Mastering Sudoku: How to Use Math for Solving Puzzles

Again, each arrangement of integers forms a unique solution to the puzzle. Permutations and combinations with the unique permutations concept also provide a multitude of possibilities of how a solution can appear. As one can see, mathematics is an critical tool of natural and rational law to play Sudoku, but a player need not explicitly be doing math. Sudoku can help improve your math as solving constraints and patterns enhances your thinking and makes math easier at a basic level.

Logic

Much of Sudoku’s mathematics includes logical rules on the number placement grid (layout), rules about what combinations are possible, and rules about the properties of numbers. These are some of the more noteworthy logical rules to remember when playing:

1. Exactly one valid solution: Every puzzle will have only one correct answer that can be arrived at without guessing or using the process of elimination. This is critical for maintaining Sudoku play as a purely mathematical game. If a puzzle has multiple solutions, the puzzle is not valid.
2. 3-try color method: This is a technique that can be used to eliminate the possibility of specific numbers. Pick three different colors and apply color to all the possible numbers that might occupy a block. Next, apply a new color according to the number of occurrences a number is blocked in at the end of all 3-trial process. If a number is occupied in all of the cells with one color, the cell with another tentative number will be correct.
3. Must-be / Cannot-be a number: The inability of a number to be in a square can help to determine what numbers must possess the square. Alternatively, knowing that a specific number is the only one that can be excluded, the number that is left must be the proper one.
4. Boxes, rows, and columns: The set of numbers for a specific row, column, and box. Should be restricted when determining where different numbers can be placed.
5. Naked and Hidden Singles Rule: When a number is a naked single, it is the only possible number in a square. If it is hidden single, the number is locked by possibilities inside a row or a column. The number is already read and the square is highlighted when the player discovers the number under a hidden single rule. One needs to search for a number that occurs only once in the column, row and the block.
6. Lone Number Rule: When 9 is spread all through the row, column and the box, it is impossible to eliminate the number. Lone Number Rule can tell the possible location for number 9 by starting from rows, columns, and boxes based on this; one position must have a 9.

Pattern Recognition

Pattern recognition in the context of Sudoku involves noticing and remembering groups of numbers with some regularly between different instances of the board. The best way to explain this is through an example where we look for a remote pair in this Fiendish level Sudoku puzzle. The first thing we do when we see that there are only two open spots to place 9‘s in is to find which rows and columns they will be a part of. If the top spot at d9 is 9, we can immediately solve the rest of column d. If the bottom spot at e1 is 9, we can finish the rest of row 1.

So we need to look for pairs of unsolved-cells with the same number between other column d and row 1 unsolved-cells. We find 9, 4, 1, and 3 in e3, e4, e5, and e8. These don’t hold so in the first space we find some trial answers for the space and proceed through the remaining unsolved cells until we find a pattern which allows us to attempt placing the 9 in d9.

Deduction

Deduction is used in Sudoku as elimination where the player uses information from the rest of the board to eliminate options for a sudoku cell. This is the one technique that requires the minimum knowledge of basic sudoku solving rules.

Deduction can be as simple as a given number eliminating shares candidates from their row or column. The operative portion of the row or column group may only have one or two empty cells (or even none), and you will eliminate a number of wrong candidates simply for this reason. Below we show giving the number 1 on the 2nd column of the sudoku board, and how the candidates for the 5th row of that column will be eliminated step-by-step.

The elimination of candidates does not require knowledge of the basic rules of sudoku, as they are very easy to understand. Their effort can be huge sometimes, one board may have up to multiple times of eliminating all the possible wrong answers from candidates, or it may need none of it. Most sudoku boards will have some initial elimination steps, but players who stick with it will often require this technique to get to the next correct number. This is to say, even if a player is at the beginning level, they can use this technique.

Elimination

Any given cell may be solved using elimination if there is only one option remaining. Throughout the process of elimination elsewhere in the grid, this option may continually evolve into different options until eventually some locked subset may be formed to allow a best fit using a Hidden Pair, Unique Rectangle, or other elimination techniques.

To continue using the example from the Unique Divide Rule section, determine how the groups of 6 and 7 in Row 6 align with their corresponding groups in Box 6.

If Top Left is 6, then Top Left of Box 6 will need to be 6. Top Left of Row 6 will then be forced to be 6. Solving for Box 6 first, but we thought we knew Top Left of Row 6, which would change to 7. Therefore if we knew Top Left of Row 6 is 7, then Top Left of Box 6 must be 6.

How to Apply These Math Concepts in Sudoku?

1. Manual methods
• Elimination
• Look at rows, columns, and boxes to see where the numbers 1-9 could go. If a there can be two or more numbers in a location, trying to figure out their sum can eliminate many other options. Example – a cell has only two options – a 5 and a 6. This means that you can be sure that the overall total will be -5 (if the cell equals 5) or -6 (if the cell equals 6).
• Naked subsets
• Simple case: Only a specific n number of cells in a column, row, or square contain the n smallest numbers that are a part of the possible choices. You can cancel out every other option. This applies to 2, 3, 4, and 5 primarily.
• Hidden subsets
• The complement of the naked subset case. If all the same 6s, 4s, whatever numbers are all concentrated in the same couple of locations, this means you’re free to cancel out any other possible location in that column, row, or box.
2. Mathematically based software solutions
• Algorithms
• To take advantage of naked and hidden subsets, computer algorithms rank these possibilities and solve problems through mathematical logic. Such algorithms can search for these mathematical conditions that simplify the solution and can inform through signals or by making the move itself.
• Rules, patterns, and logic
• Up-to-date and continuously updated Sudoku games can use mathematical and software-based logic to gradually introduce the rules and patterns of the game. This means that players that are using such apps learn without needing any prior knowledge as these rules and patterns are mathematically explained for them on an intuitive level that a player of any background can understand.

Start with the Basics

Starting with the basics includes solving cells that can only have one number in their box, line, or row two and three combined with a process of elimination. The most basic mathematical aspect of Sudoku is eliminating what a cell can be by applying the basic Naked and Hidden Single Sudoku Solving Mechanics.

Hidden and naked singles are applied using the following rules as described in the terminology section above. Hidden Single – A box contains only one cell where a number can exist, but most likely does not lay within a cage. With elimination, this one cell is seen to contain this number and it does. Naked Single – A box that has only one empty cell and is in the line and row of boxes as one of the variable numbers.

This is summarized simply by Terry Stickels as ‘guided trial-and-error’ or ‘trial and success’.

Use Logical Reasoning

A strategy to use math in Sudoku is to utilize logical reasoning when a simple pattern emerges. Pattern recognition can be more difficult on higher difficulty levels but often completing simple patterns is all that’s needed to solve the most difficult puzzles. Look for patterns of 3 in a row, horizontal or vertical 3 by 3 blocks, or minimal possibilities for open even or odd cells. Logical reasoning helps decrease variant possibilities as the pattern unfolds.

Look for Patterns

Patterns can be used in sudoku to determine possible placements for numbers. This, in turn, can help make deductions about the placement of other numbers or to confirm the placements you believe are correct.

If a number n can only appear in one of two or more opposite corners of a square, box, or row, then other numbers in the n position will soon be eliminated, and hence other numbers will fall where they lie. This will allow you to determine if the possible-location number for n is in position 1, 2, etc..

Refer to the different sections and the mathematics behind them to get an in-depth feel for reading sudoku patterns.

Use Deduction to Fill in Numbers

Use math to apply the process of deduction to the magic square where a is 1. When you have a piece of the sudoku puzzle containing a magic square, you can work out what its missing numbers are. This uses Euler’s Greek pattern to work out the missing pair of numbers. For a 3×3 magic square, the pair will be the sum of all the numbers in total divided by 6. For 4×4 squares, another step is needed, which involves partial sums. From there, you can work out what numbers are missing according to existing numbers in the pieces of the puzzle.

Eliminate Possibilities

• If you find a digit that fits in a box row or column, eliminate it from all other gaps in the box row and column.
• If you find a digit that fits in a gap, eliminate all other numbers from the row, column, and box of that gap.

If you’re really stuck on a tough Sudoku puzzle, try to follow the digits that have already been filled in and eliminate the possibilities using the aforementioned rules. In dynamic reasoning puzzles, libraries often classify possible search paths that have positive solution sequences which never generate contradicting solutions as safe paths. The technique of following safe paths may be useful in guiding your move.

A guide to using notes in solving puzzles.
1. Look at the partially filled grid
2. Consider which numbers are currently missing from a given cell and determine if there’s only one number left that could possibly go there
3. Which number(s) could go there from the partially complete grid?
4. The incomplete grid won’t always contain all the number guesses possible for a cell. If you do manage to find a number other than the one you were looking for, this makes the puzzle even easier and speeds up completion.
5. mScores of players will read one of the initial puzzles on the Sudoku.com website, then quit and go to the same day’s easier-for-beginners-for-solving puzzle. This naturally improves speed averages on the Sudokus that people (often erroneously but sometimes accurately) believe they can’t solve.

What Are Some Advanced Math Strategies for Sudoku?

Several advanced math strategies can be used to solve Sudoku puzzles. These include subset counting, x-wing, swordfish, and the jellyfish theory, and they consist of the following:

1. Subset Counting. Given a puzzle in which the locations of the givens are uniform, the various combinations of specific numbers can be quantified as subsets of that number. This becomes crucial in the event that the standard techniques do not get a player nearer to the solution.
2. X-Wing. Imagine a Sudoku’s rows labeled A, B, C, and D with eight columns. In which there are two cells with a limited number of options, say A1 and A8 can contain only two possible numbers (strained to 1-2). If this occurs, it is possible that the pair will constrain themself in the other 3 rows. The logic is the same for columns where combinations are limited to 2. This X-Wing effect occurs for numbers 1 through 9.
3. Swordfish. Swordfish is based on the same assumptions as X-Wing, except if a particular number can only be in three cells in each of three rows. Then hypothetically the same number should only be in three cells containing three alternating triangles aligned. Such a strategy requires a constraint tight enough to accurately predict where numbers might show up, but if used it quickly restricts options.
4. Jellyfish theory. The Jellyfish Theory of Sudoku is a method where you look at all the rows where all the candidates for a particular number from 2 to 9 are limited to only four cells in the row. The same process should then be performed for columns. When this occurs, you should have a potentially inaccurate visual jellyfish figure. This is a last-resort tactic if the simpler strategies are not successful.

The video tutorial on advanced sumtozero techniques gives a good demonstration of advanced strategies as an alternative to casual strategies or killer sudoku techniques.

X-Wing

In X-Wing, if a digit is restricted in two rows and either of those two columns, no other candidates for the digit exist in the intersecting rows. In this example, all the squares that have the digit `{7}` as a candidate in rows 2, and 6, and columns 2, 3 can be eliminated except for the middle cell.

Swordfish

Swordfish is a 3×3 regular fish pattern. Plus alternate squares, swordfish columns, and swordfish rows there are two instances where the rule of 3 above or 3 below a row or column including both cases and two places connecting top or bottom alternatives each of which connects a certain 3s case (3 above this and 3 below that).

A further example using available data on the SA SIPC number mentioned in the previous paragraph, is available from STEM according to Mendi in their Sudoku: From Mendi, with Method? which takes up 4x4x2 cases in the table as opposed to up 3×3 doing the same which would normally equate to those in a total of 132117 From this point on, it proceeds thinking that the 3 below after switching scalar is correct.

XY-Wing

In this advanced intermediate difficulty level sudoku strategy, approximately three cells interact with each other producing two instances of the same number, allowing this number to be excluded from another cell. Specifically, it relies on bi-value pairs with an inclusion relation. There must be a cell which includes similar pairs within its unit causing cell numbers to interact diagonally with pairing cells that form a triangle.

Let us consider another example to understand the XY-wing sudoku technique. In cell E2, there is an exclusion relationship between 2 (E8) and 2 (H1). Cells H1, H2, and E8 all connect with each other to form a triangle while all of their neighbors close in on cell E2.

If an X or Y cell = 2 turns on, there is a chain reaction that makes it crucial for either Y or X cell = 2 to exclude the possibility of a 2 in cell E2. Here Solver gives us the final 2 box which refers to cell 1E, it zeroes out signaling the puzzle is solved and the technique was correctly applied.

Coloring

Contiguous areas, which are cells connected at edges, can be two-colored. This means that one area is, for instance, red and all squares of the other color should be blue. It can be applied in three ways.

1. If any element on a row (or a column or a few rows or columns) can only have two values, those two colors can fill the possibilities in, which is now a surjection problem for the entire set. This is known as 2D coloring.
2. If there are two independent Two-Inclusion restricted sets pinning some candidates on two cells, coloring theory can be applied using a different color for each original set and filling in related cells of both subsets with a fourth color. The fourth color will form an unnecessary region cut to prove contradiction.
3. If only one number will be valid after raster decomposition, raster squares can alternate two colors, which are often either all red or all blue, to create the definition of a solution area.

The key is simpler two-way coloring.

How to Practice and Improve Math Skills in Sudoku?

You can practice and improve math skills in sudoku by manipulating all parts of a puzzle to analyze goal clarity, working memory capacity needs, the value of changed techniques, and strategic goals. Different strategies can be used in tandem based on the role of mathematics in the puzzle and its effect on the overall difficulty and whether goal clarity pursuing mathematical goals or puzzle completion is the main focus. Below are the main tips for using math to improve sudoku skills.

1. Vary methods in conjunction with different subsets of puzzle areas for analysis and learning. Switch roles between pursuing mathematical goals, puzzle completion, or strategic goals, with the latter including relief of working memory demands through reduction of goal focus.
2. Complete regular puzzles without any regard for enhancements via mathematical manipulation to increase overall practice skills and speed.
3. Learn new and more complex tactics for puzzle-solving via mathematical methods in order to enhance strategic thinking.

Start with Easy Puzzles

Easy puzzles are perfect for beginner solvers who are not yet familiar with advanced mathematics in Sudoku. The starting grid with fewer filled-in numbers means there is less information to process and try to exploit to make mathematical deductions. This makes easy puzzles quicker, while at the same time helping develop intuition about the game.

Bertram Felgenhauer (PhD in mathematics) writes in his research paper Exploring Mathematical Structures Within Sudoku that even the most difficult (unlimited solving difficulty) puzzles usually start with simple removals (<40 removals); more precisely, in the order of 27 removals. In advanced puzzles, at the late stages of solution, there are certain tricks to filling the blank spaces once the removals have been done but elucidating these are beyond the scope of basic Sudoku playing. Hence these stages of the puzzles will remain easy.

Challenge Yourself with Harder Puzzles

Another way to use math in Sudoku is to avail of the wide range of difficulty levels available within the game and to actively challenge yourself to progress to harder puzzles. Both Simple Solving and Brute Force methodologies can be employed, although the importance of Simple Solving decreases rapidly as puzzles get harder. Every time you increase the difficulty of the Sudoku you work on, you provide a real-life lab for the daily dose of brain training that Dr. Nussbaum recommends. Such crosswords are stew in your brain’s Good Will Hunting Skellig Michael.

Time Yourself

Timing is a fun and useful technique to see how long different puzzles take and how that correlates with their difficulty. Professional players will have different averages for how long they take on easy puzzles versus hard puzzles. By observing and noting how long different puzzles take, every sudoku player can get a better idea of which puzzles are likely to be easier or harder without even starting the puzzle because they have seen similar ones before.

Born in Yugoslavia in 1945, Tihomir Truhar was known as a teacher and prize-winning writer in the field of applied mathematics. Married with two sons, he committed suicide on May 17, 2001. Although Truhar’s scientific contributions were many, he is best known for doing a controlled experiment of the effects of sudoku puzzles on IQ. In 2009, Vinka Zalik began a study of the optimal time for a person to invest in starting and restarting solving a sudoku puzzle.

Speaking of how timing is a useful tool, Zalik states her observations show that each restart of the algorithm of a puzzle contributes to one in getting the right solution. Her findings suggest that for males to get good at sudoku and have optimal results on intelligence tests, they should practice on correctly timed puzzles for 6 to 10 consecutive days for a duration of 40 minutes to an hour and a half(40m-1.5h) per day. The same applies to females.

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