Mastering Sudoku: A Guide on How to Use Pencil in the Game

Are you a Sudoku enthusiast looking to up your game?

Pencil marks may be the key to unlocking those challenging puzzles!

In this comprehensive guide, we will explore the rules of Sudoku, basic solving techniques, and the benefits of using pencil marks.

From identifying candidates to tracking possibilities, we will delve into how to effectively utilize pencil marks to conquer even the most difficult puzzles.

Whether you prefer solving on paper, mobile apps, or online games, we’ve got you covered.

So grab your pencil and get ready to sharpen your Sudoku skills!

What is Sudoku?

Sudoku is a number-based logic puzzle game that became popular in the 2000s. The game is played on a partially filled grid consisting of a 9 9 square subdivided into smaller 3 3 squares. The objective is to fill the grid such that in every row, column, and 3 3 square, the numbers from 1 to 9 appear exactly once.

Sudokus always contain at least 17 clues with which to start. According to a study by Felgenhauer and Jarvis, there are 6,670,903,752,021,072,936,960 (6.67 * 10^21) possible Sudoku grids, and 5,472,730,538 (5.47 * 10^9) inequivalent sudoku grids.

What Are the Rules of Sudoku?

The rules of Sudoku (constraints, as they are properly called), mandate that each of the digits 1-9 can appear only once in each row, column, and bold region (3 by 3 subgrid)

The resulting puzzle must be numerical according to the rules, and it must have a unique solution. If it has multiple solutions, it is technically not legally a proper sudoku (though it may feel like one).

Here is an example sudoku puzzle properly set up according to the rules as we understand.

What Are the Basic Techniques for Solving Sudoku?

Solving sudoku puzzles involves three core techniques, according to Origins professor Lavanya Vasudevan. These techniques are used in progressions that extend into more advanced methods of sudoku hints application:

1. Naked singles: The simplest of techniques where pencil numbers matching a given number in intersecting boxes are removed one by one.
2. Claiming: One of the more advanced methods used the concept that a number must go in one specific box.
3. Intersections: Looking for interactions between two different boxes, e.g., that two intersecting box diagrams will be solved in the same way.

Single Candidate

Single Candidate, also known as simple candidate or naked single, is the most basic of the pencil techniques in sudoku. It is used to find cells with only one possible candidate. Anything more complicated than this will increase the number of candidates to be checked and potentially slow down the solution. This is the most important, fast, and simple way to pencil in sudoku to reduce notations required to solve a puzzle. Here is an example of how the single candidate strategy works. The open cells in red have multiple candidates, and the shaded cell has only one possible candidate.

Single Position

In this practice, the Single Position refers to a pencil mark that you jot down in a cell where it is the only one of that value in a row, column, or block. This means that the pencil mark can only be the correct answer for this cell in the puzzle. Erase all other pencil marks from the cell and continue solving. A single position mark is the least helpful of basic techniques since it requires very little deductive reasoning and is more often the result of an oversight, but it is a simple and helpful technique for new players to learn.

Naked Pairs

Naked Pairs are a common variant of locked candidates. They occur when a pair of exactly two possible candidates for the same digit are the only possible candidates in a row, column, or block. This allows you to lock these two candidates to two specific cells and remove the two digits from the possibility set in all other cells in that row, column, or block.

Hidden Pairs

Hidden pairs are similar to naked pairs, but the sets of numbers occur in more than just two blocks. They function the same – by eliminating any additional numbers outside the hidden pair from the same row and column. Hidden pairs use more advanced strategies as it is harder to spot. Here is how to find Hidden Pairs:

1. Look for cells with exact two pencil number marks
2. Look for a combination of pairs of two numbers in pencil
3. Determine if there are two pencil numbers that match the aspect of Hidden Pairs
4. Use that information to eliminate other pencil marks

How to Use Pencil Marks in Sudoku?

In pencil-and-paper versions of Sudoku, the most common ways to use pencil marks are to write the potential candidates in smaller font sizes in the individual squares around the edges. In other squares, the potential candidates for a particular number can be erased as they are eliminated (this is easier if potential candidates have already been written out along the edges).

Nancy Frey and Douglas Fisher recommend getting in the habit of starting a new puzzle by scanning the puzzle, assigning potential candidates to each box, and crossing out options as the possible choices are winnowed down.

Using Pencil Marks to Identify Candidates

Even if you use pencil marks for nothing else, nearly everyone should use them to identify candidates. Identifying candidates is when you consider a square and see which numbers are possible values for the square based on the values in the intersecting row, column, and box. This helps solve squares that would otherwise be too difficult to solve. Take the following sudoku game as an example where the row shows the values in the corresponding boxes and possible numbers for the circled squares have been penciled in.

1. Look at the marked row on the left. 1, 2, 4, and 6 are already present in that row, so the missing numbers can only be 3, 5, 7, 8, and 9. 5 is in a cell that intersects the same boxes that the circled right-side cells intercept.
2. Thus, we can tell that all right-side empty cells cannot equal 5 because the value of 5 along the left is already in those given square’s row, column, or box area.
3. The remaining possible numbers for the center cell are 3, 7, or 8. But it could be any above.
4. The remaining possible numbers for the center cell are 3, 7, or 8. But it could be any if we only use this tool.
5. For this left upper cell, we have three possible numbers, 3, 7, or 9.
6. The remaining possible numbers for the center cell are 3, 7, 8, or 9.
7. The remaining possible numbers for the center cell are 3 or 8 (with 7 and 9 being blocked by other sets of 3 and 8, respectively).
8. The remaining possible numbers for the center cell are 3, 7, 8, or 9.
9. The remaining possible numbers for the center cell are 3, 8, or 9.

This is how pencil marks help to figure out the possibilities, often significantly increasing the chances of solving even difficult-to-solve squares.

Using Pencil Marks to Track Possibilities

Though their main advantage in solving puzzles is decreasing the possibility of errors, you can track possibilities and see a puzzle more clearly by using pencil marks in the situation of simplicity or greatest difficulty. The following are typical tracking methods that natural pencil marks can be used for but are done mentally without aid from pencil marks. Notice that each one involves the ability to mentally link weak candidates to deduce stronger ones to answer Where did this number go?

Using Pencil Marks to Solve Difficult Puzzles

While pencil marks are beneficial in decreasing time for easier games, they are generally a must for difficult puzzles. Experienced players say that it is impossible to accurately recall the potential options for multiples of eight or more squares because there are so many possible interactions. Therefore, you need to map those out with pencil marks.

The most difficult puzzles are sometimes referred to as monster sudokus and you will, without a doubt, need to use pencil marks to show all of the possibilities. These are the types of puzzles known for causing temporary brain freeze and incredible frustration for all but the most ardent players. Even top players like Ulrich Voigt and Tom Collyer advise using pencil marks for even the most advanced of solving techniques. So whenever you have difficulty solving a puzzle, simply add pencil notes to help identify multiple option sets and assist in organizing the decision-making process.

What Are the Benefits of Using Pencil Marks in Sudoku?

The benefits of using pencil marks in sudoku are that they provide correct arithmetic, are a universal key for conveying puzzle information, can prevent errors, can help identify step-through options, can provide a secondary solution method, and can reduce complexity for those using algorithms.

The marking system is the only correct one for conveying puzzle information, as the `Pencil Mark Notation` page of SudokuWiki explains. The numerical notation can prevent errors by allowing the player to see which numbers are restricted in a square. They can identify the effect of a slightly more powerful critical constraint, countercryptarithmetic, to save the puzzle from a dead end.

Additionally, the system can provide a secondary, and potentially more logical solution method, and can facilitate algorithmic assistance. If no valid benefit can be found due to the puzzle’s low difficulty or if one has extreme aptitude for the game, the marking system can be counterproductive as it will lead to complex redundancy and increase solving time. Although allowing a different perspective during the rare hints of having a deadlock during the game where you have to switch to another square that does not have a pencil mark on it may be a benefit in itself.

What Are the Common Mistakes When Using Pencil Marks in Sudoku?

The common mistakes when using pencil marks in Sudoku are definitions of the numbers placed in the squares that are either too narrow, overconfidence in one’s initial board evaluation, not keeping the number log up to date, and not erasing or reassessing often enough. Singletons are numbers that can only go in the one square on a row, column, or block, as these are the most useful pencil marks in solving drawings.

One mistake is only adding these numbers when the puzzles are most difficult, when it is easier to try filling in the squares at any time with a low difficulty sudoku or waiting to place the final few numbers on the most difficult board, when there are many possibilities still on the board. Not calculating a Naked Single, when it is obvious, leaves the solver with an incomplete picture of the board. More subtly, not having notated missing possibilities for all possible routes to placing additional numbers can lead to unnecessary erasing and reevaluation.

Similar to not keeping the number log up to date, another frequently made mistake is not erasing previous markings to give oneself the best opportunities to reassess and relocate numbers to different squares. Does not mark squares again after using a process of elimination, and always take pause when there is a mismatch between marked squares in the initial number log and what is on the board grid. Erase all the marks to ensure a completed number log that allows for easy comparison of possibilities with the board.

Not Updating Pencil Marks

You should never have to erase your pencil marks to update them, because you should always write in pencil. If you change your mind about a number you are placing in the main grid and it is not a multiple candidate, just erase it and replace it with the new number. If it is a multiple candidate, erase the original number and write in the new number. Keep your multiple candidates updated every time you pencil in a new number.

Erasing Pencil Marks Too Soon

• How?
• Why?

Using a pencil on a sudoku puzzle is merely short term notation to play the game. You can erase pencil marks at any time following note-taking rules, but doing so too early will reduce the efficiency of pencil usage that players rely on. Take the time to consider if it is the right time.

Players make the mistake of erasing too soon because it can be intoxicating to know that they got a square correct. But provide yourself with a handicap or means to undo errors because the puzzle should not be stressful– just challenging.

Not Using Pencil Marks Effectively

Not using pencil marks effectively can give the impression that you need to perform trial and error on one of the two extreme ends of the spectrum. There are very simple systems out there. However, wherever you end up, the pundits will start to pound you and mention all of the elements that will actually make it easier for you. If you make a mistake during your Sudoku puzzle, it is best to erase all pencil marks and start over. Effective use of pencil marks will help prevent errors of trial and error or eliminate any errors of trial and error influencing the solution.

How to Use Pencil Marks in Sudoku on Different Devices?

You can use pencil marks in Sudoku puzzles on different devices to reduce the number of possible values in a cell and to make progress while keeping your workspace organized and easy to read. A frequent mistake on paper devices is failing to clear pencilled-in candidates when they have been fully considered, whether they are true, and for important cells that they are not false alternatives.

Paper and pencil might seem outdated, but they work just fine for most large Sudoku puzzles that are not time-sensitive and require electronic solvers. It is an inexpensive option, good for training beginners, and adds the flair of nostalgia for using notes in puzzles.

The following devices can be used to apply pencils to electronic Sudoku puzzles easily for people preferring to use a non-touchscreen interface:

• Computer systems using the computer keyboard, mouse, or webcam
• Physical computer keypads that light up e.g. Syudy Buddy using the 9 number keys
• Writing on printouts using a pen or stylus and scanning the paper
• Gliding stylus on e-paper (with high delay)

Using Pencil Marks on Paper

Marking pencil notes on paper is the way virtually all Sudoku players initially learn to play and continue to solve puzzles in newspapers and magazines. A significant degree of concentration and calculation is required to fully solve even the easier puzzles, which is why people have historically used paper to take notes and solve Sudoku.

Players can add notes in the corners of squares in order to show the possible candidates for groups of squares. Over the course of the game, the user should constantly update these to show the possible values that remain for that square. Should a particular digit be confirmed for a square, one can then erase the other possibilities.

Players should also use circled or square pencil notes to mark more detailed deductions. These may be a certain set of possibilities that occur within a group of squares, also known as a naked subset, or may occur between squares, noted as bans or locked digits. Furthermore, these notes will often include temporary digit groups known as forcing chains. The chain will start on a square with a specific pencil mark, and if this particular mark was not correct, no other squares would have any possibilities and we could eliminate all other possibilities.

Pencil marks on paper allow you to easily correct mistakes, try alternative approaches, and compare multiple possibilities. This makes them one of the best methods of learning advanced sudoku and regularly implementing them when solving a puzzle, by challenging yourself to practice makes it much easier for beginners to implement and use later on.

Using Pencil Marks on Mobile Apps

Mobile apps for Sudoku handle pencil marks effortlessly. The Simple Sudoku is particularly useful. Follow these easy steps to use pencil marks in the app:

1. Set the initial hints and givens.
2. Turn on note taking in the menu.
3. Long press on a cell to set a pencil mark.
4. Remove a wrong pencil mark by selecting an already existing one, and then select it again until it is gone.
5. Press and hold the cell until a small palette of available numbers is displayed. Select the number you wish to enter as a correct solution. (Enforce single solution in the menu to ensure a single correct solution is enforceable).
6. Rinse and repeat until puzzle solved.

Using Pencil Marks on Online Sudoku Games

Online Sudoku games like those available on web pages, smartphone apps, and tablets can be played in the same way as paper versions. The difference is you choose to leave purported pencil marks on or off, rather than working in physical pencils. Justify the use of the pencil icon when entering a number one to nine, and the game will automatically drop down to a pencil graphic.

Frequently Asked Questions

How to Use Pencil in Sudoku?

FAWhat is the purpose of using a pencil in Sudoku? Using a pencil allows you to make temporary notes and changes, making it easier to solve the puzzle.

How to Use Pencil in Sudoku?

FAHow do I make notes with a pencil in Sudoku? You can use a pencil to write small numbers or symbols in the corners of the Sudoku grid to help you keep track of possible solutions.

How to Use Pencil in Sudoku?

FACan I use a pen instead of a pencil in Sudoku? It is recommended to use a pencil in Sudoku, as it allows for easier erasing and making changes. However, you can use a pen if you prefer.

How to Use Pencil in Sudoku?

FAIs it necessary to use a pencil in Sudoku? While it is not necessary, using a pencil can make it easier to solve the puzzle and avoid mistakes.

How to Use Pencil in Sudoku?

FAHow can I use a pencil to solve more difficult Sudoku puzzles? For more challenging puzzles, you can use a pencil to make multiple sets of notes and gradually eliminate possibilities until you find the correct solution.

How to Use Pencil in Sudoku?

FAWhat kind of pencil is best for solving Sudoku? A regular #2 pencil is typically sufficient for solving Sudoku puzzles. However, you can also use a mechanical pencil for a finer point and easier erasing.