### Sudoku Lesson VIII - Completing Markings in a Stack

Let's say you've cycled through the numbers twice, have tried to solve every square, row and column, but you still have a bunch of empty, unmarked boxes, because you have numbers that fit in more than two spots for many rows, columns or squares. For example, continuing with the puzzle from the last lesson, I'm working on the third column, and I notice that 4 and 8 can only go in the top and eighth boxes, so 6 must go in the second box (3,2):

The solution of a 9 in box (1,9) leads to two numbers 1 and 9 in the middle left square:

Next, let's say that you finish checking everything else, and you just can't get anywhere -- you are stuck. Here's my advice: now you can break the mark-only-two rule and finish marking all the dots in three squares in a row or in a column, plus any three additional rows or columns. Go ahead, mark all the possible numbers in every box for every square that you are marking, but skip over those coupled pairs:

The solution of a 9 in box (1,9) leads to two numbers 1 and 9 in the middle left square:

Next, let's say that you finish checking everything else, and you just can't get anywhere -- you are stuck. Here's my advice: now you can break the mark-only-two rule and finish marking all the dots in three squares in a row or in a column, plus any three additional rows or columns. Go ahead, mark all the possible numbers in every box for every square that you are marking, but skip over those coupled pairs:

When you're finished marking a stack of squares or three squares in a row, plus three more rows or columns, you might find a coupled-triple. A coupled-triple is three boxes having the same numbers marked in a particular square, row or column. You can use the same technique for solving squares, rows and columns from Lessons V and VI to skip over these and fill in the number(s) that are left in the square/row/column. But coupled-triples are much more rare than coupled pairs. I can't find a coupled-triple in the rows/columns that I've marked in the above puzzle.

After you've so-marked, move on to the next, and most difficult lesson, but make sure that you put a check mark beside every row/column that you've fully marked the boxes in.

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