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By connecting the cells containing it, it is easy to see that in this situation only the pairs in green or yellow could be possible. Testing the pairs on the grid shows that the 5 highlighted in red would be impossible in both cases.

It can then be eliminated as a candidate to that cell. This strategy helps to eliminate a candidate from cells too.

To apply it, the player must find a digit candidate to two cells in the same row, in three different rows. These cells must also be united by column, regardless of the shape they create.

Connecting them will result in a closed chain that reveals two sets of possible placements for that digit.

The player can then put them to the test and eliminate that number as a candidate to the cells where it would become impossible in both cases.

In this grid, number 4 is a candidate to two cells in three different rows, allowing the player to use the Swordfish technique.

Connecting them reveals that whenever a digit highlighted in green is possible, the yellows become impossible and vice-versa.

Thus, there are only two sets of possibilities within this chain for number 4, the green and the yellow sets. When testing them on the grid, the player finds that the 4 highlighted in red would be an impossible candidate in both cases.

It can then be safely removed from that cell, leaving only numbers 1 and 2 as candidates to it. Forcing chains is one of the easiest advanced Sudoku strategies to understand.

Unfortunately, applying it requires very high levels of concentration and the outcome can be uncertain. When it does work, it gives the player the solution for one cell.

To apply a simple forcing chain, there must be cells with only two candidates in the grid. The player picks one and begins testing the changes to puzzle when applying each of the two digits in that cell.

The goal is to find if there is a cell that would bear the same result whichever digit is used. If so, that will safely be the solution for it.

In this example, the top highlighted cell with the candidates 1 and 2 was used to apply the forcing chains technique.

When testing for both digits, the player finds that the outcome for the highlighted cell with candidates 5 and 7 is always 5.

Aliases are seafood and sealife. Advanced Sudoku Strategies - Fish X-Wing 2 rows vs. Sashimi Fish Incomplete basic fish patterns with a fin. Being a big fan of Sir Arthur Conan Doyle, thinking using deductive reasoning reminds me of Sherlock Holmes.

Often, Sherlock would engage his friend and colleague, Dr. Sherlock would explain his analysis indicating that deduction was a learned thought process.

Similarly, Sudoku appeals to many as an exercise to maintain and improve the mind. Also like Sherlock, one particular type of deductive reasoning is used, that being the process of elimination.

Actually in Sudoku, a numeral a single digit number is never improbable. Therefore, Sudoku is even more elementary than a Holmes mystery.

At each open grid position of a Sudoku puzzle, a solver can mentally eliminate numerals by applying the basic rules of Sudoku.

If, by the process of elimination, there is only one possible numeral, then it must be the entry that will solve the puzzle.

By brute force a solver can, one at a time, analyze each open square. If it has one possible numeral, the solver enters that numeral and then goes on to examine another open square.

If there is more than one numeral possible and all but one cannot be eliminated, the solver skips that square and tries another. However, at the start of a difficult puzzle, few open squares will have a single possibility.

In fact, many puzzles will have only one or two at the start. Sometime at the start of a puzzle, using just the basic rules, there are no squares with a single possibility.

In that case, more complex logic is required just to begin. Also, even when there exists a square at the beginning, where only one digit is possible, often as the grid is filled, new locations with only one possibility cease to occur and open squares with only one possibility run out.

Again, to continue, only more complex logic can work. Using brute force becomes very tedious when there are many open squares and few having a single possibility.

Brute force has its drawbacks for sure. However, as the solution puzzle progresses and more empty squares are filled, it may become necessary and preferable to employ a brute force method.

In fact, in some situations brute force becomes easy. So, knowing how to determine a numeral by brute force is required.

The pertaining squares are shaded in yellow see Figure 1. The numerals 1, 2, 5, and 9 occur in the pertaining row, column and sector yellow squares.

This leaves 3, 4, 6, 7, and 8, or five possibilities for the upper left hand square. We cannot eliminate all but one and cannot determine the proper numeral from this analysis.

Figure 1 — Brute force analysis of the upper left hand square. The next square to the right is not much better see Figure 2.

The numerals 1, 2, 3, 5, 6, and 9 occur in the pertaining area, leaving 4, 7, and 9, or three possibilities. Figure 2 — Brute force analysis of the next square.

To illustrate the situation, a map of the number of possibilities at each position, shows where a square that has a single possibility may exist. By counting the possible numerals at each grid location and placing that count in the corresponding location, a map of possibilities is constructed.

Figure 3 — The map of possibilities of the example puzzle. It can be seen using this map, by applying the brute force method systematically, the solver would analyze 18 squares, before the first entry is found.

The solver could start at the upper left and examine each square, one at a time, across and down, applying the process of elimination see Figure 4.

Figure 4 — Applying brute force to each open square systematically. Great, we have our first entry at last. Figure 5 — Brute force analysis to determine the first entry.

Figure 6 — The map of possibilities with a new location where only one numeral is possible. Each new entry reduces the overall possibilities.

Each entry can lead to the discovery of other entries by changing what is possible. Each deduction leads to the next. Like in a mystery, one clue leads to another.

A solution is a sequence of deductive steps. If the solver applies the brute force method repeatedly in this example, the solver is stymied after only four entries.

The map of possibilities shows no squares where only one numeral is possible see Figure 7. 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. In the most extreme case where a puzzle has zero prefilled number clues, Wikipedia states there could be up to 6,,,,,,, solutions.

So, while in general every puzzle only has one solution, if you did want to do a completely blank Sudoku puzzle, yes, there are definitely many solution options.

While there seems to be some disagreement on this within the Sudoku fan community, most agree that a true Sudoku puzzle is never unsolvable.

They state that a real Sudoku puzzle has one and only one solution. If you work on a puzzle that is unsolvable, it is either an error in printing, or not a true Sudoku puzzle.

It is another type of game based on a Sudoku type of grid. There are websites and publications dedicated to purposely creating unsolvable puzzles, but those are by design.

Your best bet is to erase everything and start with a clean slate. If you really wanted to challenge yourself you could start by erasing rows or columns with repeated numbers, but that could lead to more frustration.

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