### Sudoku Lesson III - Virtual number pairs

Today's lesson is short and continues from Lesson II. As you are marking squares, you may find that the marked possibilities line up in a particular row or column. When that happens, you can project a "virtual" number across the row or up/down the column that the possibilities line up in:

As you can see in the above example, a virtual pair of 3's can be found in the bottom middle square, in boxes (5, 7) and (6, 7). Disjointed pairs of 2's and 5's in boxes (1, 7) and (2, 9) leads to the required virtual number pairs of 4's and 7's in boxes (3, 7) and (3, 8), because these are the only numbers left in the lower left square. Since a four from the lower middle square projects to the left, the solution to these boxes is easily recognized (and in fact, box (3,8) is improperly marked).

Sometimes projection of a virtual pair leads to a solution in another target square, and sometimes it uncovers another virtual pair. In the above example, an unmarked virtual pair of 7's can be solved in the top middle square, and that virtual pair leads to a virtual pair of 7's in the top left square, by projecting the virtual pair we marked in the lower left square. See if you can mark these virtual 7 pairs properly in the top middle and left squares.

Be sure to take advantage of this feature as you cycle through all nine numbers.

As you can see in the above example, a virtual pair of 3's can be found in the bottom middle square, in boxes (5, 7) and (6, 7). Disjointed pairs of 2's and 5's in boxes (1, 7) and (2, 9) leads to the required virtual number pairs of 4's and 7's in boxes (3, 7) and (3, 8), because these are the only numbers left in the lower left square. Since a four from the lower middle square projects to the left, the solution to these boxes is easily recognized (and in fact, box (3,8) is improperly marked).

Sometimes projection of a virtual pair leads to a solution in another target square, and sometimes it uncovers another virtual pair. In the above example, an unmarked virtual pair of 7's can be solved in the top middle square, and that virtual pair leads to a virtual pair of 7's in the top left square, by projecting the virtual pair we marked in the lower left square. See if you can mark these virtual 7 pairs properly in the top middle and left squares.

Be sure to take advantage of this feature as you cycle through all nine numbers.