Sudoku Solving Guide by Sarif2soon.

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Sudoku Solving Guide by Sarif2soon.

Postby sarif2soon » Mon Aug 19, 2013 6:25 pm

This is how bored I can get.

"...when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth."
-Sherlock Holmes, The Blanched Soldier.

Figure 1: Before we start.
Throughout this guide, I shall use this unsolved Sudoku as an example, showing solving methods as I go. A Sudoku is made of three sizes of squares, and to get any possible confusion out of the way (e.g. medium confused with big), I shall name them beforehand:
1-The big square. This is entire Sudoku grid.
2-The medium squares. There are nine of these and they make up the big square.
3-The small squares, which can be filled by only one number.
The methods/techniques that i'll use will rely on eliminating all other possibilities, leaving the correct one.

Figure 2.1: The only place. The most basic method.
When looking for a place to add a certain number, one must know where they cannot place that number.
There are three reasons that prevent a number from being placed in a certain square:
1-The same number already exists in the same medium square, row or column.
2-A different number already exists in the square.
3-The "either way" reason which I shall explain later on.
The figure shows which places that cannot have the number nine placed in them (coloured red).
In the left medium square of the middle row, there is only one small square which allows the number nine to be placed in (coloured green). Because there has to be one of each number in a medium square, and this is the only place where we can add a nine, this is where we shall add the number nine.

Figure 2.2: "Either way".
This figure displays the "either way" which I previously mentioned.
The right medium square of the first row shows the only two places that could have a nine placed in this medium square (coloured green).
There are two possibilities:
1-The nine should be placed in the middle small square (coloured green).
2-The nine should be placed in the right small square (coloured green).
In either of these two possibilities, another nine cannot be placed in the same row (made of small squares) shared by the three medium squares. Therefore, the middle medium square in the top row (which has available small square spots in its top and midddle rows for the nine in figure 2.1) cannot have nines placed in its top row.
This also works with three small squares in the same medium square.
I shall demonstrate the "either way" 's more direct use in another figure.

Figure 2.3: Our first number.
The Sudoku grid after adding the number.

Figure 3.1: Practical use.
The figure shows the small squares in which you cannot add a one due to ones (1) already in the same square/row/column or different number taking the spot (coloured red).

Figure 3.2: Taking direct advantage.
In this figure there are two medium squares that have only two places that you can place a one in that are aligned on the same row and column: Right of middle row and middle of bottom row, respectively (coloured green).
If we cancel out the small available squares in the same row/column as a possibility (coloured white in figure 3.1), then we will have only one place in each of the top middle and middle left medium squares which we'll fill with ones.

Figure 3.3: Adding the ones.
The grid after adding the ones.

Figure 4.1: ...
This figures denotes that the following grids have had more numbers added to them after using the previous methods only.

Figure 4.2: A little more work.
Numbers added by only using the previous methods.

Figure 4.3: No one but you.
This figure excludes every other number to focus on those eight numbers.
The small square where the row of five numbers and the column of three numbers cross (coloured green) cannot have any of the other eight numbers placed in this spot. Sudoku has only nine distinct digits which you can use, we've already ruled out the other eight which leaves us with the last number that can be used that is the seven.

Figure 4.4: Lucky number.
The number seven added.

Figure 5.1: Not much.
Current grid.

Figure 5.2: 2-2-3.
The right column of the grid made of medium squares has three squares without any twos (2).
The middle one can have a two placed in any of its columns, and the top and bottom medium squares can have a two placed in only the middle and left columns. There are two possibilities:
1-A two is added in the left column of the top medium square, leaving the bottom medium square with the middle column and the middle medium square with the right column.
2-A two is added in the middle column of the top medium square, leaving the bottom medium square with the left column and the middle medium square with the right column.
So in either of these two possibilities, the middle medium square is left with the right column.

Figure 5.3: Narrowed down.
The middle left medium square is left with only its right column as a possible spot for the two.

Figure 6.1: ...

Figure 6.2: After using methods.
Current grid.

Figure 6.3: The remaining suspect.
The top left medium square has one remaining small square without a number (coloured green). That last small square could only be filled with the last number that has not been added: Number 2.

Figure 7.1: Two added.
Current grid.

Figure 7.2: The remaining suspect II.
This is the previous method, only it's applied to a line this time. Add the last remaining number to the column.

Figure 7.3: Looking good.
The grid after adding four.

Figure 8: The extreme.
This figure takes the current grid (placed on top) and breaks it down to several grids, each for a specific number. This method solves the Sudoku at an exponential rate and should only be used when the previous methods cannot help.
I would like to draw your attention to three grids: The six, seven and eight grids.
The bottom left medium square in those three grids does not have any of those numbers.
The medium squares of the seven and eight grids have two possible small squares for the seven and eight (coloured green): Top right and middle right small squares. In the six grid, however, there are three avaiable spots: Top and middle right as well aas the top middle (in blue and background colour).
There are two possibilities:
1-The seven is placed in the top right, leaving the eight with the middle right and the six with whatever remains.
2-The seven is placed in the middle right, leaving the eight with the top right and the six with whatever remains.
In either of these two possibilities, the top right and middle right squares can only be occupied by a seven and an eight, leaving all other grids (in this case, only the six grid) with whatever remains. This technique can be applied here to the middle left, centre and bottom middle medium squares.

Figure 9: The only number for it.
Although this Sudoku grid can be solved with the previous techniques only, it is very important to show this last one.
In the middle right square, there are four empty small squares: Top middle, top right, centre and
middle right.
Top middle: Can only have a two, three, six or seven.
Top right: Can only have a two, six or seven.
Centre: Can only have a three or six.
Middle right: Can only have a six.
Due to the middle right small square having only one possible number to be placed, the only number is the correct one that square.

Figure 10: Solved.
The end. Go home.

If there are any mistakes, or things that could've been written in a better way, please notify me.
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as quiet as the forest..."
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and immovable as the mountain."

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