Tuesday, October 16, 2007

Sudoku Lesson IX - Bifurcation

WARNING - This technique pertains only to the boxes in the rows or columns that you have completely marked. You can check whether you've fully marked a box by looking at the check marks you've made at the perimeters. Do not try this on incompleted boxes, it will not work. I've borked many a puzzle by forgetting this rule. Your only other option is to completely mark the puzzle, and it may come down to that, but let's try to solve some of the remaining boxes first, shall we? Let's consider a puzzle we started in Lesson I. I've marked it completely and have solved as many boxes as I could, but now I'm stuck:

Now, pick a marked box having only two numbers marked in it. Try not to pick a coupled pair -- they won't get you very far. Make some indication of this starting box, perhaps by drawing an outline around the box. I picked (4,9):

Slash one of the dots indicating a marked number. Pretend like that number is in the box. If that number is in the box, use all of the techniques that I taught you in the previous lessons to solve another box. Slash the dot indicating the solution for that box. Continue onward until you find that you can't solve anymore boxes, or you find a conflict. In my example, I slashed the 9 dot. That led me to slash the following dots in the succeeding boxes:
7 in (4,3); 2 in (4,6); 7 in (5,6); 5 in (4,8); 4 in (1,8); 5 in (1,7); 6 in (7,8); 4 in (9,9); 2 in (6,8); 6 in (5,9); 4 in (7,6); 6 in (8,6); and 8 in (4,2). The puzzle now looks like this:

A conflict is where you have slashed the same dot for two boxes in any row, column or square. If you find a conflict, then go back to the starting box and write in the unslashed number. For example, I continue to slash dots in the above puzzle, in the following order: 4 in (8,2); 2 in (6,2). But I note that there is already a 2 marked in (6,8), thus creating a conflict:

So now, since picking a 9 in (4,9) lead to a conflict, I must pick the non-slashed dot, starting with the box (4,9), and then continue to fill in numbers as I go. I filled in 4 in (4,9); 6 in (9,9); 4 in (7,8); 9 in (5,9); 5 in (1,8); 4 in (1,7); 5 in (4,7); 4 in (8,6); 6 in (7,6); 2 in (4,8); and 6 in (6,8). The puzzle now looks like this:

If you are just stuck, go back to the starting box and put a backslash (or some other tick mark of your own choosing) on the other dot. Then repeat this process, using the other indicator, until you reach a conflict, or get stuck. If you reach a conflict, write in the slash dot number in the starting box. This did not happen for our example, the rest of the numbers followed, and the puzzle was easily completed:

If you are not so lucky, don't give up. You tried both ways, and neither path reached a conflict. Go back and mark the remaining squares, then try to solve more /s and \s to find a conflict for either branch. Mark the whole puzzle if you must.

Have you reached the end? If you have marked all the boxes in the entire puzzle and can / all the boxes without reaching a conflict, then you got lucky and picked the right branch! All the /s represent the solution to the puzzle. Congratulations, write all the numbers in.

If you mark the same dot with / and \, forming an X, then that box is solved. Write in the number, and see if you can solve other boxes with that number.

If you are still stuck, all the possible / and \ are indicated, no conflicts appear, then move on to the next lesson.