2
169
135689  13589 3689 7  58 15 4
 167 167 135678  4 368 1356  258 125 9
 14 149 1589  1589 2 159  6 3 7
 47 8 79  6 5 39  1 49 2
 1467 3 1679  79 49 29  459 8 56
 5 469 2  39 1 8  349 7 36
 9 5 4  138 7 136  23 126 136
 8 1267 167  1359 369 4  23579 12569 1356
 3 167 167  2 69 1569  4579 14569 8

Initial Markup

The standard form of markup for sudoku is a list of all candidate values in the top of each cell. This is not the only possible form of markup, and we will discuss other notation below—but this is the most useful form.

First, make sure that you understand what the candidate values represent. Look at cell (1,2) in the first row, second column. The first row contains a 2, a 4 and a 7. The second column contains a 3, a 5 and an 8. The 3×3 box in the upper left contains a 2. Therefore, cell (1,2) neighbors 2, 3, 4, 5, 7 and 8. The remaining values for that cell are 1, 6 and 9.

Now, if we look at the candidate values in this puzzle, we can determine the value of some additional cells. These are the values that can be determined right away, using elimination alone:

Note that the 3×3 boxes are conventionally numbered 123 across the top, then 456 in the middle group, and 789 across the bottom.

These aren't the only deductions that you might make at this point, and it's worth illustrating how you can scan for block interactions. If I'm looking at the threes in this puzzle, then there are three known locations in boxes 3, 4, and 7. In three of the remaining boxes, two columns or a row and a column are eliminated because of the known locations. We can see that the three in box 1 must be in column 3. The three in box 9 is in one of four cells in columns 7 and 9 and in rows 7 and 8. Finally, the three in box 6 must be in row 6. We notice this because the three in column 8 eliminates the only possible cell in row 4, and the three in box 4 eliminates the cells in row 5. So the three in box 6 must be in row 6. But this has implications for box 5. It means that the three in box 5 cannot be in row 6; therefore, it must be in row 4. In fact, it must be in cell (4,6).

You might just as easily have noticed that cell (4,6) is the only location in row 4 that can hold a three. It is very common to find a hidden single in a row or column because of a block interaction, instead of noticing the hidden single directly.

Make sure you understand how to see the block interaction in the markup. If you look at box 6, there are five empty cells. Only two of these cells could hold a three and both of those cells are in row 6. Therefore, the other cells in row 6 cannot hold a three. This eliminates three as a candidate in cell (6,4), and only one cell in box 5 can now hold a three.

 
 2
169
135689  13589 3689 7  58 15 4
 167 167 135678  4 368 1356  258 125 9
 14 149 1589  1589 2 159  6 3 7
 47 8 79  6 5 (3)  1 49 2
 1467 3 1679  (7) (4) (2)  459 8 56
 5 469 2  39 1 8  349 7 36
 9 5 4  (8) 7 136  23 126 136
 8 (2) 167  1359 369 4  23579 12569 1356
 3 167 167  2 69 1569  4579 14569 8

Maintaining the Markup

If you're working a puzzle by hand, you need to maintain the markup as you fill in more known values. There are six different values that we can fill in right now. As we enter each value, we have to remove candidates from the list in each neighboring cell. The new values appear in parentheses below, and the removed candidates in the empty cells appear in red. When you are marking up a puzzle on paper, you'll have to cross out the eliminated candidate values. Because of this, you'll often want to wait as long as possible before you mark up your puzzle.

In this particular puzzle, there are two more "easy" deductions. In cell (6,4), only one candidate value remains, so (6,4) = 9. Of course, this is also the only location left in box 5. The other deduction is (3,3) = 8. Can you see which deduction from the previous group allowed us to make this deduction now?

After making these deductions and performing the resulting eliminations, we get the puzzle below:

 
 2
169
13569  135 3689 7  58 15 4
 167 167 13567  4 368 156  258 125 9
 14 149 8  15 2 159  6 3 7
 47 8 79  6 5 3  1 49 2
 16 3 169  7 4 2  59 8 56
 5 46 2  9 1 8  34 7 36
 9 5 4  8 7 16  23 126 136
 8 2 167  135 369 4  3579 1569 1356
 3 167 167  2 69 1569  4579 14569 8

Marking Strong Links

I also use another form of markup that helps me avoid retracing my steps and also helps me spot more advanced deductions. Often, you learn that there are only two possible locations for a value in a particular box, row or column. If one of those possibilities is eliminated, then you can immediately fill in the other. There is no standard way to mark a puzzle grid with this information. When solving by hand, I will often mark in the bottom of each cell whenever I discover there are only two locations left in a box. If I notice this in a row or column (but not in a box), then I note that situation at the end of the row or column.

Often, this is situation also leads to other more advanced deductions, and it is important enough to be given a specific name: a strong link exists for a particular value between two cells, whenever the two cells are in the same row, column or box, and that value must be in exactly one of those two cells.

 
 2
169
9
13569
35
 135
3689
89
7
58
8
15
1
4
 8
 167 167
13567
35
 4
368
8
156
258
28
125
12
9
 3
8
14
4
149
49
8
15
5
2
159
59
 6 3 7
 9
47
4
8
79
9
 6 5 3  1
49
49
2
 4
9
16
1
3
169
19
 7 4 2
59
59
8
56
56
 9
 5
46
4
2  9 1 8
34
34
7
36
36
 4
6
 9 5 4  8 7 16
23
23
126
2
136
13
 8 2 167
135
35
369
3
4
3579
7
1569
1356
1
 7
 3 167 167  2 69
1569
5
4579
47
14569
4
8
46
47
3
9
24
24
5

In this grid I've marked up all of the strong links. Before we look at the results, you should note that this complete markup of strong links is unusual. I note strong links when I see them, and I do not make any attempt to make notation in numerical order, or to note every link. In contrast, the standard markup at the top of the cell lists every possibile value for that cell. If you apply standard markup to a cell, listing every possibility, you must never, ever make a less-than-complete list of possibilities for that cell. You may however, apply standard markup to some cells, but not to all cells. I often do this by noting doublets (cells with only two possible values) whenever I notice them, even before I start to do a complete markup.

There is a lot of information in the markup of strong links. Let's start with the block interactions.

Most of the strong links are in a single row or column in a single box. I always mark these inside the box, so that the notation at the end of the row or column means that there is a strong link between boxes. Sometimes the strong link exists in the row or column, but it doesn't exist yet in the box. Look at nines in column 1 to see what I mean. There are only two possible locations for a nine in column 1, and both of those locations are in box 1. Because they are both in box 1, I note the strong link in the bottom of the cells, not at the end of the column. But, wait…there are more than two possible locations for a nine in box 1. This is not a problem, it is an opportunity—we can delete the candidate nine in cell (1,3), because of the block interaction for nines between column 2 and box 1. That's why I marked the candidate nine in (1,3) red, to show that it will be deleted.

The same situation happened with the block interaction for fives between row 3 and box 2, threes between row 7 and box 9, and with the block interaction for ones between column 9 and box 9. There are also places where the strong link within a box implies the deletion of candidate values elsewhere in the same row or column. This happens for ones in box 3 implying deletions elsewhere in column 8, sixes in box 6 implying deletions elsewhere in column 9, and threes in box 8 implying deletions elsewhere in row 8. Of course, when you delete candidate values, you can make additional deductions. For example, there will be a strong link for ones in row 7 after deleting the candidate one in cell (7,8).

With this markup, you can see more advanced deductions. In box 1, the threes and fives must be in cells (1,3) and (2,3), so those cells can't hold any other value. This is a hidden pair. After deleting the candidate five in cell (2,6), cells (2,6) and (7,6) must hold one and six, so those values can't appear anywhere else in column 6. This is a naked pair, because you can read it directly from the standard markup. (One of the biggest advantages of the nonstandard markup is that it makes it easier to spot the hidden pairs.) Also, you can see implication chains by looking at the cells that have only two possible values. If two such cells neighbor each other and share one of the values, then they couple to each other. You can form chains of three or more such cells that loop back on themselves. Look at the chain in cells (3,1)(5,1)(6,2). Cells (3,1) and (5,1) both have a candidate value one, if one of those cells holds a one, then the other does not. But cells (5,1) and (6,2) share a candidate value six. Now look what happens if cell (3,2) holds a four. Cell (3,1) cannot be four, so it must be one; cell (5,1) cannot be one, so it must be six; cell (6,2) cannot be six, so it must be four—but then there would be two fours in column 2; therefore, cell (3,2) cannot hold a four. This is an xy-wing, which is a short forced chain.

 

Please send responses to my email address is mathrec at this domain

Thanks,
Steve