How to Solve Sudoku Puzzles
This section explains some of the basic strategies that are used to solve sudoku puzzles. If you want to discover these strategies on your own, feel free to jump right to the puzzles. You can always return here if you need more guidance, or want to compare these strategies to the ones you discover.
Slice and Dice
This is a good strategy to use at the very start of solving a puzzle. Once you get the hang of this technique, it will make filling in numbers seem like grabbing low-hanging fruit.
Figure 1-1: Slicing
To begin, notice that in Figure 1-1 rows 2 and 3 already have 9s in them. This means that no more 9s can be placed anywhere in rows 2 and 3. Now notice that the upper-left 3×3 box does not have a 9 in it yet. Of course, there must be a 9 somewhere in this box. And the 9 must go in row 1 because rows 2 and 3 have been ruled out. The only possibility is the blank square at row 1, column 3 where we correctly enter a 9 Congratulations, you have just sliced a row and found your first number!
In the next example in Figure 1-2, notice that columns 1 and 3 already have 4s in them. Now notice that the upper-left 3×3 box does not have a 4 in it yet. Using similar logic as in the previous example, we know that a 4 must go in this box in column 2. Uh-oh! There are two possible places in column 2: row 1 or row 3. This is where dicing comes into play. We see that row 1 already has a 4 in it, which leaves row 3 as the only possibility. A 4 is correctly placed in row 3, column 2 of the grid. You just sliced and diced a column! This basic strategy can be applied to all columns and rows within 3×3 boxes. With a little practice, you will easily notice where a puzzle can be sliced and diced.
Figure 1-2: Slicing and Dicing
Frequently there will be just one possible number for a square. For example in Figure 1-3, look at the square at row 1, column 1. If we run through all of the numbers 1 to 9 we can see that only a 6 is possible, so a 6 should be placed in row 1, column 1. All numbers except 6 are ruled out because they are already found in the same row, or column, or 3×3 box. When scanning, look for blank squares that have a lot of numbers in their row, column, and 3×3 box. Or sometimes it is necessary to methodically scan all blank squares just to find one elusive entry.
Figure 1-3: Scanning Squares
Scanning Rows, Columns, and 3×3 Boxes
Let's start with rows. Of course, every row must have all of the numbers 1 through 9. So pick a row and ask: Where in this row can a 1 go? (We are assuming a 1 is not already found in the row.) If there is only one possible square in the row, then we can enter a 1 into that square. If we find two or more squares for a 1, then we can skip it and move on. We continue by looking for a unique square in the row for a 2, for a 3, and so on up to 9. After awhile, you will be able to do this surprisingly fast.
Figure 1-4: Scanning Rows
Try this strategy with the first row in Figure 1-4. You should find that a 7 can be entered into column 2. This strategy can be applied to all rows, columns, and 3×3 boxes. You might start with just the areas of the grid that are nearly filled-in; or it might be necessary to go methodically through all rows, columns, and 3×3 boxes.
The previous strategies will not completely solve the more challenging sudoku puzzles, including the puzzles in this book. For these we will need to also take a new approach. To start, enter into each square all of the possible numbers (write small). We will call these numbers candidates. The goal of the rest of the strategies will be to rule out candidates until there is hopefully only one remaining in a square. The figures in the examples on page xiii will include small candidate numbers in some of the squares.
In Figure 1-5, notice that row 6, column 2 has two candidates: 3 and 5. It would seem that we do not have enough information to determine which is the correct candidate: 3 or 5. Okay, now look in the upper-left 3×3 box and see that there are exactly two 5 candidates in this box, both in column 2. We have just found a twin! We do not know which of these two 5s is correct, but we do know that one of the twins will be entered into column 2. As a result, all other 5 candidates should be removed from anywhere in column 2. Breakthrough! This means that row 6, column 2 must be a 3.
Figure 1-5: Twins
Figure 1-6 illustrates another strategy that can be used to rule out candidates. In this case we will determine which candidate is correct in row 1, column 8: the 1, the 4, or the 5. The first row contains a matching couple: the two squares with the candidates 4 and 5 and nothing else. Ultimately one of these squares must be a 4 and the other a 5, we just don't know which one is which. In either case, there can be no other 4s or 5s in row 1. Therefore all other 4 and 5 candidates should be eliminated from row 1. Eureka! We have just determined that row 1 column 8 must be a 1 because that is the only remaining candidate.
Figure 1-6: Matching Couples