The only way to solve a Sudoku is to solve it. There are no secret approaches beyond the consequences of its rules: each of the nine blocks (houses) has to contain all the numbers 1-9 within its squares. Each number can only appear once per row, column, and house.
What makes a good solver different than any others is not in how it is solved, but how it approaches the problem of solving itself.
What is Sudoku?
What we know as Sudoku (“single number”) has its roots in another, similar puzzle design, Magic Squares. Dating back to at least 190 BCE, a “magic square” is a puzzle where the summation of each row and column in a grid is the same without repeating numbers.
The “magical” nature of its balanced design — hence its name — held the interest of ancient mathematicians and many books, studies, and groups devoted considerable time to creating and solving different orders of the puzzle because of its possible divination usages. Starting from the more common Order 3 (a 3×3 grid), larger and more complex grids were created and rules developed to solve them quickly.
Starting with Korean Choi Seok-jeong’s work Gusuryak in 1700, but made famous by Swiss mathematician Leonhard Euler in 1783, the “Latin Square” variant was developed using letters (the Latin alphabet) and then later numbers. This was extended from the original 3×3 layout to a 9×9 playing board with nine grids. The earliest form of this variant appeared in newspapers in 1979 for a public audience under the name of Number Place, with other variants appearing earlier in published works.
In 1986, Japanese puzzle company Nikoli began publishing the puzzles under a different, and its current, name: “Sudoku” as a portmanteau of the words for “number” and “single.” By 2004, thanks to the work of Wayne Gould in developing a computer program to develop puzzles quickly, they began to appear in newspapers under the name of Sudoku.
The most common way to completely solve a Sudoku is a technique named “backtracking.” It takes its name from the action it performs: it starts by guessing a single cell for a number and then “back tracks” if the initial guess is wrong until it solves the puzzle a row at a time.
For example, given the above puzzle, the backtracking method would guess a “1” in the first cell, check if it violated any rules. If it did, it would back track and assign a “2” and check again. Once it had an answer that did not violate any rules, it would proceed to the next empty cell in the row and start the process again. It uses a type of search called “brute force,” where it will end up trying most to nearly all solutions until it finds the correct one. It is terribly inefficient, but will always, inevitably, get the job done.
However, if the goal is not to solve the entire puzzle, but provide the next possible valid move, the steps are much easier. It becomes an issue of being good enough.
Your friend, the Zone of Proximal Development
The term “scaffolding” has taken root in many educational practices. It is the process of providing support (scaffolding) for a learner to slowly take on more complicated tasks. Starting with a larger amount of help, this is slowly removed as the learner expresses mastery over a particular approach or skill until they no longer need the support. It has its basis in a more complex term, zone of proximal development.
Lev Vygotsky (1896–1934) developed the theory of zone of proximal development during the study of young children. Finding they often worked within a “zone” of completing tasks where they needed help, he found learners progress through needing larger amounts of directed help to smaller over time. Determining this “zone of proximal development,” instructors can help learners progress move quickly through dynamically adapting practices for students. Instead of using more generic test-based learning, the use of “scaffolding” (as it was later termed) can help learners master a new skill more quickly.
While not explicitly mentioned by the name “scaffolding,” many video games approach teaching a player its mechanics in the same way through a different term, tutorials. Slowly, over time, a player is taught how to control a character or perform an advanced technique through providing opportunities to use them (scaffolding) and then reducing the reminders once a player has shown they have mastered the skill.
The application Good Sudoku opens with an introduction promising to “reduce busywork” and help teach people to “solve harder and more interesting puzzles.” Instead of, as with hundreds of other collections, simply providing the puzzles in different categories, the purpose of Good Sudoku is pedagogical. It provides the tools for the player to become “good” at Sudoku through explaining its rules and common patterns as a result of its rules.
Across multiple places, Good Sudoku makes its goals explicit: it wants to teach players how to play. It provides note taking, a complex hint system, and shows players possible solutions when they select a number. It tries to make the process of completing the task easier through supporting the player in multiple ways. Instead of trying to completely solve the puzzle for the player, it is only ever one step ahead of them in possible moves. This helps Good Sudoku constantly monitor the player for its own “zone of proximal development” through watching what moves the player makes and if they ask for hints for particularly challenging patterns.
The player is even awarded achievements when they accomplish certain advanced techniques a certain number of times demonstrating their mastery of them. This is Good Sudoku reinforcing its learning. It assumes the player no longer needs the same explicit support structures once they have proved they understand how to solve particular patterns.
Great Design, Bad Mastery?
Ironically, through providing a great design for solving Sudoku puzzles, it falls into a “bad” teaching pattern itself. While it does provide a mode without the suggestions of placement for numbers in houses, most of the modes of play use this mechanic. This is part of eliminating the “busywork” of solving Sudoku, after all, but it also makes the player dependent on this approach. If the player is always given this support, why would they ever use a different collection or application? What helps players solve puzzles ever faster makes them stuck with the mechanic to help them see the openings.
The end goal of scaffolding is for it to be removed. Once a learner reaches mastery of the skill, all of the supports are removed and the learner becomes capable of independently completing the task. They should no longer need help. This is what ultimately makes Good Sudoku great at learning but bad at mastery: the tools it uses to teach are always there. The help with the numbers are there. The hint system is there. Yes, on harder difficulties using these decreases the player’s score on a particular puzzle, but this is not much of a disincentive for most players.
If a player can be become good enough at Sudoku for their own goals, what would prompt them to move on? This problem is not one limited to Good Sudoku, of course, but haunts all uses of hint and learning systems. What is to stop Good Sudoku players, for example, from completing the tens-of-thousands of puzzles in the Easy category? The theories surrounding scaffolding do not have an easy answer, and neither, ultimately, does Good Sudoku. It is interested in helping learners progress to complex tasks, if they challenge themselves first, but it also cannot prevent learners from stopping at a particular level of scaffolding and ending their own progress toward greater mastery.
Like Sudoku itself, learning is only ever a forward action. The only way to do it is to do it step by step.