas.

We use elimination to remove impossible numbers and narrow down choices. Step by step, we place numbers where only one option is possible and continue until the grid is complete.

Detailed Explanation:

Filling missing numbers in Sudoku logically

Checking row, column and box
The first step in filling missing numbers in Sudoku is to check the row, column, and 3×3 box of an empty cell. Each number from 1 to 9 must appear only once in these three areas. So, if some numbers are already present, those numbers cannot be used again in that cell.

For example, if a row already contains numbers 1, 3, and 5, then these numbers cannot be placed in the empty cells of that row. The same checking must be done for the column and the box. By looking at all three together, the solver can find which numbers are missing and possible for that position.

This step helps in reducing confusion and gives a clear idea of possible numbers for each empty cell.

Using elimination method
Elimination is one of the most important methods in Sudoku. The solver removes all numbers that are not possible for a cell. After eliminating impossible numbers, only a few options remain.

Sometimes, after elimination, only one number is left as a possible option. In such cases, that number is placed in the cell. This is called a definite placement. By repeating this process, more and more cells can be filled.

Elimination makes the puzzle simpler and helps in solving it without guessing.

Finding single possible number
In some cases, a cell can have only one possible number that fits according to the rules. This happens when all other numbers are already present in the row, column, or box.

This method is known as finding a single candidate. It is one of the easiest and most useful ways to fill missing numbers. By focusing on such cells first, the solver can quickly make progress.

Logical techniques used

Looking for missing numbers in a group
Another method is to look at a complete row, column, or box and find which numbers are missing. Once the missing numbers are known, the solver checks where each number can fit.

For example, if a row is missing numbers 2 and 7, the solver checks which position is suitable for each number based on column and box restrictions. This helps in placing numbers correctly.

Using cross-checking method
Cross-checking means checking a number across rows and columns together. If a number cannot be placed in certain cells due to restrictions, it may have only one possible place left.

This method is useful in solving difficult parts of the puzzle. It helps in finding correct positions by combining information from different parts of the grid.

Step-by-step solving approach
Sudoku must be solved step by step. The solver should not try to fill all cells at once. Instead, start with easy placements and move to more difficult ones.

Each correct number placed helps in solving other parts of the puzzle. As the grid fills, fewer options remain, making it easier to complete the puzzle.

Avoiding guessing
Logical solving means avoiding guessing. Every number should be placed based on clear reasoning. Guessing can lead to mistakes and confusion later.

By using logical methods like elimination and checking, the solver can ensure that each step is correct. This leads to an accurate and complete solution.

Conclusion

Missing numbers in Sudoku are filled by checking rows, columns, and boxes, and using elimination to find correct options. Logical methods like single candidate, cross-checking, and step-by-step solving help in completing the puzzle without guessing. This approach ensures accuracy and clarity.