If you have come to a point where obvious clues have dried up, before moving into unknown territory and beginning bifurcation (more on that later), you should ensure that you have actually found all the numbers that you can. The first step towards achieving this is to pencil in all possible numbers in each square. It takes less time than you would think to rattle off “can 1 go”, “can 2 go”, “can 3 go” while checking for these numbers in the cell’s region, row, and column.
It never hurts to repeat the one basic tenet of the Sudoku puzzle: if something is true for one element then it has to be true for the other two associated elements. Let’s look back to something that we looked at earlier: twins. When you discovering the rule about “twins” the grid wasn’t so crowded as it is in section of the Sudoku grid below.
This time the twins are mixed with other numbers. It’s not obvious, but the two 1’s in the top region are twins. While you don’t know which cell is correct, you do know that the 1 in that region will exclude any other 1’s in column 3 right the way down to the bottom cell. Using the twins strategy eliminates two 1’s in that column of the bottom region. Two 1’s in one region helped to eliminate 1’s in another remote region.
The more numbers that you can eliminate from a region the easier it will be to determine where these eliminated numbers go on the grid. Some cells have obvious number choices and this makes it easy for you to start solving the puzzle based on scanning and placing the numbers in the right cell.
It’s important to show you this, because while nothing is actually solved by this action, eliminating those 1’s could make all the difference in proving a number. You will be looking for things to help you move on in these kinds of crowded conditions. In a tough or diabolical puzzle it might allow you to proceed through to a solution without guessing.
Now you should look for matching pairs or trios of numbers in each column, row, and region. You have seen matching numbers before: two squares in the same row, column, or region which share a pair of numbers. You can see the concept in the following illustration.
In this sample row from a grid at column 1 there is a 1 and an 8 and at column 6 there is also a 1 and an 8. This matching pair is telling you that only either 1 or 8 is definitely at one or other of these locations. If that is true then neither of these numbers can be at any other location in that row. So you can eliminate the 1 and 8 in any other cell of the row where they do not appear together.
As you can see, this immediately solves the cell at column 5. This rule can be applied to a row, column, or region. Don’t hesitate to try to use this rule on any Sudoku puzzle that you attempt to solve. It may not always work but you want to get into the habit of applying a variety of solving strategies to any puzzle that you put your pencil to.
Three Numbers Exclusively
The number sharing rule can be taken a stage further. Say that you have three cells in a row that share the numbers 3, 7, and 9, and only those numbers. They may look like 3 7, 3 9, 7 9, or 3 7, 3 9, 3 7 9, or even 3 7 9, 3 7 9, 3 7 9. In the same way as the pair example worked, you can eliminate all other occurrences of those numbers anywhere else on that row (or column or region). It will probably take a minute or so to get your head around this one, but like the pairs, where you were looking for two cells that held the same two numbers exclusively, here we are looking for three cells that contain three numbers exclusively.
Sometimes, the obvious simply needs to be stated, as in the case of two cells that contain 3 7 and 3 7 9. If the 3 and the 7 occur only in those two cells in a row, column, or region, then either the 3 or the 7 must be true in either one of the cells. So why is the 9 still in that cell with what is so obviously a matching pair? Once that 9 has been eliminated, the pair matches, and can now eliminate other 3’s and 7’s in the row, column, or region. You could say this was a “hidden” pair.
You may find such hidden pairs in rows, columns, or regions, but when you find one in a region, only when it has be converted to a true matching pair can you consider it as part of a row or column. Hidden trios work in exactly the same way, but are just more difficult to spot. Once you have assimilated the principle of two numbers sharing two cells exclusively or three numbers sharing three cells exclusively, you will be well on the way to solving the most difficult Sudoku puzzles.
Step up the Action
It’s important, if you want to successfully solve Sudoku puzzles that you take the time to attempt both easy and difficult puzzles using the concepts that you learn here. But what do you when all of the other methods have failed? In a nutshell, what you have to do is pick a likely pair of options in an unsolved cell and attempt to solve the puzzle using one of them. The method is called “bifurcation” which simply means taking a fork.
Since many books about Sudoku have been published there has been an amazing amount of discussion on the Internet about extending the more satisfying elimination methods to solve Sudoku puzzles. So much so that solvers have come up with schemes for most puzzles without resorting to guessing methods at all. At this point it should be emphasized that Sudoku puzzles that may resist the methods discussed so far represent only a very small minority of the puzzles that are presented in most magazines, newspapers, and puzzle books. These puzzles will be among the diabolical and possible some tough puzzles at the end of a book. They are valid puzzles, and many advanced Sudoku solvers have devised logic schemes (and computer programs) for solving them.
However, the wonderful thing about Sudoku puzzles is that you don’t have to be a genius or a computer programmer to solve even the most diabolical example.
If you are meticulous and patient, and have mastered the gentle and moderate puzzles, then you can solve each and every Sudoku puzzle.
Hundreds, if not thousands, of people who do Sudoku puzzles are just ordinary, intelligent people. All it takes is a bit of time to sit down and enjoy the process of solving a puzzle.
Continue to Chapter 9...
