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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. 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. 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.
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.
Therefore, this digit will be the solution for that particular cell. Note that when testing for number 1, the player could also have made a chain by going right to the cell with 1 and 4 as candidates.
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.
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.
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