Sunday, September 02, 2007

Sudoku Lesson IV - Cycle Once and Take Advantage of Pairs

Sometimes you'll find that two pairs of numbers are marked in a pair of boxes. When these boxes line up in the same row or column, or are in the same square, they are known as "coupling pairs." During the second projection cycle, you should take advantage of such marked pairs. In this lesson, I will show you how to do that.

In the above puzzle, I've marked all of the boxes in the lower three squares, for illustrative purposes only. DON'T DO THIS, as you should only mark boxes only if there is two occurrences of a number in the box. This marking is only to help you understand the principle of coupling pairs.

The numbers 2 and 5 in the bottom row, boxes (2, 9) and (6, 9) form a coupling pair in the bottom row. Consequently, boxes (4, 9) and (5, 9) can have all occurrences of 2 and 5 removed. This forms a second coupling pair of the numbers 7 and 9 within the lower middle square. Any other occurrences of 7 and 9 can be removed from that square. Box (5, 8) forms yet another coupling pair with box (6, 9) using numbers 2 and 5, and a new coupling pair is revealed in row 7. Here is the resultant marking:

I've taken the liberty of removing the markings of 2 and 6 from the 7th column, since they must go in the 7th column in the top right box.

Remember in Lesson II when I told you to mark only boxes when there were two and no more than two possibilities? This is so that you can establish and recognize coupling pairs. I cheated above, just to show you how the coupling pairs come about, but don't cheat. Why is that so important? I'll tell you in the next two lessons. In the meantime, cycle through the numbers two times, like I've been telling you, but use these pairs to your solving advantage.

As you project and find solutions in a box, if the box already has a dot for a different number, find the other spot in that square with the dot and fill in that number too. It's like getting two-for-one.

In the above example, I marked all the 1s and 2s. As I started to mark the 3s, I found that the top square had only one spot for a 3, in (3, 2), due to the upward projection of the virtual number pair of 3 from the bottom left square. So now, I can put a 3 in (3, 2) and a 2 in box (3, 3). After getting a two-for-one, then try to figure out why coupling pairs are so cool.