Ok, so you've cycled through all the numbers twice, and have done all the projections possible. Cycle through all the numbers twice means that you should try to project the number 1 into all nine squares, then try 2 through 9. As you find solutions, you fill them in. This will change the available spots in each square, so you might be able to solve other numbers that you couldn't the first time through. That's why you start over and do the projections again a second time. I find that when you cycle a third time, you may only solve a paltry few extra numbers, and you time is better spent using other techniques to solve numbers in each square. Thus, you can cycle through a third time, but in my experience, that's going to be a waste of time. Instead, you should proceed with the next few lessons, which will provide you with some powerful Sudoku tools to solve the puzzle.
Sometimes it looks like you're going to get stuck early, with no way to get even a few boxes solved. When this happens, step back and take a look at the forest. Let's take our early puzzle example, which I've marked only using projection, and by cycling through all of the numbers only twice. Here's where we stand:
Pick any three squares in a row or a column that you haven't solved a particular number in yet. See if you have pairs (not necessarily coupling pairs) marked in all three squares. If so, then if there are two pairs in the same two rows or same two columns, then another different number, marked in a third pair in the last square in that same row or column as the other two pairs is a solution. Remember, a square is a set of nine boxes, illustrated with the bold lines surrounding it. There are nine squares in a sudoku puzzle.
Confused? Let's look at an example. Take the number 8 for example. Look at the middle row of squares. Can you see what I'm talking about? I can mark an 8 in the two middle boxes of the center square at (5,5) and (6,5), because 8 is marked in the top and bottom rows in the left and right middle squares, at (3,4), (3,6), (7,4) and (7,6). This is because, if the left middle square has an 8 in the top row of the middle squares, i.e.
, row 4, then the right middle square must have an eight in the bottom row 6, and vice-versa. Therefore, the only place that an 8 can go in the center square is in the middle row 5, so this is how I mark the center square:
Continue to look for potential solutions using this tool. Sometimes there will only be two boxes that you will mark, and sometimes you will find only one box, which you will solve.