Saturday Sudoku XI
In this walkthrough, when I use the word square, I will be referring to an individual cell that will eventually contain one number. I will use the word box to refer to a 3x3 set of nine squares.
In some but not all of the steps, I mention removing duplicate numbers. I often left that out to save on space and to keep from getting overly repetitive, but you can assume that once a number is filled in, you should remove that as a possibility from all other spaces in its row, column, and box.
1. The starting grid with numbers 1-9 inserted into all blank spaces.
2. After going through all the pre-filled numbers and removing the duplicates from their columns, rows, and boxes, this is left. There is only one possibility left in the top left-hand corner, a 3.
3. After filling in that 3 and removing the duplicates from its row, column, and box, we find that in the top left box, there are two squares with only 7 and 8 as possibilities. We thus know the numbers 7 and 8 must go in those two squares and none of the others in that box.
In the top row, there are two squares containing only the numbers 2 and 8, so the numbers 2 and 8 must go in those squares and nowhere else in the top row.
4. After removing the 2s, 7s, and 8s as mentioned in the previous step, we have two squares with only one possibility left.
5. After filling in the 2 and 6 and removing their duplicates, we have three more single-possibility squares. We also have only one possible spot left for a 6 in the top right box and only one spot for a 6 in the bottom row, so can fill those in as well.
6. Removing the duplicates from the boxes, columns, and rows for the 4, 8, 9, and 6es gives us two squares containing only one possibility, a 5 in the center box and an 8 in the lower left box. There is also only one 4 left in the center box and only one 6 left in the middle left box.
7. After filling in the numbers from the last step and removing their duplicates, we have two 7s left alone in their squares, in the top and middle left-side boxes. There is only one 8 possibility remaining in the top left square, one 8 possibility in the second row from the top, and one 5 possibility in the center column. Those numbers can all be filled in.
8. Removing the duplicates for our new filled-in numbers gives us just one empty space in the top middle box, which means there is only one possible number to put there - a 3. There is only one space left for a 7 in the upper right box and one space for an 8 in the center box. So next we will fill in those numbers and remove their duplicates from the appropriate rows and columns.
9. In the center box, the numbers 2 and 3 each only appear in the same two squares, so we know those two squares must hold the numbers 2 and 3. We can remove the 7 from the middle left square in that box.
10. Now there is only one 7 left in the center box and one 7 in the fourth column from the left.
11. After filling in the 7s and removing their duplicates (plus a 1 I'd missed in a previous step), there are two squares with only one option left. There is also only one space left for a 6 in the bottom center box and one space for a 7 in the bottom right box.
12. There are two squares in the lower right box with only 1 and 3 as options, so we know none of the other squares in that box can contain a 1 or a 3. In the bottom row, only one of the squares has 2 as an option and the other only has 4 available, so we can fill in those spaces.
13. The bottom left box now has only one square in which 4 can go and one square with only 5 as an option. The fourth row from the top has only one square in which 2 can go and one square with only 3 as an option.
14. The far right column now has only one square in which 3 can go and has one square with only 2 as an option. In the next column over, there is only one square in which 5 can go and one square only has 1 as an option. The bottom right box and the center box each have only one square in which 3 can go. The center column only has one place 5 can go. And the third row from the bottom only has one place where a 3 can go.
15. The top left, top right, middle left, and middle right boxes now all have one square with only one possibility left. Fill those in and, as always, remove the duplicates as possibilities for the rest of the squares in their boxes, rows, and columns.
16. Now each remaining empty space has only one possible answer left. Fill in those spaces and we're done.