Chapter 2: The Structure of Sudoku
Sudoku has evolved into many different forms with the same basic rules. But Classic Sudoku (about which this book is written) consists of a 9x9 grid that I call the 'Matrix', made of 81 equally sized squares that we call 'Cells'. The Matrix, shown in Figure 2-1, is further subdivided into 9 groups of Cells arranged in 3 rows and 3 Columns; we call these groups of cells 'Boxes'. Thus, the 'Primary Components' of Sudoku are the Rows, Columns & Boxes (yellow highlight) and the 'Elements' of Sudoku are the Cells (pink highlight) and the Digits 1-9.
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Figure 2-1: The Sudoku Matrix: Row, Column, Box, Cell
Note that the Boxes are delineated by heavier lines than the Cells within them. In Figure 2-1, the Row, Column and Box highlighted in yellow are for reference and to demonstrate the following point: Notice, In Figure 2-2, that if you turn the Matrix 90 degrees, it looks exactly as it did before it was moved (except for the orientation of the highlighted Components)...what were Rows in Figure 2-1 are Columns in Figure 2-2 and what were Columns in Figure 2-1 are Rows in Figure 2-2. While the Box in Figure 2-1 is still a Box in Figure 2-2, it is in a different location and has a different orientation.
Figure 2-2: Figure 1 Turned 90° to the Left
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While this may seem to have been a ridiculously obvious maneuver, I believe you will agree that it makes a point that you will not soon forget. The Rows and Columns are entirely interchangeable. What works for the Rows, works for the Columns as well. Collectively, Rows and Columns are called 'Lines'. Every Row intersects with 9 Columns and every Column intersects with 9 Rows. The intersection of a Row and Column 'creates' a 'Cell', as indicated by the pink Cell in Figures 2-1 & 2-2.
Notice, too, that the Boxes are arrayed in groups of 3. Each linear grouping of Boxes is called a 'Chute'. The Horizontal Chutes are called 'Bands' (Figure 2-3) and the Vertical Chutes are Called Stacks (Figure 2-4).
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Figure 2-3: A Band of a Sudoku Matrix
Note that there are left, middle and right Boxes and left middle and right Columns within those Boxes. Notice, however, that the nine Columns are uniquely identifiable within a Band, but the Rows are not, unless other information is provided. We refer, in general, to the Upper, Middle and Lower Rows in each Box and in each Band unless they can be numbered. Bands are very important Components of the Sudoku Matrix, as are the Stacks as shown in Figure 2-4. A lot of very interesting and important things transpire in the Chutes.
When talking about the Chutes, we may refer to the left Column of the middle Box (or Column 4) of the Band, shown in Pink, or the upper row of the Band, as shown in yellow (except for the pink cell in the Left Column of the Middle Box of the Band). While it is useful to refer to the Chutes, it is far more common to discuss Primary Components; the Rows, Columns and Boxes, as you will soon understand.
Figure 2-4: A Stack of a Sudoku Matrix
Similarly, we may discuss the right column of the Upper Box of the Stack, as shown in pink. Analogous to the Bands, the Rows of the Stacks can be identified uniquely, but the Columns cannot, unless we know which Stack we are discussing.
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Sphere of Influence:
Figure 2-5 illustrates the 'Sphere of Influence' (SoI, yellow highlight) of a Cell (pink highlight).
Figure 2-5: Sphere of Influence
The Cell is the intersection of the sixth Row and second Column, and it is located in the middle Box of the Left Stack (which is the left Box of the Middle Band). Every Cell is simultaneously in a Row, Column and Box. While the shape of the Spheres of Influence are different and unique for each Cell, (compare Figures 2-1 & 2-2) each contains exactly the same number of Cells: 21.
The digit added to the illustration in Figure 2-6 emphasizes that the SoI are related to the Digits, but in reality are fundamental components of the underlying structure of the Matrix.
Figure 2-6: Sphere of Influence
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Describing a cell as being located in the Middle Column of the Upper Box in the Left Stack of the Sudoku Matrix can be a bit tedious, especially when we are describing multiple maneuvers during a Solution. The most common way to describe locations in such a Matrix is to treat it like a Cartesian Coordinate System in which each Row is numbered 1-9 and each Column is numbered 1-9. In such a coordinate system, the (x,y) or Cartesian coordinate system, each cell is identified by a unique pair of numbers which represent the Row and Column that intersect at that cell. Cell (1,1) would be at the upper left corner, cell (9,9) would be the lower right corner and cell (5,5) (Figure 2-6) would be dead center in the Matrix. This is a common system for designating location in a plain Grid and its use is familiar to anyone with an exposure to 6th grade geometry.
But, in this author's mind, Sudoku is in need of a more distinctive form of notation, yet, one that is just as simple and understandable as the Cartesian system we are so familiar with, yet which brings to focus the unique characteristics of Sudoku. For this, I have created the Matrix shown in Figure 2-7.
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Figure 2-7: Sudoku Matrix Nomenclature
The Rows and Columns are each numbered 1-9. Although I toyed with the idea of using the system employed in Chess (where the Rows are numbered and the Columns are represented by lower case letters), I found that it was confusing to switch from numbers to letters. And while the chess system is good for knowing exactly what the Row and Column designations are, other systems work just as well (the x,y designation, for instance where the position is the key: Row listed first, Column listed second). But the most distinctive feature of the Sudoku Matrix is its division into Boxes of equal size and with numbers equal to the digits...9. A chess board, by contrast, is not divided into Domains. Within each Box are 9 Cells arranged as if a reflection of the order of the Boxes themselves, in Rows and Columns of 3. There is something elegantly simple about all this and it adds to, I believe ...if not creates... the exquisite beauty of Sudoku.
The Boxes are indicated with Capital Letters from A-I reading from Left to Right and Top to Bottom. I have numbered the cells in the Boxes 1-9. To designate a specific Cell in the Matrix, I use the Letter designation of the Box followed by the number designation of the cell within the Box; without any punctuation or spacing. Thus, the pink cell in Figure 2-8 would be named A9; and the green cell would be named H2.
This nomenclature accomplishes several objectives. First, and foremost, it provides a system that is elegantly simple and easy to understand. Secondly, it brings to focus one of the more unique characteristics of Sudoku: the Boxes. I have grown to appreciate more and more the pivotal role that the Boxes play in Sudoku. With a nomenclature that places these at the center of focus, the Sudoku-ka cannot help but see the importance of the Boxes in the solution of Sudoku. I am sure the reader will appreciate this organization and nomenclature even more when we begin our discussion on Sudoku Notation in THE NEXTchapter.
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In a sense, the Sudoku Matrix is like a playground for the Digits. Think of it as a football field or a tennis court for Digits. It just sits there waiting for the players to arrive and begin their game. Once the players take the field, however, the fun begins!
Figure 2-8: Sudoku 1: Givens
Every Sudoku is provided certain Digits at the start, and these are called 'Givens'. These are what actually 'define' a particular Sudoku. The other Cells lacking Digits are said to be Vacant Cells. In 1986, the Japanese company Nikoli began creating Sudoku that had symmetrical distributions of Givens. It is what is called 'Rotational Symmetry'. Interestingly enough, it was only after this modification was made in Japan that Sudoku really took off as a popular puzzle. If you examine Sudoku 1 you will find that the distribution of digits in the upper left box (Box A) and the lower right box (Box I) are something like upside down mirror images of each other as far as the distribution of digits in the boxes. In other words, if you took Box I out and rotated it 180* it would have the same pattern of Givens as Box A, although the Digits would be upside-down. The 2 & 7 are actually in the same relative place, but the symmetry refers to the pattern of Givens and NOT to the actual digits used! Compare, for instance, Boxes C & G. Nonetheless, this knowledge can help in solving Sudoku. Similar matches are also found in Boxes B & H and Boxes D & F. In Box E, however, the symmetry is within the Box. Notice, too, that the Symmetry applies to the Chutes as well. The middle Chutes (Middle Band and Middle Stack) have their own internal symmetry, while the extreme Chutes (Left & Right Stacks; Upper & Lower Bands) are symmetrical to each other.
It is not essential to have symmetrical distributions of Givens in Sudoku, however. In fact, originally this was not the case, and some enthusiasts think that Sudoku created without symmetry are more challenging.
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It is the number and distribution of Givens that create the levels of difficulty of Sudoku. But difficulty is related more to the distribution of Givens, rather than their number. Sudoku with only a few givens can actually be quite easy to solve, while some with many givens can be quite difficult. Sudoku are typically rated on 3-5 levels of difficulty: Easy, Medium, Hard, Very Hard, and Fiendish. The latter is also referred to as 'Devilish', 'Evil', or the like.
Taking the Givens, one begins to search for Solutions for each of the Vacant Cells. When a Solution is Discovered, it is written in the Cell and the search is continued until all Cells contain Digits. If these are all the correct Digits, then we have the Ultimate Solution to this particular Sudoku.
Another twist on the rotational symmetry issue is that each Sudoku is really four Sudoku as demonstrated in the next three Figures. Figure 2-9 is just Sudoku 1 rotated 90° counter clockwise. Figure 2-10 is just Sudoku 1 rotated 180° and Figure 2-11 is Sudoku 1 rotated 90° clockwise. If you examine the four together it would take a keen eye to tell that they are the exact same Sudoku. For all practical purposes, however, they are completely different Sudoku, and your chances of ever seeing two of the same Sudoku in different orientations is vanishingly small. Even if you did, the chances are you would never make the connection and it is unlikely that solving one form would be useful in helping to solve another. While this information does not help us solve Sudoku, it is interesting nonetheless.
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Figure 2-9: Sudoku 1 Turned 90° Counter Clockwise
Figure 2-10: Sudoku 1 Turned 180°
Note here that the Pattern of Givens is exactly the same as in Figure 2-8.
Figure 2-11: Sudoku 1 Turned 90° Clockwise
Note here that the Pattern of Givens is exactly the same as in Figure 2-8.
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