How to Solve Sudoku Puzzles
Did you know that sudoku didn’t originate in Japan? The beginning concept was a mathematical diagram called a Magic Square, first seen in China over 2,000 years ago. In this puzzle, you had to draw a square, and fill each position with a number only once, such that all rows, columns, and both diagonals added up to the same sum.
The basic concept that later led to Sudoku came from Swiss mathematician Leonhard Euler in 1707. He changed the rules to make it a challenge of permutations, calling it Latin Squares. In his puzzle, each square was filled with a letter from the Latin alphabet, combined with a letter from the Greek alphabet, and no combination could be repeated in any row or column.
French newspapers between 1890 and 1920 tried to introduce Magic Squares as a puzzle, by removing portions of the grid, and people had to work out what was missing. It involved calculation, instead of just determining positions of missing numbers. They simplified it to only require the numbers 1-9, but because of the calculation challenge, it didn’t catch on.
Much later, in 1979, Howard Garnes introduced a puzzle he called Number Place in Dell Puzzle Magazine. He took the row and column permutation concept that required no math, and added an additional constraint, a set of 3×3 sections that also had to be filled with 1-9.
When the puzzle arrived in Japan, Nikoli puzzle magazine named it Sūji wa dokushin ni kagiru (literally “the numbers must occur once only”), later shortened to Sudoku (“number only” or “number single”). They also added the convention that starting numbers had to be symmetrical in the grid placement, for a touch of visual elegance. This final form became extremely popular all around the world.
The rules are fairly simple. Fill out the entire grid with the numbers 1-9 so that:
- Each row, column, and 3×3 region contains all 9 digits
- No row, column, or region may have any number repeated.
For our example puzzle, we’ll focus on a single grid, however the solving techniques are largely the same for the variations on Sudoku you can find at the bottom of this page. They may introduce extra regions, extra 9×9 grids, change the numbers to symbols and letters, or even add new placement restrictions.
It is important to remember that at its core, Sudoku is a puzzle about the process of elimination. Whether you are eliminating all the possible squares that a number can be until only one location remains, or if you are eliminating the possible numbers that can be in any given square, the concept is the same.
There are many ways to visually scan the puzzle to find possible moves. Here, you see scanning in a single direction, checking rows across sections. Notice that these 6’s eliminate the top two rows of the third section. This means that the green square is the only possible place that a 6 can be placed in that section.
Here we can see two more single-direction scans. The yellow highlights show the spaces in the columns already containing a 7. That leaves only one space in the bottom middle section able to hold a 7.
Meanwhile, the bottom two rows with the red highlight show that the bottom rows of the first section on the bottom has only one space that can contain an 8.
Sometimes, you only need one column or row for a single-direction scan. Looking at the column containing this 6, there is only one space left in the bottom middle section for another 6 to go.
The second most common technique is to scan in two directions, again, mostly looking for spaces you can eliminate in a given section.
This time, we look down these two columns with the 6’s, and use the 6 along the bottom row to eliminate all squares but one in the section on the bottom left of the puzzle. This tells us where a 6 must be placed.
In this section, we see that the only numbers not used yet are 3 and 9. Since the 3 is already in the highlighted row, the green square must contain the 9, and the other square is the 3.
As you’ve seen, single and double-scanning will get you through a majority of the cells in most puzzles.
However, sometimes, you need to resort to pencil marks, where you examine what numbers are still possible in the remaining cells. You will be writing in what isn’t already duplicated in that cell’s row, column, and section.
Typically, you’ll only fill in the marks in areas where there are a lot of numbers around it, because the process is a little time-consuming. Here, I went a little overboard and filled in the possible numbers in the rest of the puzzle. As you can see, we found two places where there’s only one number left.
Now, in updating the pencil marks from the new placements, it looks like we revealed another square with a single number, but before we get to that, I want to point out another technique.
Notice that in this column, the top section has only 1 and 9 as possible numbers remaining. Because they are the only two cells left in that section, those numbers MUST go there.
As a result, neither 1 nor 9 can be anywhere else in the column, so we can eliminate those marks as possibilities. While I mark those two 9’s out, I’ll also add the 4 on the right side of the puzzle.
This same principle applies within a section, as well. Here, we see that the 3 and 8 are the only numbers left for these two cells.
So you can eliminate 3 and 8 not only along this row, but also where they exist in the section, because they cannot be duplicated in the same 3×3 area.
That’s going to give us a 9 in the highlighted row. I’ll also fill in the 9 revealed as the last number possible near the top right of the grid.
There is another way the pencil marks can be useful.
Look at the green cell in this section. Notice that while there are several possible numbers for that position, it is now the only cell in this section that can contain a 3.
So, we can place a 3 in this position.
Sometimes, when scanning for your next move, you’ll notice a minor error you made several steps ago. I’m sure astute observers caught this error when I first started talking about pencil marks. The middle section could have had the 1 eliminated from the cell in this row.
It’s okay – these things happen. As I said, filling in pencil marks, while useful, is also time consuming, and it is easy to miss eliminations you could have made, only to catch them later in the puzzle.
Since there are only two cells left in the bottom middle section, we’ll put the 2 in the red space, and add the 1 just below it.
In this column, we see I missed another elimination, so that space only has a 7 left as a possibility. That means the space in the middle contains the 5.
This gives us several more places with only one number left, and a square on the right which is the only possible position for a 7 in its section.
Sudoku has many variations, but the solving techniques are pretty much the same, so I’m including them on this page.
First, Jigsaw Sudoku simply changes the shape of the 3×3 regions into irregular areas. There are still 9 of them, each containing 9 squares. So your goal here would be to make sure there are no duplicates in any row, column, or thick-bordered region. If a book is printed in color, you can add extra clarity for the different regions.
Next, we have several which simply add extra regions. The main difference is that you must ensure that the numbers 1-9 also do not repeat within the new region(s).
These each have 1 extra region. From left to right, they are Asterisk Sudoku, Center Dot Sudoku, and Girandola (or Firework) Sudoku. Notice that numbers do not repeat in any row, column, 3×3 section, or the colored extra region. This acts as an extra clue when you want to determine whether a number can be placed in a given cell.
Sudoku X has 2 extra regions. To solve the puzzle, a number cannot repeat along any row, column, 3×3 section, or either diagonal.
Windoku has 4 extra 3×3 squares mixed into the puzzle. This means you’re making sure that numbers don’t repeat in any row, column, or the 13(!) different 3×3 sections.
Offset Sudoku is certainly a colorful variation. This puzzle has 9 extra regions, one for each square in the 3×3 sections, distinguished by different colors.
So now, there is no repetition along any row, column, 3×3 section, or in any set of same-colored squares.
Rather than extra regions, you can also add extra Sudoku grids for a new challenge. Each Sudoku puzzle is solved as normal, with no number repeating within any row, column, or 3×3 section. However, where the different Sudoku grids cross, those 3×3 sections are shared between more than one puzzle. This means that repeats are possible in rows or columns that are longer than 9 squares, because there is more than one Sudoku grid using it. That repeat will NEVER happen within the same 9×9 Sudoku grid, though.
Butterfly Sudoku has 4 different 9×9 Sudoku grids. I’m using colored squares here to make it easier to see the different 9×9 Sudoku areas.
Sohei Sudoku also uses 4 Sudoku grids, but the blending isn’t quite as extreme as Butterfly Sudoku.
Cross Sudoku squeezes the grids a little closer together, which gives us an additional Sudoku grid, so we have the 4 outer puzzles, and 1 in the center.
Flower Sudoku brings the 5 grids still closer together.
Because of how the offset makes the puzzle look, it can seem like Windmill Sudoku has more grids than it actually does. Look carefully at the numbers, and you’ll see that it is still 4 outer grids, with 1 in the center.
If you look at any other 9×9 set, you’ll see the repeats that means the numbers are from separate puzzles.
Gattai-3 uses 3 Sudoku grids. This is one possible arrangement.
The final multi-grid puzzle I’ll be showing here is Samurai Sudoku, in which the 5 Sudoku grids form an X shape.
There are other multi-grid variations of Sudoku as well, with only 2 or 3 puzzle grids, but there are enough combinations there that I won’t show them here, as you get the basic idea.
Another way to add flavor to a Sudoku puzzle is simply to replace the numbers. Using symbols or letters can make the puzzle look more interesting, or less like a math problem, while still requiring the same techniques to solve.
Here we have Picture Sudoku, which you may sometimes see in children’s activity books. Because Sudoku is a positional puzzle, and doesn’t require any math, this can be a good way to challenge young minds.
It’s also great for people who like to doodle, or feel a mental block when faced with the numbers in a normal Sudoku puzzle.
While you could simply use any 9 random letters as a Sudoku puzzle, Godoku (sometimes called Wordoku) goes one step further.
It can turn a completed puzzle into a miniature word search. The challenge for the puzzle creator is to make sure the hidden word doesn’t have anagrams that can form other words. I did not do that in this case, so you can easily spot several smaller words.
If I were to use a puzzle like this with multiple words in a book, I would either specify the number of letters in the correct result (8), or I might say how many possible words exist in the completed puzzle, and let the solver find them all.
Not counting unusual Scrabble words or those which are part of a longer word, or anything less than 3 letters, I count 16, most of them 3-letter words.
