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.

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:

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.

Sudoku Lesson VII - Solving Rows and Columns

Let's continue with the puzzle from Lesson VI. Much like solving squares, solving rows and columns can take advantage of coupling pairs to skip over them and find numbers that remain.

I like to take the following approach when solving rows and columns:

1. Try to find a solution for the blank boxes first. Often, a result of the cycle is that there are some boxes left blank. Many times, these boxes might only have one number that fits. If so, write it in, then eliminate dots for that number in that box's row and column, and put in dots in other squares that have not yet been marked. If you eliminate a dot, be sure to check the square for a corresponding dot, and write the number in there too, then wash, rinse and repeat before going back to the rest of the row/column that you were checking.


For example, let's attack column 4. Note that 1 can only go in (4,5). We put that in, and then mark the left and right middle squares for 1:




















All that is left is to put a 9 in box (4,7), and column 4 is done, check mark it, then erase the 9 dot in (8,7), put a 9 in (8,8), and put a 9 in lower left square at (1,9), thus solving the 2 in (3,9).























2. Next, check all of the one-dot boxes. If the dot indicates the only number that fits, write it in and eliminate any other dots for that number in that box's square. Then follow the process for 1. above.


3. Next, check all of the two-dot boxes. Remember the rule for coupling pairs from Lesson VI.

If you happen to mark all the boxes in a row or a column, put a check mark beside the row or column to indicate that you've completely marked the row/column. This helps prevent wasting time, and will help you later when you bifurcate. For example, I'm going to mark rows 3 and 9:

Feedback on Sudoku

How do you like my Sudoku lessons so far? Are you learning from them, or do they just bore you?

I've often been accused of leaving too many steps out between one proposition and the next, in effect, taking incomprehensible quantum leaps. Please tell me if this is the case, so that I can further explain and make my expression simpler, easier to understand, and hopefully useful. I plan to submit the ten lessons to a publisher, so your help is greatly appreciated.

A reader writes: "You lost me at 'cycled through all the numbers twice.' That's some long commute you have, yes?"

G-man: Yes, I developed these lessons and my method to solve Sudoku on my one-hour commute. 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. The other techniques are demonstrated in the subsequent lessons. I've updated lesson V per your query.

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