I decided to learn how to solve an advanced Sudoku puzzle.
Early in 2019, I downloaded a Sudoku app to test if this kind of puzzle could be a relaxing pastime. I immediately wanted to learn more.
Now, some 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 lots of things that I never even knew there was to know about this beautiful little puzzle.
So, why not share some of these learnings?
Who knows — Sudoku might be something for you to dive into as well!
What’s a Sudoku?
Contrary to what I first thought, the Sudoku puzzle wasn’t invented in Japan — even though it got its name there.
“[…] the modern game of Sudoku as we recognize 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.”Source: Wikipedia
The little puzzle itself is deceptively simple:
As most of you know, there are 9 rows (r1-r9) and 9 columns (c1-19). This grid forms 9 squares with 9 cells each.
The idea is therefore to fill out each cell with a digit between 1-9 without repeating a digit in the rows, in the columns, or in the 9-cell square.
The finished puzzle will therefore 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, there are a few given digits that will fit the final solution.
Based on these given numbers, you have to find your way through the puzzle until all the digits are in their right place — and you’re done!
This is exactly what I knew about this little puzzle before downloading my first Sudoku app 18 months ago.
But there were more to learn still.
How do you solve a Sudoku?
The first thing I actually learnt about Sudoku was notation. Most digits cannot be filled in directly, so you often have to notate the possible cells for a digit.
Where can a 2 go in box 1 (the first square of 9 digits)? In this example, it can actually only go in two places, so I annotate these two as potential candidates for 2 in that box.
As an example, let’s say that we can find the same restriction in box 1 for the digit 5. Since the 2 and the 5 share only 2 cells in box 1, those two cells have to be either 2 or 5. Alas, I now know that they can’t contain another digit except for 2 or 5.
Since I now know that 2 and 5 will be in both of those positions in in box 1, I now also now that 2 and 5 can’t be anywhere else in that row, either.
So, even though I can’t actually place 2 and 5 in box 1, annotation helps me see several other restrictions that might help me annotate (or remove annotations) for other numbers in other cells of the grid.
Sometimes, the digits in the puzzle allows you to place another digit directly (called “a naked single”), but mostly you’ll just have to annotate cells and work your way through by looking for restrictions.
Different Sudoku notations
Perhaps due to the special circumstances of the pandemic, the Youtube channel Cracking the Cryptic gained lots of traction. And it ended up in my feed, too.
Via the Youtube show, I immediately understood the power of using different notation techniques.
For instances, noted digits in the center of the cell will mean that the cel will contain one of those digits and no other digits.
Noted digits along the edges of the cell means 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 possible cells for a number in either a box, a row, or a cell.
And finally, there are situations where it make sense to use different colors 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 be able to solve more advanced puzzles quickly.
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 literally testing out certain configurations to see if they work.
If the bifurcation work, that generally tells you nothing, because you might not now if you’re following the path of the unique solution. However, if it does break, then you might be able to at least eliminate a specific candidate.
I personally dislike using bifurcation because it’s mostly just strenuous work instead of applying 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 nunchucks.
How to master the Sudoku
By learning how to efficiently scan and notate a Sudoku puzzle, even a beginner can quickly move up to solving intermediate puzzles.
When it comes to really difficult puzzles, just knowing and understanding the more advanced solving techniques is only a small part of the skill. The real challenge here is to actually be able to discover exactly where in the grid these advanced techniques should be used.
And even if you’re able to solve a very difficult puzzle, there’s the question of how long time 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 to get the hang of solving more advanced puzzles, you can’t help but to discover the adjacent universe of Sudoku variations.
Some of the more popular variations include chess Sudokus (Knight Sudoku, King Sudoku, and Queen Sudoku), Killer Sudoku, and Thermo Sudoku.
Again, this is not the right place to outline and explain each and every 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 solves 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 simply invites a wide array of highly satisfying applications of logic. Typically, none of the advanced solving techniques are lost, but with a variation, you get to play around with additional and sometimes even more satisfying techniques.
Also, these variations allow for more creative freedom for the puzzle setters.
The Sudoku setters
When the beginner downloads a Sudoku app, chances are that the app will generate computer-created puzzles. In terms of logical beauty, creative novelty, and satisfactory solving paths, it’s hit or miss with a computer-generated puzzle. Mostly miss.
Beautiful puzzles, whether they are classics or variations, are typically created by a master setter — by hand.
The master Sudoku setter will reverse-engineer the puzzle to make sure to challenge you and to lead you through the puzzle in a creative way. And there’s a whole global community of highly talented solvers who 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 own 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 and yet highly unique type of puzzle is the work of a bonafide genius.
The complexity of a Sudoku
The unwritten rule of Sudoku is that each puzzle should only have one unique solution. If this isn’t correct, the Sudoku puzzle is literally “broken”.
So, in a classic Sudokus, how many starting clues must be provided to ensure that the puzzle has only one final solution?
Well, the community have found several puzzles with 17 starting digits that are solvable with a unique solution. But they have never ever found a puzzle where the same is true with only 16 given digits at the start. But no-one has been able to prove that there are no solutions to a puzzle with 16 starting digits.
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
The answer? There are no unique solutions to puzzles with 16 or less starting digits. But we still don’t know why, we just know that there aren’t any.
Breaking into a handcrafted Sudoku
In variations of Sudoku, it’s possible to use far less than 16 digits in the start (sometimes none!) which has put a lot of emphasis on the given clues in 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 very well be more unique and more difficult than solving the rest of puzzle. Some techniques for breaking into a variation can be insanely difficult to figure out!
Sudoku puzzles have an ancient feel to them, much like chess or go. But the numeric puzzle is actually is a fairly recent phenomenon.
“The Times of London began publishing Sudoku puzzles in 2004, and the first U.S. 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 into Sudoku
I can’t think of no better way to start than to explore the Youtube channel Cracking the Cryptic. The show’s hosts, Mark Goodliffe and Simon Anthony, have both represented the UK in the World Puzzle and World Sudoku Championships.