I decided to learn how to solve advanced Sudoku puzzles.
Early in 2019, I downloaded a Sudoku app to test if this puzzle could be a relaxing pastime.
Eighteen months later, I’ve fallen pretty deep into this numeric rabbit hole!
From knowing next to nothing about Sudoku, I now know more than most about this beautiful little puzzle.
Here’s what I’ve learned:
- What I Knew of Sudoku Before I Started This Journey
- The Origin Story of the Sudoku Puzzle
- Notation, Notation, Notation
- Other Sudoku Notations
- More Basic Sudoku Techniques To Learn
- Some More Advanced Sudoku Techniques
- And There’s Also Bifurcation (But Not Really)
- Different Sudoku Variations
- The Genius of Masterful Sudoku Setters
- The Complexity of a Sudoku
- How To Get Started
- More Creative Projects
What I Knew of Sudoku Before I Started This Journey
As many of you, dear readers, may already know, the foundation of a Sudoku consists of nine rows (r1-r9) and nine columns (c1-c9), forming a neatly structured grid. This grid contains nine distinct squares, each containing nine individual cells.
This intellectual exercise aims to meticulously populate each cell with a numerical value ranging from 1 to 9. The challenge, however, lies in ensuring that no integer is repeated within any single row, column, or 9‑cell square, thereby demanding careful thought and strategic planning.
Consequently, upon successful completion, the fully solved puzzle will feature a well-balanced distribution of each digit from 1 to 9, with nine instances of every number.
However, one cannot rely on these constraints to achieve the desired outcome. Each unique puzzle starts with predetermined numerical placements congruent with the eventual solution. These given numbers serve as a foundation upon which the solver must build.
Utilising these initial clues as a guiding compass, it becomes the solver’s task to navigate the complexities of the puzzle, meticulously placing each digit in its rightful position. Once all the numbers have been expertly arranged, the puzzle is complete.
This is what I knew of Sudoku before downloading my first Sudoku app some 18 months ago.
But there was more to learn.
The Origin Story of the Sudoku Puzzle
Sudoku puzzles have an ancient feel, much like chess or go. But the numeric puzzle is a relatively recent phenomenon.
“The game first appeared in Japan in 1984 where it was given the name “Sudoku,” which is short for a longer expression in Japanese – “Sūji wa dokushin ni kagiru” – which means, “the digits are limited to one occurrence.”
Source: The history of Sudoku
Contrary to what I thought, the Sudoku puzzle wasn’t invented in Japan — even though it got its name there. Unfortunately, the inventor of the Sudoku puzzle died before getting to experience his invention became a global phenomenon.
“The modern Sudoku was most likely designed anonymously by Howard Garns, a 74-year-old retired architect and freelance puzzle constructor from Connersville, Indiana, and first published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku). Garns’s name was always present on the list of contributors in issues of Dell Pencil Puzzles and Word Games that included Number Place, and was always absent from issues that did not. He died in 1989 before getting a chance to see his creation as a worldwide phenomenon. Whether or not Garns was familiar with any of the French newspapers listed above is unclear.”
“The Times of London began publishing Sudoku puzzles in 2004, and the first US newspaper to feature Sudoku was The Conway (New Hampshire) Daily Sun in 2004. Within the past 10 years, Sudoku has become a global phenomenon. The first World Sudoku Championship was hosted in Italy in 2006 and the 2013 World Sudoku Championship will be held in Beijing.”
Source: The history of Sudoku
Fans of the famous biologist Richard Dawkins will be pleased to note that the Sudoku puzzle is a fascinating case study for memes!
“Scientists have identified Sudoku as a classic meme – a mental virus which spreads from person to person and sweeps across national boundaries. Dr Susan Blackmore, author of The Meme Machine, said: ‘This puzzle is a fantastic study in memetics. It is using our brains to propagate itself across the world like an infectious virus.’”
Source: So you thought Sudoku came from the Land of the Rising Sun
Notation, Notation, Notation
The first thing I learned about Sudoku was — notation. Owing to the inability to place the majority of digits immediately, denoting the cells for potential numbers becomes essential.
Consider, for example, placing the number two within box 1 (the left upper-corner 9‑digit square). Suppose the number two can be positioned in only two locations, prompting me to annotate those as prospective candidates.
Suppose we encounter a similar constraint within box 1 concerning the number five — and it happens to be the same two cells!
Given that both numbers share only two cells in box 1, it becomes evident that these cells must accommodate a 2 or a 5. Consequently, I can confidently ascertain that no other digit besides 2 or 5 can occupy these cells.
This technique is known as a “naked pair.”
As a result, even though I cannot definitively assign the numbers 2 and 5 within box 1, the notation leads to several other constraints that may prove instrumental in annotating (or eliminating annotations) for various digits in corresponding grid cells.
In most Sudoku puzzles, methodical notation is the only way to solve the puzzle.
Other Sudoku Notations
Perhaps due to the particular circumstances of the pandemic, the Youtube channel Cracking the Cryptic gained lots of traction. And it ended up in my feed, too.
I immediately understood the power of using different notation techniques via the Youtube show. For instance, noted digits in the centre of the cell will mean that the cell will contain one of those digits and no other numbers.
Noted digits along the edges of the cell mean that those digits are candidates, but there might still be unnoted digits that might go into that cell.
A variant of the edge notation is called Snyder notation:
Snyder notation, named after the renowned Sudoku expert Thomas Snyder, is a refined and systematic technique Sudoku enthusiasts use to improve their puzzle-solving proficiency. This method entails strategically annotating small pencil marks within each cell to signify the possible candidates for that specific location. By doing so, solvers can better visualize patterns and restrictions, ultimately enhancing their ability to ascertain the correct placement of digits.
The Snyder notation approach involves denoting only pairs of potential candidates within 3×3 boxes when the candidates can occupy exactly two cells within that box.
The technique hinges on the premise that identifying these pairs of candidates will reveal critical information about the grid’s constraints and, in turn, the placement of other digits. This focused and streamlined notation practice aids in reducing clutter and confusion, enabling solvers to more effectively recognize opportunities for progress and solve the puzzle with greater ease and efficiency.
Knowing how to use different types of notation will quickly take the newbie solver to solve more advanced puzzles rapidly.
More Basic Sudoku Techniques To Learn
There are several other Sudoku techniques that solvers utilise to progress through and ultimately solve puzzles.
Some of these methods include:
Some More Advanced Sudoku Techniques
However, I quickly learnt that really good Sudoku puzzles require more advanced techniques to be solved.
Some of these methods include:
And There’s Also Bifurcation (But Not Really)
Bifurcation, commonly referred to as “trial and error” or “guessing,” is a technique employed by some Sudoku solvers, particularly when confronting complex and challenging puzzles.
This method involves selecting a cell with a limited number of candidates (ideally a bi-value cell with only two possibilities) and tentatively assigning one of the candidates as the correct value.
The solver then solves the puzzle based on this assumption, carefully observing the resulting implications.
If the initial guess leads to a contradiction or invalid solution, the solver backtracks to the original bifurcation point and proceeds with the alternative candidate.
Although bifurcation can be an effective approach to solving difficult puzzles, many Sudoku enthusiasts consider it less elegant and less intellectually satisfying compared to the systematic application of logic-based techniques.
In short: Don’t resort to bifurcation.
Different Sudoku Variations
Once you start solving more advanced puzzles, you can’t help but discover the adjacent universe of Sudoku variations.
The Sudoku variations invite a wide array of highly satisfying logic applications. None of the advanced solving techniques is typically lost, but with a variation, you can play around with additional and sometimes even more satisfying techniques.
Also, these variations allow for more creative freedom for puzzle setters.
The Genius of Masterful Sudoku Setters
It’s not the solvers who are the superstars in Sudoku; it’s the setters.
Whether they are classics or variations, beautiful puzzles are typically created by a master setter — by hand. Make no mistake about it: setting up a Sudoku puzzle is hard work.
The master Sudoku setter will reverse-engineer the puzzle to challenge you and lead you through the puzzle in a creative way. And a whole global community of highly talented solvers holds these famous master setters in extremely high regard.
And these setters sometimes have their unique styles; in certain Sudokus, you can recognise the setter’s approach to setting puzzles, especially in variations where the creative freedom for the setter is much greater.
A few notable examples include:
Setting a beautiful puzzle is the work of a master. And setting a beautiful yet highly unique puzzle is a genius’s work.
The Complexity of a Sudoku
One rule of Sudoku is that each puzzle must only have one unique solution. A Sudoku puzzle is literally “broken” if there are multiple solutions.
How many starting clues must be provided to ensure a puzzle has only one final solution? The community has found several solvable puzzles with 17 starting digits and a unique solution. But no one has been able to construct a Sudoku where the same is true with only 16 given numbers at the start.
Researchers in Dublin decided to test all possible 16-digit puzzles.
“Nevertheless, the resulting calculation is still a monster. The Dublin team say it took 7.1 million core-hours of processing time on a machine with 640 Intel Xeon hex-core processors. They started in January 2011 and finished in December.”
Source: Mathematicians Solve Minimum Sudoku Problem
There are no unique solutions to puzzles with 16 or fewer starting digits. But we still don’t know why; we only know there aren’t any.
How To Get Started
I can’t think of a better way to start than to explore the Youtube channel Cracking the Cryptic. The show’s hosts, Mark Goodliffe and Simon Anthony have represented the UK in the World Puzzle and World Sudoku Championships.