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Seeking a "chain" is nearly the last resort in solving a Sudoku puzzle. This is my attempt to explain how it's done.
First, the disclaimer: Finding a "chain" is confusing, time-consuming, over-whelmingly full of dead-ends, and thus frustrating. I suppose I say this to make you feel "okay" when you become confused, frustrated, and upset about the time you're wasting!
In practice, you will probably NOT follow the exact order listed below, but I find it the best way to describe the necessary skills.
I recommend that you print these rules.
1. Be sure you've exhausted all other methods for teasing out solutions and eliminations
2. Use an ink pen to circle the tiny numbers in all empty boxes that have only two possibilites - which I call "doubles:
»»3. Focus on one of the "doubles"
4. Mark its "zone", i.e., its GROUP, ROW, and COLUMN. (This very messy step is used for learning purposes only.)
Selecting a "source": Steps 5-7
5. Focus on ONE of the numbers in the "double". I will call this number the "source". Do a quick glance to see if the source appears in any double OUTSIDE the zone. If not, jump to step #20!
6. Briefly scan for a "way out" for source. If there is none, skip to step #20!
7. IF a "way out" exists, look for any "possible elimination". This is any tiny number INSIDE THE ZONE that is the same as the "source." They do NOT have to be found only in doubles.
If none exist, skip to step 20.
If you find even one, you FINALLY can make a faint mark on the source.
For learning purpose only, you should make tiny marks on all the "possible eliminations".
Selecting "Targets": Steps 8 - 11
8. Scan OUTSIDE the zone for doubles that contain the source number. These numbers are potential "targets"
Time saver: do not bother with doubles that were previously used as a source.
9. For each potential target, decide if ...
(1) it is linked to one of the "possible eliminations" that you marked in step 7 and if
(2) on cursory examination - it might be an ending of a chain. This is done by seeing if its "buddy" (the other number of the double) can be linked to ANY double ANYWHERE.
As always in Sudoku, two numbers are "linked" if they are in the same row, column, or 3x3 group.
CHAINS are extended when the "buddy" in a double can be linked to another double.
10. IF both conditions of step #9 are true, then faintly mark the number. This number is now a "target" (which - sigh - you will fail to hit 98% of the time.)
11. Continue finding all targets. (steps 8-10) They are always OUTSIDE of the zone.
12. After finding all targets, look back at the source.
13. Consider the OTHER number that's with the source, its "buddy".
14. Beginning with the "buddy", seek a path of linked doubles that will END UP on any one of the targets. This is what you'll spend most of your time doing, because it's often difficult to know if failure is caused by impossibility, or by lack of trying.
15. Erase marks from targets as they become known failures. There no simple method to predict which ones this will be, or in what order it will occur. You WILL however recognize it after a little practice.
16. IF YOU SUCCEED at a target, you can cross out any linked "possible eliminations" that are INSIDE the zone.
17. If you succeeded, see if simpler methods can produce further progress. Hopefully, you'll finish the puzzle!
18. Given up working with the particular "source"? Erase marks from all targets.
19. Learning only: Erase your "possible eliminations"
20. If you only tried one number in the blue box as a "source", you need to
Erase any marks you made for targets or possible eliminations.'
Go back to step 5, but now swap "buddy" and "source".
otherwise:
21. Still here?
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