The Magic of Pairs

Triangulation and Elimination works well for sudoku puzzles through medium difficulty but harder puzzles requires other techniques. Seeking pairs of numbers lifts the level of skill needed to find the locations of numbers in the puzzle. Whenever, I attempt a hard or expert puzzle, I always start from the beginning noting pairs of each number. Here's a hard Sudoku puzzle I'll use as an example:

First, I'll use triangulation to find as many numbers as possible but will put in pairs of numbers as well. Let's start.

As I work through the puzzle, at number four I can use ghosts in row six, columns four and five to eliminate the four in row six, column seven and insert a four in row five, column 7:

At five, I can place a 5 using the 5 in row seven, column three and ghosts in row eight, columns four and five in row nine, column nine. Then using elimination, place a 3 and 7 in the remaining spaces of row nine:

Now I see several things:

• Eliminate a 3 and place a 3 in row one, column two. Whenever you can eliminate one number of a pair, then the remaining number must be placed in that space.
• I can triangulate all of the 7s and in doing so eliminate the 1 in row four, column nine leaving the 1 in row six, column seven which eleminates the 1 below leaving the 1 in row seven, column nine.

From now on I'm going to just show the puzzle as I find pairs and use triangulation and elemination to place numbers without explaination: