Solving Sudoku: Eliminating Numbers for a Winning Strategy

Are you a fan of puzzles and logic games? If so, you’ve probably heard of Sudoku. But do you really know what Sudoku is and how to play it?

We will explore the rules of Sudoku, different strategies to solve the puzzle, and specifically focus on the importance of eliminating numbers.

By mastering the art of elimination in Sudoku, you can narrow down possibilities, reduce difficulty levels, and increase your chances of finding the correct numbers.

Keep reading to discover how to eliminate numbers effectively and avoid common mistakes in this popular brain-teasing game.

What Is Sudoku?

Sudoku is a logic-based number game akin to a cross between a number puzzle and a crossword puzzle. Here’s a complete game dictionary definition of Sudoku which details how one plays and wins in the game:

Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 subgrids that compose the grid (also called “boxes “regions or “blocks”) contain all the integers from 1 to 9. The game may be played on a 9×9 board with smaller numbers, but it is commonly played with all cells displaying unpaired digits from 1-9.

The development of the modern Sudoku game is usually credited to Howard Garns, a retired architect and freelance puzzle constructor from Connersville, Indiana, who had the puzzle published in Dell Magazines in 1979 with the name “Number Place”. Later moved to Japan, the Puzzle was popularized under the name “S ji wa dokushin ni kagiru”., which translates as “The numbers must remain single”. It caught on in the Japanese market, and it became known throughout the world as Sudoku.

Today the modern Sudoku game has a worldwide player base, a number of annual national and international championships are organized, and the game has become popular in newspapers, magazines, and online websites. It is thought that there are 6,670,903,752,021,072,936,960 distinct Sudoku solutions.

How To Play Sudoku?

Sudoku is a logic-based number-grid puzzle where you must fill in the cells with the numbers 1 through 9. Play starts with predefined numbers called givens or clues already filled in spots, which helps determine the placement of other numbers by deduction. A new problem is released each day for the following solution. The goal of solving a sudoku puzzle is to fill in the empty cells with the numbers 1 through 9 in such a way that every row, column, and 3×3 box contains all the numbers only once. The same number cannot appear more than once in the same row, column, or box.

What Are The Rules Of Sudoku?

The rules of sudoku are that each row, column, and 3×3 square region contain every integer from 1 to 9. If there are numbers in place of some of the boxes (which there typically are), there should typically be 17 to 30 given digits to solve the puzzle.

For the general version of Hypersudoku with the four inside boxes, it is the same thing, except those four boxes must follow the same rule as the rest of the board. In the case of the Very Hard variation – if there are no numbers on the board, there must still be only one solution. Logically, this means there must be enough numbers given on the board such that only one solution exists.

What Are The Strategies To Solve Sudoku?

The strategies to solve a sudoku puzzle are the solution steps that people execute in order to complete the puzzle. The usual steps include the following, though in what order they are completed depends on the starting number of givens and the personality of the solver.

  • Filling In
  • Scanning
  • Cross-hatching
  • Basic Solving Techniques
  • Naked Pairs and Triplets
  • Locked Candidates
  • Crosshatching Advanced Techniques
  • Pointing Pairs, Triples & Quads
  • Box/Block Linking
  • Advance Techniques (The Deadly Patterns)
  • Sudoku X-Wing
  • Sudoku Swordfish
  • Solver techniques

The most important initial solving strategies are filling in known digits and scanning for known digits before resorting to making notes in the sudoku puzzle cells. The expert solving techniques (especially Locked Candidates and Box/Block Linking) can be powerful and easy once you have learned the principles. If beginners work to improve their proficiency at earlier steps they may find that they often are able to solve sudoku with the simpler strategies before needing to apply the more advanced techniques in everyday sudoku solving practice.


In sudoku, you can eliminate possibilities in each cell until you find the actual number in each 9×9 box. This is the first and main strategy used in typical sudoku puzzles. There are two keys to cell elimination (spotting and crosshatching). Spotting is to identify cells with only one candidate number. Whenever you find these in the grid, write them in to help solve other cells in the same row, column, or box. Crosshatching starts with identifying all the candidate numbers in each cell by considering the rows, columns, and boxes of each cell. As you find the numbers in your candidate grid that could be in only one box within a row or column, eliminate them from the candidate list.


Scanning in Sudoku is a search technique where you scan a grid row by row and column by column in order to find out possible numbers. In this way, you reduce the number of possibilities to figure out how numbers are placed in a local area. This search technique is important in solving the Sudoku grid when few but all vertical and horizontal alignments are already fixed.

It does not completely eliminate the void, but makes it easier to visually eliminate numbers in local areas.


Crosshatching is the process of crossing out (or hatching) with lines numbers that have already appeared in a column and row. Look at columns and rows with the most numbers to get of. When there is only a possibility for an unwanted number to either be in a column or a row simultaneously then using crosshatching can show that no number is left.

In the very difficult example in Step 3 of How to Eliminate Digits in Sudoku? we use the same concept of the low placement of 7 so the low columns are eliminated. If there is a 2 in the high row, there is no possibility for a number in the row to be left, eliminating 9 from the high column and thereby eliminating the possible 9 in the R3C7 square for Step 3.

Penciling In

When eliminating numbers from Sudoku, pencil in values to see how they impact the rest of the puzzles. This allows you to better visualize the spread of solutions stemming from possible iterations.

Penciling in refers to writing down the possible candidate values for each cell in Sudoku and progressively eliminating numbers based on interactions with other numbers. This technique is important when more than one possibility exists in the same cell. Penciling can help you immediately see the consequences of one value being placed.

What Is The Importance Of Eliminating Numbers In Sudoku?

The importance of eliminating numbers in sudoku is that finalizing an empty cell (i.e. solving it by entering the missing number) is only possible after all but one number from the missing numbers are actually eliminated. Every number eliminated narrows down eventual proof that a given move to a final number placement is the correct move.

In cell by cell solving from the start of a new sudoku puzzle, ultimately you are required to eliminate all but one of the remaining possibilities. Work continues until your current puzzle position requires picking a number by filling in the final cell, usually a 50%-50% attempt that does bounce back and forth with repeated moves before definitively eliminating the non-final number and finalizing the cell’s placement. Expert Sudoku strategies of X-Wing, Swordfish, Jellyfish, etc. tend to speed your solving and number elimination processes towards completion. Row and column current possibles tables are useful new tools for making elimination decisions.

Helps To Narrow Down Possibilities

Checking out any of the visible repeating blocks with two or more cells helps to narrow down possibilities. In utilizing elimination strategies, people often do not actually need to eliminate a number but simply need to limit the possibilities.

In this same standard Japanese latent Latin rectangle game, by seeking out and solving overlapping patterns, an individual cell can have its own set of limitations applied, sometimes eliminating certain numbers altogether despite them never actually appearing in the cell.

For example, the inability to form a certain number within its row or column both apply as restrictions as well as eliminate certain possibilities for other cells.

Once the game is so restricted that there is only one potential number remaining, then the need to eliminate further choices comes into play. “+ Unlocking Cells is the most effective way of eliminating numbers in sudoku. When all other strategies have been applied and a complex pattern of numbers has been reduced to a simple puzzle where two or more numbers can be input, a player should choose a number to enter.

Reduces The Difficulty Level

By eliminating numbers, you reduce the difficulty level of a sudoku puzzle. The more givens there are, the easier the puzzle will be to solve. Japanese Sudoku Grand Master Maki Kaji recommends that the number of givens on a sudoku puzzle be 22 and that the highest number of givens possible blocks should not exceed 24 as these would be too easy and boring. Opt for puzzles with more givens to ensure they are more challenging.

Increases The Chances Of Finding The Correct Number

Removing extra numbers reduces the complexity of the Sudoku puzzle. As the complexity of the puzzle decreases, the easier it becomes to spot valid patterns that can aid in solving. Try eliminating extra numbers through guesses until you can solve the solution – numbers can always be placed back in their original locations if guessed incorrectly…

If you get to the end of the game you should be able to find a solution. The logic of the game ensures that a solution does exist. If you still can’t, 99% sure you screwed up the prior steps…

If you math it out incorrectly, chances are you won’t even detect your misstep – and you’ll sit there forever wondering how there could be a solution. So before you try to remove numbers from a solved puzzle, make sure you can solve the puzzle yourself after doing so…

How To Eliminate Numbers In Sudoku?

You eliminate numbers in Sudoku by removing the number markers (bi-value, multi-value, claimed, or candidate) for the unwanted digit or by removing the actual digits from the cell.

Every completed and legitimate Sudoku puzzle will be constructed with all single digits in the correct cell position. Removing the pencil or name of the unwanted digit is the same as eliminating the cell for that digit.

You then win and eliminate the remaining pencil marks in the cell until you have those single cells from which to build your puzzle. Enjoy the feeling of logical emptiness after eliminating all non-required numbers. On the 21st Mar 2007, we at Webster’s Online Dictionary, proved that no numbers existed but 1 for a 9×9 Sudoku by showing that the millions of AI-generated puzzles resulted in only that one number.

Look For Single Candidates

Single candidates are cells inside the 3 3 boxes that only have one candidate. By implication, this must mean the candidate is correct, even if F2 = 7 in the example, altering the row and column. It is easy to see how elimination sudoku becomes simpler to solve when a subset of numbers can be confidently removed. Use this rule on sudoku puzzles whole-cell combinations can be eliminated correctly to remove numbers.

Use The Scanning Technique

The Scanning Technique is a way of keeping numbers out of specific cells by looking at restricted zones and locked candidates in other zones. The Scanning Technique helps determine a certain pattern that would prevent numbers in specific points. Scanning is necessary for advanced levels whereby the search space is reduced using certain rules in order to solve the puzzle. For a more clear explanation, Cracking the Cryptic provides an analysis in the following video.

Utilize Crosshatching

Crosshatching, sometimes called box counting, is another good sudoku strategy to eliminate numbers. Crosshatching focuses on boxes as opposed to columns and rows. Take a single box and look at the rows and columns that intersect and try to determine which numbers are not possible. Work through all of the numbers in the box to determine their possible locations either in a row or column. Ruled-out numbers can then be erased, leaving only the valid placements for each number.

Suppose in the 4-number box in the bottom-middle of the grid (label the boxes by row-column number as we did with rows and columns) we have filled in an 8 in the lower center position. We can now eliminate an 8 from the fourth column and an 8 from the eighth row. This is a key strategy for eliminating and determining two number.

Simple Sudoku has the ability for the user to color entire rows and columns with a right-click and drag-and-drop. A user can select the color to paint the columns and rows that have the number they are focusing on. This system then shows where the possibilities for the number in the box they are looking at are. In orange is the box with the number you are looking for, in blue are the columns and rows that do not have the number you are looking for, in red are the columns and rows that have the number you are looking for.

Below is our Simple Sudoku grid mimicking this approach of erasing possibilities for one number in the numbered box area by visualizing how our number of interest traverses the rows and columns.

Use Penciling In To Keep Track

How to eliminate numbers in Sudoku? The process to eliminate numbers in Sudoku involves using penciling in to test out choices, deductive reasoning, inductive reasoning, and process of elimination with careful note-taking of possibilities. Penciling in to track elimination and choices can be helpful for medium and hard puzzles.

Penciling in in the context of a Sudoku grid consists of noting down numbers that are possible at a particular cell. When trying to fit in a number in a cell, write it in lightly and in small print. If you then determine you will not be able to utilize that number there, erase it and/or cross it out.

This process allows you to keep track of the options you are considering without erasing the original puzzle. However, overusing penciling in will make a Sudoku puzzle more complicated and mentally draining, so keep it in reserve as a method of last resort. Here is an example of penciling in.

In the following subset of a Sudoku puzzle, note that neither of the cells in the bottom row can have a 3 because cell E6 has a 3. You can pencil in the numbers 7 and 9 in the bottom row as shown. That does not provide enough information to confirm a 3 in any of the three available options, so one might pencil all three options in.

What Are The Common Mistakes To Avoid While Eliminating Numbers In Sudoku?

  • Guessing – You have no logical or formulaic reason for asserting which moves might make progress toward a solution. If you are still guessing and you have eliminated numbers, then you have not eliminated them all.
  • Picking the easiest number to eliminate – It is often quicker to eliminate the least frequent number if you are in a hurry. However, for success in reducing the complexity of the rest of the board, you will be better served if you focus elimination efforts on the most frequent number first. You will then find that there are more settings of the board in which that number is the last, and you will be using less brute force.

Not Checking For Duplicates

  1. Because some of them may be concealed, the most effective technique to solve Sudoku puzzles is to make common notes.
  2. Ensure you write numbers in each section of the puzzle. Writing down the numbers doesn’t make you a failure. Working hard at the puzzle does not mean you don’t need to trust yourself or have faith in your abilities. It simply means that you are open to experimenting to locate a solution.
  3. There are numerous unique solutions to a Sudoku puzzle, and it may be very incorrect to dictate there is merely only one way to solve the puzzle. The most rational method would be to avoid having too many numbers all over the grid. This may degrade the appearance of the original appearance of the grid without making it easier to solve.
  4. It is acceptable to make a rough estimate from time to time if it is feasible to do so. The game of Sudoku has a great deal to do with mathematical calculations. The majority of the time, only expert players get the chance to enjoy playing it.
  5. It is difficult to win if the eyes and mind are not attentive. The most effective technique to solve Sudoku is to look at the entire diagram and the numbers published by other players.

Current research indicates that playing Sudoku is an effective method to help reduce back pain. The study, which was performed by Harvard University in the spring of 2021, took 1,000 people off work on account of lower back pain and split them into two groups for two weeks. One group exercised regularly while the other group did Sudoku.

At the end of testing, it was determined that the individuals who did Sudoku experienced 25% less backache than those who exercised. These findings were revealed in the Journal of the American Medical Association’s report. The verdict was then expanded to determine that relaxing and light activities like Sudoku with a small number of steps are safer for back health. These are thus applied in therapy.

Not Using Pencil Marks

If you are trying to eliminate a variety of possibilities at once, pencil marks can further highlight them and help find elimination strategies. Assign every possible value in a square and put negative assumptions in the pencil marks. Remember that seeing the pencil marks gets more difficult as there are less numbers. You’ve probably already gone past this step, but sometimes the difficulty in elimination lies within what you haven’t noticed.

Skipping Steps

The degree of difficulty of a certain situation is determined more by the correct deduction of which digits can be entered and confirmed in a space, rather than identifying which specific digit to enter. This is because there are generally relatively few combinations for the latter when there are more than two open squares where only one digit can be filled in. Sophie Goldsworthy recommends trying to solve the puzzle by skipping steps to reduce time. This means trying to deduce the possible moves in a space without first identifying which moves to try. Use crossing out and pencil marks to mentally focus more on possibilities rather than their completion.

Alex Bellos in his book Sudoku, recommends developing this skill by beginning with a grid where the majority of answers are already easily visible. You can then noodle out which numbers can potentially go where. You will probably make minimal progress with such a simplified grid which means you will lose interest. The aim of this exercise is not to complete the grid but to improve your skill at identifying the most fruitful branches to explore.

Frequently Asked Questions

How to Eliminate Numbers in Sudoku?

What is the purpose of eliminating numbers in Sudoku? The goal of Sudoku is to fill in a 9×9 grid with numbers so that each column, row, and 3×3 section contain all of the digits from 1 to 9. Eliminating numbers helps to narrow down the possibilities for each space in the grid.

How to Eliminate Numbers in Sudoku?

How do I start eliminating numbers in Sudoku? Begin by looking for spaces with only one possible number. Then, use that number to eliminate possibilities in other rows, columns, and sections. Repeat this process until you have filled in all the spaces with only one option.

How to Eliminate Numbers in Sudoku?

Can I use a pencil to eliminate numbers in Sudoku? Yes, many people find it helpful to use a pencil when playing Sudoku. You can write in the possible numbers for each space in small, light markings. Then, as you eliminate possibilities, you can erase the incorrect numbers.

How to Eliminate Numbers in Sudoku?

What if I get stuck and can’t eliminate any more numbers? If you reach a point where you cannot eliminate any more numbers, try looking at the spaces that have the most possibilities and work from there. You can also use logic and trial and error to fill in the remaining spaces.

How to Eliminate Numbers in Sudoku?

Can I use any strategies to help me eliminate numbers in Sudoku? Yes, there are many strategies you can use to eliminate numbers in Sudoku. Some popular ones include the X-Wing, Swordfish, and Y-Wing techniques. These techniques involve looking for patterns and using logic to eliminate numbers.

How to Eliminate Numbers in Sudoku?

Is there a specific order to follow when eliminating numbers in Sudoku? No, there is no set order for eliminating numbers in Sudoku. It is important to remember to constantly check for new opportunities to eliminate numbers as you fill in more spaces. You may need to go back and reevaluate previously filled spaces as well.

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