Elimination

Elimination looks at the empty spaces in a row, column, or block, determines which numbers are missing, and attempts to fill the spaces based on the location of the missing numbers elsewhere in the puzzle.

It's best to start with the row, column, or block that has the fewest open spaces. The fewer the spaces, the more likely we'll be able to figure out how to fill them. Here's the puzzle we did two rounds of triangulation on:

There are two empty spaces in the eighth row and the missing numbers are 6 and 7. Because there is already a 7 in the last column, there can only be a 6 in that column and, therefore, a 7 in the last empty space in the row (Or visa versa, because there is already a 6 in the sixth column, a 7 can be placed in the eighth block.):

In the fourth row, 3, 4, and 5 are missing. With 4 and 5 in the fourth column, we can put in a 3:

4 and 5 belong in the remaining spaces but there isn't any way to eliminate either of them. They are ghost numbers but knowing that they belong there is good information so keep it in mind. With difficult puzzles, it is helpful to pencil them in. We now know the location of seven numbers in block five. We can pencil in ghost numbers 1 and 8 as well in this block:

But there is already an 8 in the fifth column so we can put in the numbers 1, 4, and 8 in the sixth row:

This is where the line between triangulation and elimination blur. We can either shade columns six and seven and put an 8 in the second block and 1 in the eighth block because we've eliminated all the other numbers. Or see that in the fourth column 1 and 8 are missing but we can put a 1 in the eighth block because there is already an 8 there and put 8 in the second block. The result is the same. And we've eliminated all the numbers but 6 in the eighth block. Here's the current situation:

It pays to check frequently if we can triangulate any numbers. Going through the numbers yields a 6 in the seventh block, an 8 in block three, and 9s in blocks three, six, and nine:

A second round yields 1 in block three and 7s in blocks seven and nine:

That leaves only 3 and 4 in the last column but no way to eliminate either number. We'll put in ghost numbers:

1 and 8 are missing in the second column. With a 1 in the last row, we can put 1 and 8 in the column. In addition, we now know that 5 and 8 are ghost numbers in the fourth block:

Only 1 and 4 are missing in the seventh block. Because there is a 1 in the last row, we can put in the 1 and 4:

With 4 in the last row, we can alternately eliminate 3 and 4 in the last column:

1 and 5 are missing in the first row. We can place them based on the 1 in the fifth column. In the fourth row, 4 and 5 are missing. With 5 in the fifth column, we can place 4 and 5 in the fourth row:

I wouldn't bore you with the last few steps. Here the finished puzzle:

The position of the numbers can change the difficulty and approach. You may encounter a sudoku puzzle for which you use elimination first. Here is an example: