Sudoku Explained
By Giulio Zambon
Copyright 2011 Giulio Zambon
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Introduction
Welcome to Sudoku Explained. The purpose of this book is to help you become a Sudoku champion. Not everyone has the pattern-recognition abilities to become a champion, and I cannot guarantee that you will succeed, but I can point you in the right direction. The rest is up to you.
The WorldWide Web is full of explanations about Sudoku. You can find the description of many strategies and lots of examples. What I believe you will only find in this book is a systematic and consistent description of all significant strategies coupled with commented solutions of puzzles.
For this book, I have chosen forty Sudokus I generated with the program described in my first Sudoku book (Sudoku Programming). They all have thirty-three clues and look like medieval crosses, because the central row and the central column are always already solved.
Their difficulties range from easy to diabolical, measured as the number and complexity of the solving strategies they require. It is true that, in general, puzzles with more clues are easier, but what is true statistically, doesn't apply to each individual puzzle. I have seen very easy puzzles with only seventeen clues, while the last puzzles in this book, despite their thirty-three clues, are as challenging as they come.
Getting Started describes the terminology I use throughout the book.
The Strategies explains in detail fifteen strategies grouped in four levels of complexity, from 0 to 3.
The Puzzles presents the Sudokus. Puzzles 1 to 10 can be solved with level 0 strategies; 11 to 20 require level 1 strategies; 21 to 30 are at level 2, and 31 to 40 are at level 3.
The Solutions provides the solutions of the puzzles shown in the preceding chapter. For the puzzles at level 1, I also list a sequence of strategies capable of solving them, while for the puzzles at level 2 and 3, I explain the application of the more complex strategies in detail.
Have fun with Sudoku!
Getting Started
In this chapter, I describe the terminology I will use in the reminder of the book to explain Sudoku solving strategies and to illustrate the examples. Although you will probably be familiar with most of it, I suggest that you still go through this chapter.
A Sudoku puzzle is a grid of 9 x 9 cells partially filled with numbers between 1 and 9, as shown in Figure 1 1.
Figure
1 1: A Sudoku puzzle
You identify each cell through its row and column indices (see Figure 1 2), and the term line indicates either a row or a column.
The nine groups of 3 x 3 cells delimited by thicker borders are called boxes. You identify them with an index between 0 and 8 as shown in Figure 1 3. The term unit indicates either a line or a box.
To solve a Sudoku puzzle you need to complete the grid with numbers between 1 and 9 in such a way that each number only appears once in each row, column, and box.
Figure
1 2: Cell numbering
Figure
1 3: Box IDs
The Strategies
Not all strategies are created equal. When solving a Sudoku puzzle, it makes sense to use simpler strategies as long as they result in candidate removals and cell solutions, and only then move to more complex strategies.
I group Sudoku-solving strategies in four levels of increasing complexity, which I number from 0 to 3.
Level 0 Strategies
Level 0 strategies are the strategies on which Sudoku puzzles are based. They are those that every Sudoku beginner knows and applies. So much so, that they often are not even named. They are: naked single, unique, and cleanup (my naming convention).
The strategy 'naked single'
If a cell only has a single candidate, that candidate must solve the cell. This is obvious: if there are no other possible candidates in a cell, the only one present must be it.
The strategy 'unique'
If a candidate for a particular number is only found in one cell of a unit, it must solve the cell. For example, suppose that looking at all cells of box 4 you find out that the only candidate 2 of the box is in (5,3). As there must be a 2 in every box, it means that the 2 in (5,3) must be it, and you can remove all other candidates from that cell.
The strategy 'cleanup'
When you solve a cell, the digit that solves it cannot be a candidate anywhere else in any of the units to which the cell belongs. For example, if (0,4) solves to a 3, you can remove the candidate 3s from all the cells of row 0, column 4, and box 1.
Usage of Level 0 strategies
When removing a candidate from a cell, it could be that it was one of only two candidates in that cell. If that is the case, by removing it, you leave the cell with a single candidate, which means that the 'naked single' strategy applies. For example, if (2,7) contains two candidates, 1 and 3, and you remove the 3, it is clear that 1 must solve the cell.
It could also be that the candidate you have removed was one of only two candidates for a particular number in a unit. In that case, the 'unique' strategy applies to the remaining candidate. For example, if you remove a 3 from cell (2,7), it could be that only one candidate 3 is left in column 7, say, in (8,7). In that case, 3 must solve (8,7), otherwise there wouldn't be in column 7 any 3 at all.