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?
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
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
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!
SUDOKU EASY TO HARD PRESENTED BY WILL SHORTZ,
New York: St. Martin's Grffin, 2005.