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Not all sizes of fish were discovered at the same time. Size 2 was already familiar to many players before they realized that the same trick could also be performed with more than 2 rows and columns.
Here is a list of names given to fish of different sizes. If you know the naked pair has either a two or a six as the possible answer, check to see where those numbers are used already.
Naked pairs do not have to align within the grid. They can be naked pairs and be scattered within the square too. No matter where they fall the point of the strategy is that you know there are only 2 possible numbers that can be placed in those cells and that you need to use the process of elimination to find the right one.
This same idea can also be applied to what more advanced players will call naked triplets or threes and naked quads. Example of how hidden pairs can distort the true options for a cell.
This is a great way to open up your grid and get a good feel for where to place numbers. The hidden pair strategy is a way to eliminate clusters of numbers from two cells which leaves you with simple options for the rest of the cells.
Then look through your square, columns, and rows to rule out those numbers as options. In the above example, you can see that the hidden pair appears to have a multitude of options.
But if we apply the rule of looking through columns and rows we can see that the actual value of those cells is limited to being either a 6 or 7.
Again, this strategy can be used in triplets or quads but that could take more practice. If the numbers are aligned in the same column or row they are called a pointing pair.
The pointing pair tells you the number must be used in that line and can be ruled out from other possible cells.
This is another strategy to help eliminate possibilities and make the entire puzzle more easily solved. Are you noticing a theme with these strategies?
Intersection removal is no exception to that line of thinking. If any number occurs as a possibility two or three times in any one unit row, square, or column you can then remove that number from any intersecting other units.
The key to using this strategy is to really fully understand what a unit is in the game. If the pair or triplet of numbers intersects with another row, column or square it can be eliminated as a possibility for that intersecting unit.
Another way to methodically use the process of elimination to get to the final result. This one just takes a little more focus on the entire grid than previous strategies mentioned.
Take a look at your rows and see if there are any pencil marks that are exactly the same in two spots. Match up that row with another row that mirrors it.
The pencil marks must be exactly the same in the same two spots. You can see an example below to get a better idea. As you can see, the parallel rows create an X giving this strategy its name.
Looking at the example above you can now see that each of these rows has to have a 4 in it. You also know you can only place the 4 in either the slots that are dark blue, or light blue, since doing anything else would cause a repeated number in the row or column.
This will guide you to the right choice to erase and the right cell to place the X formation numeral. This is a strategy that takes a lot of thought but it does work very well.
It will help you develop the skills required to move beyond focusing on just one square or one row or column.
It helps you see the bigger picture. Remember how the X wing involved 2 possible numbers in two rows?
The blue lines show you the slots where a 5 matches up and crosses the blue line itself. In short, the blue lines are showing you where you have the possibility of placing a five.
If there is not a somewhere that a blue and red line intersect, you can eliminate five as a candidate in that cell. The blue lines will not tell you WHICH cell the five goes in, it just shows you what to eliminate.
Brute force will no longer work. Figure 7 — The example puzzle grid and map of possibilities after four entries. Clearly the brute force method is lacking.
It is tedious and frustrating. It may be the first strategy learned, but should not be the first strategy used when starting a new puzzle.
This is at the other extreme of the tedium spectrum. Finding the one missing numeral in a row, column or sector is easy.
For example, if there is only one open square in a row, the solver can look across the row for a existing numeral 1 in the filled positions.
If found, the solver looks for a 2. If found, the solve looks for 3 and so on. If a numeral cannot be found in the row, then is must be the numeral in the last open square of the row.
Another time to use brute force is when at an impasse stymied. It is sometimes useful to try brute force to overcome the impasse and continue solving the puzzle.
If there are only two or three open squares in a row, column or sector, it may work and the solution may progress again. Also at impasse, it sometimes can be visually detected that a particular location may be a candidate for a brute force analysis.
As the solver becomes acquainted with more complex techniques, the solver is less likely to be willing to endure the tedium, and will consider brute force at an impasse only as a last resort.
And unfortunately, It seldom succeeds. Here is the method I use when starting a new puzzle. By projecting particular rows and columns, that contain the same numeral, into a target sector, a single position in the target sector, where that numeral can occur, can sometimes be visually identified.
It is a mental construction and is best explained visually. Figure 8 — Projecting rows that contain a 2 to the right.
The 2 is entered to the grid. So next, I try projecting two columns down into the lower right target sector see Figure 9. Unfortunately, there are two locations where a numeral 2 can occur two shaded green.
I make a mental note of that. If I find another 2 in one of these two rows later, I can determine which of these two locations has the 2 in it, just from memory.
Figure 9 — Projecting columns that contain a 2 down. Figure 10 — Projection quickly finds two more locations. Projecting the last location found with another column down , finds another solution digit in the middle bottom sector see Figure The row that it is in, is a row where I made the mental note from before.
Thus, I have demonstrated the wisdom to continue projecting with numerals that have been found recently. The time to remember things is shortened.
Figure 11 — The mental note pays off. At this point in solving the puzzle, we have found the numeral 2 in eight of the nine sectors.
The projection method will always find the remaining location in the last sector see Figure Figure 12 - Finding the ninth of a numeral in the last sector.
There is always only one square. For this puzzle, I never consider a 2 again. Arbitrarily, I use numeral 4 in the projection technique next.
In this example, the 4 was nearby in the last target sector, and very little thought is given to selecting the numeral to project. As an exercise, the reader is encouraged to try the projection method with these four numerals.
I tend to be a visual thinker. Many people have that tendency. After a little practice, this visual technique becomes very quick and does not require a lot of brain power.
Note that in this example after completely filling several numerals, there exists a sector and a row shaded yellow that have only one open square see Figure It is extremely easy to apply brute force to fill these two open squares shaded green.
Figure 14 — Only one open square in a sector and one in a row. Out of all the advanced Sudoku strategies, this is the one that players avoid the most and use only as a last resource as it takes guessing as a premise.
Basically, the player must take a cell with two candidates, choose one, and try to solve the grid.
If it works, great. If not, the player must return to the starting point and choose the other. The point of this strategy is that as soon as an incompatibility arises, the starting candidate can be eliminated which turns the remaining into the solution for that cell.
The theoretical approach of advanced Sudoku strategies is relatively easy to understand and to apply. The biggest difficulty is to find the right patterns on the grid and to know which technique to use in each case.
Nevertheless, only experts reach the levels in which they are required and at that point knowing and being familiar with these strategies becomes a requirement to progress in the hardest levels of these puzzles.
The X-Wing The X-Wing method is one of the most basic advanced Sudoku strategies. Example In this example, the number 5 forms the necessary pattern to apply the X-Wing strategy.
The Swordfish This strategy helps to eliminate a candidate from cells too. Example In this grid, number 4 is a candidate to two cells in three different rows, allowing the player to use the Swordfish technique.
Forcing Chains Forcing chains is one of the easiest advanced Sudoku strategies to understand. Example In this example, the top highlighted cell with the candidates 1 and 2 was used to apply the forcing chains technique.
The XY-Wing The XY-Wing is a strategy to remove candidates. Example In the example above, the stem cell contains the digits 2 and 9 highlighted in orange and connects to the branches, each with one of these digits as candidates purple squares.
Unique Rectangle Type 1 Any Sudoku puzzle must have only one possible solution. Example In this table, the player faces a possible deadly pattern with the candidates 2 and 3, in which the placement of those digits can become indifferent in the end and result in two possible solutions for the puzzle.
Nishio The Nishio strategy takes its name from professional puzzle player Tetsuya Nishio who is credited with inventing it. Sudoku Genius.
Classic Numbers Puzzle. Sudoku Unique Rectangle: Types and Patterns.