### Puzzlepedia^{™} Logic Puzzles

# Sudoku

Fill the grid with the numbers 1-9 so that the same number does not exist in the same column, row or box.

Sudoku is a logic-based number placement puzzle. The objective is to fill a 9x9 grid so that each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9 only once. The puzzle starts with a partially solved grid.

Sudoku was invented by an American architect, Howard Garns, in 1979 and published by Dell Magazines under the name "Number Place". It became popular in Japan after it was published by Nikoli and given the name Sudoku, meaning single number.

### Solution Methods

**Single Solution Method**

The Single Solution method is the first step that most people use to solve Sudoku puzzles. Many of the easy levels can be solved entirely using this method. This method is also known as the Naked Single, Sole Candidate, or SiSo method.

The first thing you need to do is focus on an empty cell. Use pencil marks to mark every possible number that could go into that empty cell. You will need to look at the row, column and sub-square to determine which numbers can go into the empty cell.

In the first example to the left we are examining the yellow square. By looking at the row, we can eliminate numbers 2 and 3. By looking at the column we can eliminated 7, 5, and 6. By looking at the sub-square we can additionally eliminate number 8. Now, we know that the yellow square may only contain the numbers 1, 4 or 9. You can use pencil marks to indicate this information.

If you determine that only one number can fit in a cell, then you have solved it. In the second example we can eliminate everything except the number 9, so we have found the solution.

When you have examined each cell once, you will need to go back and try again. Usually the entire puzzle can be solved in 2 or 3 iterations.

**Single Cell Method**

The "Single Cell" method is the second step that most people use to solve Sudoku puzzles. By using this method along with the Single Solution method, all the easy levels can be solved.

The first thing you need to do is use the Single Solution method to place pencil marks into each cell corresponding to the possible values for that cell. Now, focus on one row. Look at the pencil marks for each empty cell in that row. If there is a pencil mark that only appears once, then that must be the solution for that cell.

In this example, we are inspecting the top row. Notice how the pencil mark 3 only appears once in that row. It must therefore be the solution for that cell.

You can repeat this process for each row, column and sub-square.

A variation of this method is to look for places where a number is forced. In the second example on the left, the green square must be a 3 because the 3 has nowhere else to go in the middle sub-square.

**Locked Candidates Method**

The "Locked Candidates" method is a more advanced method that can be used when the Single Solution and Single Cell methods cannot solve the puzzle on their own. This may be required for medium difficulty puzzles. This method is also known as the Single Box or Block-Block method.

In the first example on the left, look at the center sub-square. Notice how the number 2 can only appear in the top row. Since it must appear in the top row of the center sub-square, we know that it cannot appear in the top row of the adjacent sub-square. Thus we can eliminate the number 2 as a possibility from the cells marked with yellow.

In the second example, look at the fourth row. Notice how the number 2 must occur in the top row of the middle right sub-square. Since it must occur in one of these two spots, we can eliminate 2 as a possibility in the cells marked with yellow.

**Naked Pair Method**

The Naked Pair method is a more advanced method that can be used when other methods cannot solve the puzzle. This may be required for medium difficulty puzzles. This method is also known as the Disjoint Subset method.

In both of these examples, look at the cells marked with green. Notice how the numbers 1 and 7 are both the only possibilities. These are a naked pair. Since 1 and 7 must occur in these two cells, we can eliminate 1 and 7 from the cells marked with red.

A Naked Pair can become a Hidden Pair if there are extra possible values for those cell. In this third example on the left, the cells marked with green make a Hidden Pair with the numbers 1 and 7. We can safely remove 3 and 8 as candidates for the green cells because we know that they must contain either 1 or 7.

**Disjoint Chain Method**

The "Disjoint Chain" method is a more advanced method that can be used when other methods cannot solve the puzzle. This may be required for the hard difficulty puzzles. This method is also known as the Naked Triples or Hidden Triples method.

Look at the green squares in this first example. Notice how the numbers 1, 2 and 3 are the only candidates for these cells. This is called a chain. Since the numbers 1, 2 and 3 must appear in these three cells, they cannot appear in any other cells in this sub-square. In particular, we can remove the numbers 1 and 3 from the red cells.

This second example is essentially the same as the first. It has just been rearranged a little bit to demonstrate that the chain can appear in a row or column just as easily as it can appear in a sub-square.

Additionally, you could have a chain of 4 or more cells, although these are generally difficult to identify.

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