Chapter 3: Basic Strategies
Blocking:
One of the most basic concepts in Sudoku is that of 'Blocking'. Simply put, when a Digit occupies a cell, it blocks that Digit from appearing throughout that Cell's Sphere of Influence (SoI). More commonly, it would be said that a Digit Blocks its appearance in the same Row, Column or Box in which it occurs. This is best seen with Figure 2-6 from Chapter 2, where the 9 in the center of the Matrix blocks all the Cells in the SoI of Cell E5. The same would hold true for any Cell and its SoI, as in Figures 2-1, 2-2 & 2-5 of Chapter 2. While this is easy to visualize when the Cells are highlighted, it is far less obvious when working an actual Sudoku. But, it is not always necessary to visualize the entire SoI when searching for solutions, as many applications require only considering the Rows (Bands) or Columns (Stacks). It is much more useful to consider Blocking when there are two of the same Digit in a Chute. But two of the same Digit (we shall refer to these generally as 'Siblings') anywhere in the Matrix could be useful in Discovering a Solution
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Discovery:
Another basic concept in Sudoku is that of Discovery. Perhaps this is what makes Sudoku psychologically thrilling. Every Solution is a Discovery. All of the strategies involved in solving Sudoku are processes designed to Discover Solutions for individual Cells and eventually to arrive at the Ultimate Solution...the completion of the Sudoku. In the system presented in this book, every (correct) Digit entered into a Vacant Cell is a Discovery! But, as we shall soon realize, there are other Discoveries to be made as well.
Intersections:
The most basic and fundamental strategy in Sudoku is what I call 'Intersections' and represents the Intersections of Rows and Columns in a particular Box, as shown in Figure 3-1.
Figure 3-1: Intersections Strategy
Notice, here, that I have not highlighted the entire SoI of the Given 8s, only their relevant Rows or Columns. Thus, if there are 8's in Rows 1 & 3 and Columns 1 & 2, then an 8 MUST go in Cell A6. This strategy is the backbone of Sudoku Solutions and you cannot solve Sudoku puzzles without it, whether you are aware of it or not. As we shall see later, it is not necessary to have the actual digit in a particular Row or Column in order to use this strategy. A quick example will illustrate the point.
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In Figure 3-2, we see that cells C4-6 are all occupied with digits 1, 3 & 5 (but it could be any three digits except 8). Since there is already an 8 in Row 1, the only place an 8 can go in Box C is in Row 3. Thus, even though we cannot determine where in C7-9 the 8 will go, we know that it must go in this Row, and our Intersection Strategy can still be used.
Figure 3-2: Phantom Numbers
I refer to these situations as 'Phantoms' or 'Phantom Digits', and they are a powerful tool in solving Sudoku using the Standard Strategy. Other authors refer to these spooky creatures as 'Ghosts', and when you begin seeing Ghosts, it can become an exhilarating experience! In the Sudoku Rev(eo)l(au)tion, we turn Phantoms into work horses. You could say that the Rational Sudoku is the Ghost-Busters of the Sudoku spook-house!
The 'Intersections Strategy' is the first strategy that you will use in solving a Sudoku puzzle. It can be used on every cell with every digit. Others refer to this strategy as 'Cross-Hatching'. But by whatever name you call it, it is an essential strategy in solving Sudoku.
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Scanning:
The term 'Scanning' is used in a variety of ways in the Sudoku World. Here we use it to indicate a systematic search of the Sudoku Matrix to Discover Solutions. The most frequent and common use is to search for a particular Digit throughout the Matrix. That is referred to as 'Scanning on a Digit'. When discussing an actual Sudoku, we would say that we performed a 1-Scan or a 6-Scan if we were Scanning on the 1 Digit or the 6 Digit, respectively. A 'Full Scan' is a complete Scan of the Matrix using all Digits.
Typically, when I begin a Scan of the Matrix, I start with a 1-Scan and progress systematically to a 9-Scan, or vice versa...starting at 9 and moving to 1. Some authors suggest that you begin by Scanning the most frequently occurring Digit in the Matrix, then progress to the least populated Digit. There is nothing wrong with this Strategy, particularly when you have mastered the process, but such hopping around can lead to oversights and omissions, so I recommend, especially for inexperienced Sudoku-ka, that they begin with a systematic complete Scan.
Once the Matrix is completely Scanned, I will repeat the process (again systematically from 1-9). If I Discover a significant number of Solutions on the second Scan, I will repeat the process again, and again, until I no longer Discover Solutions by Scanning. For the easiest Sudoku, this is the only Strategy needed to reach the Ultimate Solution. For the most Difficult Sudoku, Scanning is generally useful only for the first Cycle, and sometimes for a second. After that, you must move on to other Strategies. This is, in fact, the Benchmark for moving on to other Strategies.
All is not over, however, for Scanning. You can revisit The Scan at any time during a Solution and often with surprising results. When you have worked through more advanced strategies for a while and are making no further progress, it is always good to return to The Scan as a systematic review.
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Unique Solutions:
Whenever 8 of the 9 digits are in place in a Row, Column or Box, there is, of course, a Unique Solution (US) to the remaining Vacant Cell...the missing Digit. This is the second most basic and important strategy in solving Sudoku.
Figure 3-3: Unique Solutions
Box C contains 8 of the 9 digits. Thus the remaining Digit, 8, is the only one that satisfies this Cell. If you are a beginner, these two strategies are about all you need to know to solve easy puzzles. For the harder puzzles, some different strategies are required.
But it is not necessary to have 8 of 9 Digits in order to have a US. In Figure 3-4 we can see that there must be an 8 in C8 since 8 cannot go in C6, leaving a US in C8. This also means that there is a US in C6: C6=5. While this may seem trivial, it is a powerful tool for Solving Sudoku. And while this example is quite obvious, in a Matrix bristling with Digits, these may be hard to spot.
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Figure 3-4: Unique Solutions Variant
In fact, if all of the Digits in Row 2 of Box C were missing (1, 3 & 5), there would still be a US at C8. This is Blocking at its best. In fact, all Solutions in Sudoku should be US in some form or other...as was our 8 in Figure 3-1!!
Intercalation:
Another Basic Strategy is also a variation of the US is what I call 'Intercalation'. Others refer to it as simply 'Counting'. This is demonstrated in Figure 3-5.
Figure 3-5: Intercalation
C8=8 is a US in this case, even though there are multiple Vacant Cells in each Component of this Cells SoI. While this is easy to see when there are no other Digits in the Matrix and the SoI is highlighted, in real life, these situations can also be difficult to spot.
This is a most laborious Strategy for Discovering Solutions, but in some instances, it is all that is left. As a general strategy, it is very inefficient when compared to Scanning and other Strategies. Each Cell must be evaluated separately. Notice that only the Cells in the SoI of the Cell of Focus (C8, in Pink) directly influence it. Understanding SoI is a strong position when using Intercalation.
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Least Squares:
Least Squares has nothing to do with the statistical procedure of the same name, so there is to need to for math phobes to freak out. I like to use this term because it drives the mathematicians crazy! It simply designates a process of finding the Primary Component with the least number of Vacant Cells (Squares). The rationale is simple: it is easier to solve a Sudoku with fewer Vacancies than with more Vacancies. This becomes a powerful tool when combined with a systematic evaluation of Vacant Cells as we will encounter in Chapter 6.
The Standard Strategy:
If you examine the Literature on Sudoku you will find that there is a 'Standard Strategy', although no one calls it that. It is as if the Sudoku Gurus (Sudoku-sensai) all got together and agreed: 'Well, this is the way you solve Sudoku!' I call it the 'Strategy of Attrition', and it is described in detail in Chapter 5. It basically boils down to this:
After you have Discovered all the Solutions possible by Scanning, you take an inventory of the 'Potential Solutions' (or 'Possible Solutions,) for each Cell. You record these Potential Solutions in the Margins ('Marginal Notes') and begin a process of elimination to reduce these Potential Solutions down to a US.
The Standard Strategy comes to us, I presume, from the Japanese Gurus. It is a very Zen approach to solving Sudoku...completely experiential. It has also been suggested that the 'Possibility Matrix' approach is a mathematician's way of finding the Solution. However it became entrenched, it has led to a plethora of bizarre and obtuse 'Strategies': The X-Wing, the Y-Wing, the XY-Wing, the WXY-Wing and the WXYZ-Wing; the Sword-Fish, Jelly-Fish, Finned X-Wing, and Sashimi-Finned X-Wing; Naked Pairs, Hidden Pairs, Remote Pairs and the Death Blossom!! It begins to sound like the Ptolemaic Universe before Copernicus came along! Epicycles upon epicycles! All of these concepts become obsolete under the Rational Sudoku! (Sorry Guys!!)
The Standard Strategy works (just as the Ptolemaic Universe Works), but it is awkward and cumbersome and, in real life, very messy. In Chapter 5, I solve an 'Evil' Sudoku using the Standard Strategy and in Chapter 6 after that, I solve the same 'Evil' Sudoku using the Rational Sudoku Strategy. The Sudoku-ka can see for themselves the difference and decide which is more suited to their tastes. At any rate, this brings us to the heart of the Sudoku Rev(eo)l(au)tion.
Pairing:
Short of Discovering a Solution, Discovering a Unique Pair is thought to be the next-best-thing. The next-best-thing to a Solution is a set of Twin Pairs. Once you Discover one member of a set of Twins, the other Member is Determined also. It is a Quantum Effect: When one member is Discovered, the other Member is Discovered Simultaneously!!
Blow-Up:
Often the Sudoku-ka will be working merrily along toward the Ultimate Solution when they will Discover a second copy of a Digit in a Row Column or Box. When this happens it is like a little explosion of disappointment and sometimes frustration. We call this a Blow-Up. Sometimes fixing this is simple...sometimes it is very difficult. Repairing a Blown-Up Sudoku, however, is beyond the scope of this book. If it cannot be resolved easily, it is best to start over or move to the next Sudoku.
This concludes our discussion of Basic Strategies. The Sudoku-San may be disappointed at this point, because I have not discussed the plethora of Standard Sudoku Strategies mentioned above. But as I stated...these concepts become obsolete under the Rational Sudoku. We will now move on to Scanning.
Read the other two parts of this series: Beginner and Intermediate.