The Science of Sudoku

The Science of Sudoku

Ever since 2005, Sudoku has seen a fantastic burst in popularity that remains widespread to this day. Originating in 1979 as “Numbers in Place,” it eventually reached Japan, where it earned its name (which means “single numbers”). Its unique number grid setup is easy enough for children to understand but complex enough to require real thought to solve. 

And the science behind Sudoku is almost as fascinating as the game itself. Over the years, mathematicians and other scientists have studied this game and found many unique concepts fuel its success. Understanding these ideas can help you discover new methods for solving a Sudoku puzzle and provide those who love science with a deeper understanding of their favorite game.

Table of Contents

The Basic Concepts Behind Solving Sudoku

Sudoku centers primarily on logic and trial-and-error. Solving a puzzle requires individuals to carefully understand the game rules and extrapolate number positions based on their relative position to others. Logic dictates that certain numbers cannot be in specific spots, allowing the solver to guess which is the most likely to be in a spot. 

Confused? Well, when you look at a Sudoku puzzle, you examine the numbers placed in the grid at the start (these are known as “clues” and will be referred to as such throughout) and trace them throughout the rest of the puzzle to see where they cannot be in other sectors. Remember: only one like number can exist within the meeting point (called “squares” throughout this article) of horizontal and vertical lines.

Every Sudoku puzzle or grid can be solved using these unique concepts. By understanding the essential elements of each of these ideas, you will quickly become much better at Sudoku. However, some people may doubt whether logic and trial-and-error can be considered science. However, they are part and parcel with certain scientific and mathematical theories, particularly those that fuel the creation and solving of Sudoku puzzles.

Logic and Trial-and-Error as Science

But are logic and trial-and-error science? Absolutely. Logic is defined by stringent rules specific to a unique situation. For example, if a nine is positioned in one grid, logically, you can deduce that no other nine is in that grid based on Sudoku’s rules. You can then state that no other nine will be positioned in the same row or column as that nine from that idea. 

Trial-and-error is a simplified version of logical deduction. This method requires you to make decisions based on the best information available to you at the time. For example, you may find that a square could logically have a 2, 5, or 8. However, examining the other grids provides you with no other clues. So, you put an eight into the square for now and work on the assumption that it is correct. 

If the number turns out to be incorrect, you work backward from where you made a mistake and try another number to see what happens. While it may seem like random guesswork, it is based on the logical and mathematical deductions you’ve made throughout the rest of the puzzle. And much of science uses trial-and-error to find things as complex as new medications and medical treatments.

These science concepts fuel the basic process of solving a Sudoku puzzle. If you understand these ideas, you can solve a Sudoku puzzle of any size, from one as small as a 4×4 to the standard 9×9 and beyond to more extensive 20×20 options. But what of the actual mathematics behind this game? What ideas and concepts make it such a popular game? Let’s read on to find out more!

The Deep Mathematical Concepts Behind This Game

The basic science concepts that fuel Sudoku are primarily mathematical. A heavy amount of symmetry is involved, which focuses heavily on balance. Grid theory is also crucial for the science of Sudoku, as is group operation and the associative properties of group operation.

These terms are not as complex or difficult to understand as they may seem. We’ll discuss them below in context to Sudoku to help make it easier for you to know what we mean. Later on, we’ll discuss a few unique puzzle-solving methods that utilize a deeper understanding of these ideas to help make your gaming so much easier.

Symmetry and How It Applies to Sudoku

Symmetry can be defined in a few different ways. The most common definition is “(the)…arrangement of elements or parts on opposite sides of a dividing line.” However, symmetry can also be defined as a similar arrangement on a plane or about an axis. And it can also be described as a relationship in which each part corresponds to the other to create a balance. 

Confused? Let’s look at a straightforward example to give you an idea of what we mean. A high level of symmetry often defines physical beauty. For example, please take a look at your favorite celebrity and examine their face and body. If you were to place a line directly down the center of their body, there’s an excellent chance that the two sides would look identical. Likewise, you could likely put a mirror between the line, and the reflection would perfectly replicate this beautiful person. 

In Sudoku, symmetry is defined by the balance of numbers on each line and within each grid. So, for example, in a 9×9 puzzle, each row and column would have one through nine without duplicates. And the 3×3 grids within the puzzle would contain each number only once as well. Obtaining this perfect symmetry and balance is the core drive behind each game of Sudoku. 

Interestingly, Sudoku is that you could take a valid puzzle, rotate it 90 degrees or more, and produce a new and unique puzzle. However, Sudoku specialists and science experts wouldn’t consider that one of the almost unlimited number of potential Sudoku puzzles. Instead, they would claim it is essentially the same as the original but can serve as a new variant. 

Grid Theory in Sudoku

Many mathematical and science fields use grids as a way of arranging data. They work perfectly as a visual way of sorting through vast reams of information and help present it in a way that makes immediate sense. In Sudoku, grid theory primarily focuses on various grouping concepts and properties that help to define the symmetry that each Sudoku puzzle contains. 

Without going into difficult or complex mathematics, it is enough to know that each integer (number or clue within the grid) has a corresponding number within other grids. This concept is called the associativity property of the group operation. Concepts of inverse identity (wherein numbers are not alike, such as how nine is not the same number as six) also help fuel this unique puzzle. 

Just as importantly, concepts of rotations and reflections help to create a valid Sudoku puzzle. These ideas help to transform the initial Sudoku grid (typically a 3×3 starting point) into an appropriate and effective total puzzle. By applying these common transformations (used in many other types of grid theory), you can create an almost endless number of potential Sudoku puzzles. 

Square Transformations in Sudoku

After creating a valid 3×3 grid, Sudoku puzzle creators often start by rotating this initial grid 90, 180, or 270 degrees in any direction. Doing so immediately alters the positions of the numbers to create a sense of symmetry with the previous grid. Puzzle makers can then apply more rotations to that transformed grid to fill out the puzzle. However, they also need to use reflections as well. 

Reflections flip the grid horizontally or vertically either through the center of the square or along the diagonals from the bottom-left to the upper-right or the bottom-right to the upper-left. When combined with rotations, it is usually possible to create unique and valid Sudoku grids that will keep puzzle fans engaged for hours at a time. Fans can even make their own grids in this way. 

That said, various other permutations can be taken to help transform a grid and make it harder to solve. For instance, puzzle difficulty is decided by the number of clues available in the grid. These clues are carefully chosen to make sure that they follow the lines of symmetry and balance. This uniquely diverse range of possibilities creates an almost staggering number of available Sudoku puzzles. 

The Possible Number of Sudoku Puzzles

Fans of Sudoku may worry that they’ll eventually run out of puzzles and that their favorite hobby may become dull. After all, even a game as complex as chess has just 90,040 possible games. While that number is still incredibly high, there’s a good chance that many players could run into repeated moves that feel a bit too familiar. So is there any risk of this issue affecting Sudoku fans?

Thankfully, science seems to suggest that running out of grids is not likely within the life of the universe. Why is that? In 2005, mathematicians Bertram Felgenhauer and Frazer Jarvis calculated the number of possible 9×9 Sudoku puzzles. And they found that it was a staggering 6,670,903,752,021,072,936,960. That’s six sextillion or 9! × 722 × 27 × 27,704,267,971.

That’s nearly one trillion unique puzzles for every person on the planet! Ironically, this number was computed using logic and trial and error and has been verified multiple times. Now, if you consider that Sudoku also has many grids of varying sizes, you can see that the possible puzzles are unlimited for all intents and purposes. This helps to make Sudoku a very engaging game for millions of fans.

The Science Involved With Solving a Puzzle

The complex mathematics behind Sudoku often forces players to use a rigorous number of rules and concepts to solve a puzzle. These concepts are just as much science as the rest of the ideas in Sudoku. And they can be used to logically choose which number goes in a cell and provide you with a puzzle solution. Follow these steps in order, and you can solve any Sudoku puzzle on the market.

The Forced Cell Concept

This idea is probably something you’ve already used without realizing it even had a mathematical or science-based name! The forced cell concept centers around the assumption that a number is fixed and cannot be changed. Often, this idea centers on the clues that start the puzzle. However, it can also apply to numbers that you’ve logically deduced must be in a specific cell. 

Once you have a fixed cell that you can use as a base, you can scan the columns and rows shared by this cell and all the other grids it touches within the puzzle. If this number is fixed and cannot be changed, you can logically deduce that no number of that type can go within these rows or columns. Then, you can check each other grid to see if it contains that number elsewhere. 

Does this idea seem fairly basic? At its core, it is a reasonably complex logical deduction that fuels the core of Sudoku solutions. Most of the numbers that you find will be based on this type of trial-and-error elimination, allowing you to finish the puzzle. Once you no longer have any clues that fall within the forced cell concept, you can move on to other ideas to fuel your success. 

The Only Cell Concept

This idea is also relatively simple at its core but complex in its application and mathematics. It starts by choosing a specific value (such as seven) and looking through the columns, rows, and grids of a puzzle, and gauging which have this value and which do not. So, for example, if you look through the sixth column of the total grid and discover it lacks a seven, you know one needs to go here. 

However, if you also find that it does contain a seven, it cannot be placed in this column. You can then go through each of the rows in the grid and discover whether they have a seven. For example, if the sixth column and fifth rows have no number, you can logically deduce that the seven may go in this square. 

In a certain way, this method is similar to the forced cell concept. However, it focuses on specific numbers rather than on the connections between the cells. And it will require a certain amount of trial-and-error that may be frustrating to some people. However, it is a powerful way to solve puzzles that you’re struggling to figure out using other methods. 

How Learning About This Science Helps Your Game

If you fully understand the math and science behind Sudoku, you can likely solve puzzles more effectively. Why is this the case? Let’s say you completely solve a single 3×3 grid within your larger puzzle. You can then use the concepts of transformation to logically deduce other potential positions for numbers throughout the rest of the grid. Though challenging, this process is possible. 

Just as importantly, understanding this math can help to deepen your appreciation of this beguiling game. Though it might seem simple on the surface, a deeper dive reveals that there is a lot of math and science just beneath the hood of this game. So, why not download a new Sudoku puzzle and try out these concepts yourself? You might just find yourself becoming a Sudoku master in no time.