Step-By-Step
How to Solve Sudoku

A sudoku puzzle consists of a 9 × 9–square grid subdivided into nine 3 × 3 boxes. Some of the squares contain numbers. The object is to fill in the remaining squares so that every row, every column, and every 3 × 3 box contains each of the numbers from 1 to 9 exactly once.

Solving a Sudoku puzzle involves pure logic. No guesswork is needed—or even desirable. Getting started involves mastering just a few simple techniques.

Take the example on this page (in which we’ve labeled the nine 3 × 3 boxes A to I as shown). Note that boxes H and I already have 8’s filled in but G does not. Can you determine where the 8 goes here?

 A B C D E F G H I

 5
 8
 6
 1
 2
 5
 2
 8
 6
 2
 4
 8
 1
 3
 5
 3
 9
 8
 1
 2
 4
 4
 5
 6
 7
 3
 8
 5
 2
 3
 8
 1
 7
 8
 3
 6
 5

The 8 can’t appear in the top row of squares in box G, because an 8 already appears in the top row of I—and no number can be repeated in a row. Similarly, it can’t appear in the middle row of G, because an 8 already appears in the middle row of H. So, by process of elimination, an 8 must appear in the bottom row of G. Since only one square in this row is empty—next to the 3 and 6—you have your first answer. Fill in an 8 to the right of the 6.

Next, look in the three left-handed boxes of the grid, A, D, and G. An 8 appears in both A and G (the latter being the one you just entered). In box A, the 8 appears in the middle column, while in G the 8 appears on the right. By elimination, in box D, an 8 must go in the leftmost column. But which square? The column here has two squares open.

The answer is forced by box E. Here an 8 appears in the middle row. This means an 8 cannot appear in the middle row of D. Therefore, it must appear in the top row of the leftmost column of D. You have your second answer.

In solving a sudoku, build on the answers you’ve filled in as far as possible—left, right, up, and down—before moving on.

For a different kind of logic, consider the sixth row of numbers—4, ?, 5, 6, ?, ?, 7, 3, 8. The missing numbers must be 1, 2, and 9, in some order. The sixth square can’t be a 1, because box E already has a 1. And it can’t be a 2, because a 2 already appears in the sixth column in box B. So the sixth square in the sixth row has to be a 9. Fill this in.

Now you’re left with just 1 and 2 for the empty squares of this row. The fifth square can’t be a 1, because box E already has a 1. So the fifth square must be a 2. The second square, by elimination, has a 1. Voilà! Your first complete row is filled in.

Box E now has only two empty squares, so this is a good spot to consider next. Only the 4 and 7 remain to be filled in. The leftmost square of the middle row can’t be a 4, because a 4 already appears in this row in box F. So it must be 7. The remaining square must be 4. Your first complete box is done.

One more tip, and then you’re on your own.

Consider 3’s the boxes A, B, and C. Only one 3 is filled in—in the ninth row, in box C. In box A you don’t have enough information to fill in a 3 yet. However, you know the 3 can’t appear in A’s bottom row, because 3 appears in the bottom row of C. And it can’t appear in the top row, because that row is already done. Therefore, it must appear in the middle row. Which square you don’t know yet. But now, by elimination, you do know that in box B a 3 must appear in the top row. Specifically, it must appear in the fourth column, because 3’s already appear in the fifth and sixth columns of E and H. Fill this in.

Following logic, using these and other techniques left for you to discover, you can work your way around the grid, filling in the rest of the missing numbers. The complete solution is shown below.

 5
 8
 6
 3
 7
 4
 9
 1
 2
 1
 3
 7
 9
 5
 2
 8
 6
 4
 2
 4
 9
 8
 1
 6
 5
 7
 3
 8
 7
 2
 5
 4
 3
 1
 9
 6
 6
 9
 3
 7
 8
 1
 2
 4
 5
 4
 1
 5
 6
 2
 9
 7
 3
 8
 9
 5
 4
 2
 3
 7
 6
 8
 1
 7
 2
 1
 4
 6
 8
 3
 5
 9
 3
 6
 8
 1
 9
 5
 4
 2
 7

Remember, don’t guess. Be careful not to repeat a number where you shouldn’t, because a wrong answer may force you to start over. And don’t give up. Soon you’ll be a sudoku master!

From SUDOKU EASY TO HARD PRESENTED BY WILL SHORTZ, VOLUME 2.
New York: St. Martin's Grffin, 2005.