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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. Figure 15 — Brute force will work again. Continuing in this example, since the last numeral discovered was the 7, we apply the projection method using 7 and completely fill all that can be that numeral.

This leaves only 5, 6, and 8. The strategy of alternating between visual projection and brute force, fills the rest even easier than before.

No other technique is needed. A solver can gain proficiency at this technique and puzzles with the difficulty of this example will quickly become less than challenging.

By using this alternating method, it can become almost mindless or robotic to fill Sudoku puzzles of lower caliber.

Some find this activity of solving Sudoku puzzles, without intense reasoning, comforting or relaxing. I sometimes do this myself.

It can clear my mind or I can unwind after a period of intense contemplation. Also, one can fill enormous amounts of time, oblivious to its ultimate duration.

It can be an excellent nullification or numbing waste, while suffering from things like modern air travel. But we are not here for basket weaving, Watson.

Challenges of the mind and more complex logic are our interests here. A more complicated technique that is sort of a simultaneous combination of brute force and projection is the next approach to be demonstrated.

First, we will introduce some new nomenclature. Since the basic rules of Sudoku are similar for rows, columns and sectors, the logic that applies to one, for example rows, often applies to both of the others i.

Having symmetry in the rules is a benefit. To avoid always referring to them individually when we mean all three, we give the three structures a common name.

Rows, columns and sectors are all types of a Sudoku Class or just a class. So, if a rule applies to any class, the meaning is it applies to rows, columns and sectors.

Or, there is a singular possibility. This shorthand should help. And, when a condition where only one open square exists in a class, the technique switches to brute force to find the last entry of that class.

As before we start by employing the alternation method. By using the alternation method, we only completely place the numerals 3 and 7.

Figure 16 — More difficult puzzle. With only the alternation method, the solver is stymied. Figure 17 — Brute force candidates and the map of possibilities.

Because a numeral can only appear once in a class, and there is the same number of numerals as the number of positions of a class, then all of the numerals must occur in the class.

Franken Fish Fish patterns that include box constraints. Mutant Fish Fish patterns with mixed sets of constraints.

Kraken Fish A fish pattern with indirect connections to a candidate which can be eliminated. 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.

In the image, you can see that with this strategy the cells highlighted in blue are the ones that might have a five. The cells that cannot have a five are highlighted in red.

Again, this is a highly advanced and somewhat complicated strategy to use for solving your puzzle. Start in numeric order and go up from there.

If you have 1 as a candidate in any of those areas you now know you can eliminate it as a possibility. In the image above you can see this more clearly.

The 2 has been taken and cannot be reused in that row, square, or column. For a Sudoku game you really just need 9 symbols, and numbers just happen to be the most popular version.

You could just as well use shapes and still present the same type of puzzle with the same rules and the same outcome. In fact, there are Sudoku versions where letters or symbols are used instead of numbers just to change things up.

However, Sudoku does require mathmatically logical thinking. Sudoku is solved with logical reasoning heavily based on concepts used in mathematics.

In most cases, there is only one solution for a Sudoku puzzle. However, if the puzzle is blank there are definitely multiple solutions.

Out of the advanced Sudoku strategies, the forcing chains method is usually a last resort as the chains can be very long and complicated and they do not always produce results.

The XY-Wing is a strategy to remove candidates. It can be applied when there are three cells in the grid, each with only a pair of candidates that share at least one digit among them e.

With a bit of mental effort, the player can picture a Y when connecting them, with one cell working as the stem and the remaining as the branches.

The next step is to trace lines in each row and column of the cells to form a square or rectangle. If any of the shared digits are candidates within the lines connecting the cells or at an intersection point, they can be safely removed.

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.

If any of the cells on the red paths contained one candidate shared by the cells on the extremities of the lines, it could be eliminated, but this is not the case.

However, the cell at the intersection of both branches of the Y contains a shared digit by both number 1 , allowing the player to eliminate it as a candidate to that cell.

Any Sudoku puzzle must have only one possible solution. However, at the most extreme levels, the players might find themselves with two.

This pattern happens when there 4 cells with the same pair of candidates facing each other. To avoid having multiple solutions, the player must apply the strategy of the unique rectangle.

During the game, and when facing the prospect of a deadly pattern, the player must check if one of those four cells has any other candidates to them.

If so, he or she can eliminate the problematic digits from it and leave the remaining candidates. 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.

Since one of the cells also contains 8 as a candidate, the player can use the unique rectangle method to eliminate the pair 2 and 3 from it and avoid the deadly pattern in the end.

In theory, the unique square technique should be the least used out of the advanced Sudoku strategies, even in the hardest levels, as this pattern only occurs when the puzzle is poorly designed albeit not wrong.

The Nishio strategy takes its name from professional puzzle player Tetsuya Nishio who is credited with inventing it.

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.

· Strategy. For me, the greatest appeal in solving Sudoku puzzles is the thought process used. Invariably, deductive reasoning is employed when solving a Sudoku puzzle. If this, then that – is the basic thought process. Being a big fan of Sir Arthur Conan Doyle, thinking using deductive reasoning reminds me of Sherlock Holmes. Advanced Sudoku Strategies - Fish X-Wing 2 rows vs. 2 columns Swordfish 3 rows vs. 3 columns Squirmbag 5 rows vs. 5 columns Finned Fish Fish patterns with additional candidates in a single box. Sashimi Fish Incomplete basic fish patterns with a fin. Franken Fish Fish patterns that include box .
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