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.
18 months later, I must confess that 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’s a Sudoku?
Contrary to what I thought, the Sudoku puzzle wasn’t invented in Japan—even though it got its name there.
“[…] the modern game of Sudoku as we recognise it today was invented by Howard Garns, a freelance puzzle inventor from Connersville, Indiana, USA in 1979 when it was published in Dell Pencil Puzzles and Word Games magazine. The puzzle was known as “Number Place,” since it involved placing individual numbers into empty spots on a 9 x 9 grid. 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
Unfortunately, the inventor of the Sudoku puzzle died before getting to experience his invention becoming 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 little puzzle itself is deceptively simple:
As most of you know, there are nine rows (r1-r9) and nine columns (c1-19). This grid forms nine squares with nine cells each.
Therefore, the idea is to fill each cell with a digit between 1-9 without repeating an integer in the rows, columns, or the 9-cell square.
Therefore, the finished puzzle will contain 9 of each digit between 1-9.
But, you can’t just fill these digits in based on those constraints alone. At the start of each unique puzzle, a few given numbers will fit the final solution.
Based on these given numbers, you must find your way through the puzzle until all the digits are in their proper place — and you’re done.
The above is all that I knew about Sudoku before downloading my first app 18 months ago.
But there was more to learn still.
Basic Sudoku Notation
The first thing I learned about Sudoku was notation. We cannot fill most digits in straight away, so you often have to notate the viable cells.
Where can a two go in box 1 (the first square of 9 digits)? This example can only go in two places, so I annotate these two as potential candidates for 2 in that box.
For example, let’s say we can find the same restriction in box 1 for digit 5. Since the two and the five shares only two cells in box 1, those two cells have to be either 2 or 5. Alas, I now know they can’t contain another digit except for 2 or 5.
Since I know that 2 and 5 will be in both of those positions in box 1, I know that 2 and 5 can’t be anywhere else in that row.
So, even though I can’t place 2 and 5 in box 1, annotation helps me see several other restrictions that might help me annotate (or remove annotations) for different numbers in other grid cells.
Sometimes, the digits in the puzzle allow you to place another number directly (called “a naked single”), but mostly you’ll have to annotate cells and work your way through by looking for restrictions.
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; you only add candidates if there are only two viable cells for a number in either a box, a row, or a cell.
And finally, there are situations where it makes sense to use different colours to notate cells. This is helpful when looking for symmetries or non-numerical, cell-related restrictions.
Knowing how to use different types of notation will quickly take the newbie solver to solve more advanced puzzles rapidly.
More Advanced Sudoku Techniques
Quite soon, I ended up trying to solve more advanced Sudoku puzzles.
One such technique is bifurcation (which I hate, by the way). This means testing out specific configurations to see if they work.
If the bifurcation works, that generally tells you nothing because you might not know if you’re following the unique solution path. However, if it does break, you might be able to eliminate a specific candidate.
I don’t particularly appreciate using bifurcation because it’s strenuous work instead of satisfying logic.
Other advanced solving techniques are X- and Y-wings.
“An XY-Wing is a group of three cells, one sharing a unit with the other two, each having only 2 candidates. The two cells that share a unit with the first are called the Wings. Each of the wings must share one candidate with the first cell, (that’s part of sharing a unit) but of different values. If the second candidates in the wings are both the same, and both share a unit with a common candidate in a fourth cell, that candidate can be eliminated.”
Source: XY Wings
There are not too many advanced solving techniques to learn, but still too many to describe in a meaningful way here.
But a few of the most known are variations of the X- and Y-wing, the swordfish, forcing chains, the jellyfish, the Kraken, and the nunchucks.
How To Master the Sudoku
By efficiently scanning and notating a Sudoku puzzle, even beginners can quickly move up to solving intermediate puzzles.
Knowing and understanding the more advanced solving techniques is only a tiny part of the skill for challenging puzzles. The real challenge here is to discover precisely where the solver should use these advanced techniques in the grid.
And even if you can solve a challenging puzzle, there’s the question of how long it took you. Solving on time is not my idea of fun, but that’s where it’s at if you want to be more competitive.
Different Sudoku Variations
Once you start solving more advanced puzzles, you can’t help but discover the adjacent universe of Sudoku variations.
More popular variations include chess Sudokus (Knight Sudoku, King Sudoku, Queen Sudoku), Killer Sudoku, and Thermo Sudoku.
Again, this is not the right place to outline and explain each one of these countless variations.
But there’s something to be said about the mere existence of these variations. After all, the most prominent Sudoku channel on Youtube, Cracking the Cryptic, is not focused on classic solutions at all but rather on various variations.
How come that amongst enthusiasts, these variations are often more popular than classic Sudokus? Wouldn’t it be reasonable to assume that these enthusiasts were purists?
The answer, I dare say, is 100% about the satisfaction you get from applying different types of logic.
The Sudoku variations invite a wide array of highly satisfying logic applications. Typically, none of the advanced solving techniques is 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.
Masterful Sudoku Setters
When the beginner downloads a Sudoku app, the chances are that the app will generate computer-created puzzles. In terms of logical beauty, creative novelty, and satisfactory solving paths. Mostly miss.
Whether they are classics or variations, beautiful puzzles are typically created by a master setter — by hand.
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.
It’s not the solvers who are the superstars in Sudoku; it’s the setters.
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.
Make no mistake about it: setting a Sudoku puzzle is hard work.
Setting a beautiful puzzle is the work of a master. And setting a beautiful yet highly unique type of puzzle is the work of a genius.
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’s more than one solution.
How many starting clues must be provided to ensure that 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.
Handcrafted Sudoku Puzzles
In variations of Sudoku, it’s possible to use far less than 16 digits at the start (sometimes none!), which has put a lot of emphasis on the given clues at the beginning of a puzzle.
In essence: The fewer or more outlandish the initial clues, the more challenging the “break-in” becomes.
In some handcrafted variations, the “break-in” might be unique and more complex than solving the rest of the puzzle. Some techniques for breaking into a variation can be insanely difficult to figure out!
Sudoku puzzles have an ancient feel, much like chess or go. But the numeric puzzle is a relatively recent phenomenon.
“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
How To Get Started
I can’t think of any 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.