# Sudoku Puzzle Techniques

A Sudoku grid consists of 81 squares divided into nine columns marked a to i, and nine rows marked 1 to 9. The grid is also divided into nine 3 × 3 sub-grids named boxes marked with squares 1 through 9.

## Scanning technique

The easiest way starting a Sudoku puzzle is to scan rows and columns within each triple-box area, eliminating numbers or squares and finding situations where only a single number can fit into a single square. The scanning technique is fast and usually sufficient to solve easy puzzles all the way to the end. The scanning technique is also very useful for hard puzzles up to the point where no further progress can be made and more advanced solving techniques are required. Here are some ways of using scanning techniques:

#### 1. Scanning in one direction:

In our first example we will focus on box 2, which like any other box in Sudoku must contain 9. Looking at box 1 and box 3 we can see there are already 9s in row 2 and in row 3, which excludes the two bottom rows of box 2 from having 9. This leaves square e1 as the only possible place into which 9 can fit in.

#### 2. Scanning in two directions:

The same technique can be expanded by using information from perpendicular rows and columns. Let’s see where we can place 1 in box 3. In this example, row 1 and row 2 contain 1s, which leaves two empty squares in the bottom of box 3. However, square g4 also contains 1, so no additional 1 is allowed in column g. This means that square i3 is the only place left for 1.

#### 3. Searching for Single Candidates:

Often only one number can be entered in the box because the remaining eight are already used in the relevant rows, columns, and boxes. Looking at square b4, we can see that 3, 4, 7 and 8 are used in the same box, 1 and 6 are used in the same row, and 5 and 9 are used in the same column. Omitting all of the numbers above leaving 2 as the single candidate b4.

#### 4. Remove numbers from rows, columns and boxes:

There are more complex ways to find numbers using the process of elimination. In this example, the 1 in square c8 implies that either square e7 or square e9 must contain 1. Whatever the case, 1 of column e is in square 8 and therefore it is impossible to have 1 in the middle column of square 2. the remaining squares for 1 in square 2 are squares d2.

#### 5. Search for missing numbers in rows and columns:

This method can be very useful when the row (and column) is almost complete. Let’s look at row 6. Seven of the nine squares contain the numbers 1, 2, 3, 4, 5, 8 and 9, which means 6 and 7 are missing. However, 6 cannot be in the cube of h6 because there is already a 6 in that column. Therefore, 6 must be in box b6.

## Analysis technique

As the levels of the Sudoku puzzle get more and more difficult, you will find the simple scanning methods described above are not enough and more sophisticated solving techniques must be used. Hard puzzles require deeper logical analysis which is carried out with the help of pencil marks. Sudoku pencil marking is the systematic process of writing small numbers in squares to show which numbers match. After pencil marking the puzzle, the solver must analyze the result, identify the special number combinations and deduce which numbers should be placed where. Here are some ways to use analysis techniques:

#### 1. Eliminate the box using the Naked Pairs in the box:

In this example, the c7 and c8 squares in box 7 can only contain 4 and 9 as shown by the red pencil mark below. We don’t know which one, but we do know that both boxes are filled. Additionally, box a6 excludes 6 because it is in the left column of box 7. As a result, 6 can only be in box b9. The cases where the same pair can only be placed in two boxes are called Discrete Subsets, and if the Discrete Subsets are easy to see then they are called Naked Pairs.

#### 2. Eliminate squares using Naked Pairs in rows and columns:

The previous solving techniques were useful for summing numbers in rows or columns, not squares. In this example we see that the d9 and f9 squares in box 8 can contain only 2 and 7. Again we don’t know which one, but we do know that both squares are filled. The remaining numbers to place in row 9 are 1, 6 and 8. However, 6 cannot be placed in box a9 or in box i9, so the only possible place is box c9.

#### 3. Remove squares using Hidden Pairs in rows and columns:

The Discrete Subset isn’t always obvious at first glance, in this case it is called the Hidden Pair. If we look at the pencil marks on line 7, we can see that 1 and 4 can only exist in square f7 and square g7. This means that 1 and 4 are Hidden Pairs, and the squares of f7 and g7 cannot contain any other numbers. Using a scanning technique, we see that 7 can only be in the d7 box.

#### 4. Eliminate the box using X-Wing:

The X-Wing technique is used in rare situations that occur in some very difficult puzzles and be a Sudoku Master. Scanning column a we see that 4 can only be on square a2 or square a9. Likewise, 4 can only be in the i2 or i9 squares. Due to the X-Wing pattern in which the boxes are in the same row (or column), a new logical limitation occurs: it is clear that in rows 2, 4 can only be in box a2 or in box i2, and not in any other box even. Hence 4 is excluded from squared c2, and squared c2 must be 2.