How to Solve a Sudoku Puzzle?

What is a Sudoku puzzle?

Sudoku or Su Doku is a Japanese word ("Suji wa dokushin ni kagiru") meaning something similar to "Number Place" or "Number Puzzle" or "Single Number" or "the numbers must be single" or "the numbers must occur only once"; The name Sudoku is a trademark of puzzle publisher Nikoli Co Ltd in Japan. Some people spell the name as "Su Doku". Sudoku became popular in Japan in 1986, Britain in late 2004, and internationally in 2005.Sudoku on Google Trendsexplains exactly when Sudoku became popular in the online world or the world wide web. Sudoku is often described as Rubik's cube of 21st century. The speciality of the Sudoku puzzle is that the completion rules are simple, yet the reasoning required to completion can be very difficult. A Sudoku puzzle contains a 9X9 grid, which comprises nine 3X3 sub-grids (also called as boxes or blocks). Some of the entries in the grid are filled with numbers from 1 to 9, whereas other entries are left blank. Click here for an example Sudoku puzzle. A Sudoku puzzle is solved by assigning numbers from 1 to 9 to the blank entries such that every row, every column, and every 3X3 sub-grid contains each of the nine possible numbers 1 to 9. Sudoku grids are special cases of Latin squares; Latin square of order n is an nXn square containing each of digits 1...n in every row and column. The numerals in Sudoku puzzle are used for convenience, and any kind of arithmetic relationship between the numerals is irrelevant. Any set of symbols, letters or shapes can be used instead of numerals. Many publishers like ESPN, Dell Magazines, Viz magazine, use leading actors in place, images of television series, positions in baseball fields instead of numerals.

Variants

9X9 grid comprising nine 3X3 sub-grids is the most popular variant of Sudoku puzzle. However Sudoku puzzles are also available in variety of other dimentions. It is very common to find 12X12 grid Dodeka Sudoku puzzles, comprising nine 4X4 sub-grids; 4X4 grids with 2X2 sub-grids; 5X5 grids with pentomino subgrids; 6X6 grids with 2X3 sub-grids; 7X7 grids with six heptomino sub-grids; 25X25 grids with twenty five 5X5 subgrids (Giant behemoths),and 21X21 (roughly) grid Samurai Sudoku puzzles with five overlapping 9X9 sub-grids. Multitude of additional variants can be obtained by enforcing more constraints. Once common constraint is requiring the numbers in the main diagonal to be unique.

Another interesting variation is the combination of Sudoku with Kakuro on a 9X9 grid, also called as "Cross Sums Sudoku". Kakuro Sudoku puzzles come with clues which are generally cross sums. The sums might be expressed using cryptic alphabets or numbers. A good example is NUMBER+NUMBER=KAKURO, which has a unique solution 186925+186925=373850. Another good example is SUDOKU=IS*FUNNY, whose solution is 426972=34*12558.

HyperSudoku is another variation of normal Sudoku; it looks like normal Sudoku, but has additional constraints. HyperSudoku contains 9X9 grids with 13 3X3 overlapping sub-grids. In addition to the nine 3X3 sub-grids in normal Sudoku, there are four more 3X3 sub-grids overlapping with normal nine 3X3 sub-grids in normal Sudoku. The additional four 3X3 grids are highlighted in blue color Fig 1.

As mentioned before, using alphabets or symbols instead of numerals with no functional difference are also found. We are also planning to introduce a variant that is a mix of crosswords and normal sudoku puzzles. Sudoku versions of Rubik's cubes known as Sudokube or Sudoku cube or Sudocube is also available. Fig 2 is an example Sudokube. Solving Sudokube is more difficult than solving Rubik's cube. There are many variations of Sudokube, some of them when solve have the number arranged in order, while others don't. The orientation of numbers is a vital information in Solving Sudokube. It is also important that one should understand how the cubes move around and interact with other cubes in order to solve both Sudokube and Rubik's cube. I will soon write a article on solving sudokube and link it here.

Algorithm used to solve Sudoku puzzles

There are many techniques that can be used to solve Sudoku puzzles. We have attempted to explain all of them in the increasing order of difficulty below. Naked Single, Hidden Single, Block vs (Column/Row) Interactions, Block vs Block Interaction, Naked Subset, Hidden Subset, X-Wing, Swordfish, XY-Wing, XYZ-Wing, Coloring, Remote Pairs, XY Chains, Forcing Chains, Nishio, and Trial and Error are the list of well known techniques used to solve Sudoku puzzles. The difficulty level of a Sudoku puzzle depends on total number of filled in cells, number of iterations of solving, and algorithm used to solve the puzzle. Our online web interface uses most of the above mentioned alogrithms for solving Sudoku puzzles.

Naked Single (aka - Sole Candidate)

Naked Single also known as sole candidate is the most simplest technique used. Naked Single is the empty cell that can take only one value, when the contents in block, row, and column containing the cell are analyzed. For example, in the partial Sudoku puzzle below; the cell at row four and column five can only take number 7. All other numbers have already occurred in the same row or column or block one ore more times.

	6       8
1	?             4    3	2
	9       5

Hidden Single (aka - Unique Candidate)

Hidden Single also known as Unique Candidate is another simple technique often used. If a cell is the only one in a row or column or block, that is free to take a particular value, then it has to take that value. This is because all numbers from 1 to 9 must occur in each row, column and block. For example, consider the partial Sudoku puzzle below; the cell at row six and column five has to take number 4 because number 4 cannot occur in any other cell in that block. In this example we consider blocks, however you can easily find similar situations with respect to rows and columns.

	4
4	5       ?
	4

Naked Single and Hidden Single are the only techniques other than Forcing chains, which can fix a single number to a cell. All other techniques described below can only associate or eliminate a set of numbers to or from a cell or group of cells, thus enabling us to use Naked Single or Hidden Single.

Block vs Row/Column Interactions

As mentioned before, this technique will eliminate numbers from a group of cells. If a particular number can only occur in a particular row or column inside a block, then it cannot occur in any other cells in the same row or column outside the block.For example, consider the partial Sudoku puzzle below. Number 3 has to occur atleast once inside block 2, and it has to either come in cells marked as A or B. This eliminates number 3 from all other cells in the row containing cells A and B, outside of block 2. So we can conclude that number 3 cannot come in cells marked in black colour.

3	1 a 7          5 b 9
	x       x       x
	x       x       x

Here are some free online web sudoku puzzles than can be solved using Block vs Row/Column Interaction strategy. Click on the link to load the example puzzles in the free online sudoku puzzle solver.

Block vs Row/Col Interaction Free Online Sudoku Puzzle 1
Block vs Row/Col Interaction Free Online Sudoku Puzzle 2
Block vs Row/Col Interaction Free Online Sudoku Puzzle 3
Block vs Row/Col Interaction Free Online Sudoku Puzzle 4
Block vs Row/Col Interaction Free Online Sudoku Puzzle 5

Block vs Block Interactions

Consider a particular number as a candidate for two different blocks; if the particular number can only appear in two different rows or columns in these two blocks, then the same number cannot appear in the same row or column outside of these two blocks. Consider the partial Sudoku puzzle below; number 3 can only appear in the places marked as *, in block 4 and block 6. This eliminated 3 as a candidate from the cells marked in black.

3
* *    * *    1 2	x x x x x x	*    * *    * 9    8
		3

Let us consider another general example. In the partial Sudoku puzzle below, numbers 3,6 and 7 can only occur in places marked as *. This eliminates 3,6, and 7 as a candidate from the cells marked as x. We can arrive at the same conclusion just by looking at row 6. There are only 3 empty cells in row 6 and numbers 3,6 and 7 must appear in them. Since all these three empty cells happen to be in the same block, the cannot appear in other cells of that block.

* * * * * * 4 1 2	x x x x x x	* * * * * * 9 5 8

Here are some free online web sudoku puzzles than can be solved using Block vs Block Interaction strategy. Click on the link to load the example puzzles in the free online sudoku puzzle solver.

Block vs Block Interaction Free Online Sudoku Puzzle 1
Block vs Block Interaction Free Online Sudoku Puzzle 2
Block vs Block Interaction Free Online Sudoku Puzzle 3
Block vs Block Interaction Free Online Sudoku Puzzle 4

Naked Subset (aka - Naked Pair)

If the same two numbers appear as candidates for two cells in a same row or block or column and if those cells can't take any other values other than these two number, then we can elimiate these two numbers from other cells in the same row or column or block of that sudoku puzzle. The same logic when applied to three numbers is called Naked Triplet; four numbers is called Naked Quad. Let us look at an example; in the partial Sudoku puzzle below, cells marked as 'a' and 'b' can only take numbers 2 and 3; they can't take any other value other than 2 and 3. Thus we can elimiate 2 and 3 as a candidate from other cells in the same row (marked as x).

5       6
1 a x 7    8	x 9 4	7       b x x
		8       6       5

Let us consider another example of the same type with three numbers. In the partial Sudoku puzzle below, the cells marked as 'a', 'b', and 'c' can only take numbers 2,3 and 4; they can't take any other value other than 2, 3 and 4. Thus we can eliminate 2, 3 and 4 as a candidate from other cells in the same row (marked as x).

5       6		6       5       8
1 a x 7    8	x 9 x	7       b x c
		8       6       5

The same algorith can be extended to four numbers, and even 5 or 6 plus numbers in bigger sudoku grids (eg 12X12 grids). The rules to be kept in mind are 1)the number of cells housing the naked sets must be same as number of candidates in the naked set. 2)No cell can take any number other than those in naked set as a potential candidate and all cells don't have to take all numbers in the naked set as potential candidate. By naked set I mean 2 and 3 in the first example and 2,3, and 4 in the second example.

Here are some free online web sudoku puzzles than can be solved using Naked Subset strategy. Click on the link to load the example puzzles in the free online sudoku puzzle solver.

Naked Subset Strategy Free Online Sudoku Puzzle 1
Naked Subset Strategy Free Online Sudoku Puzzle 2
Naked Subset Strategy Free Online Sudoku Puzzle 3

Hidden Subset (aka - Hidden Pair, Unique Pair)

Hidden Subset is similar to Naked Subset. If the same two numbers appear as candidates for two cells in the same row or block or column and if they don't appear as candidates for any other cells in the same row or column or block, then these two cells cannot take any value other than these two numbers. Thus we can eliminate all candidates other than these two numbers from these cells. Let me explain it with better with an example. In the partial Sudoku puzzle below candidates 2 and 3 can appear only those cells marked as 'a' and 'b'. Thus we can remove remove all candidates other than 2 and 3 from cells a and b.

2		3
3 a	2                      3	b             2
2		3

The same logic can be applied to columns and blocks. In the above example cells a and b could accomodate only two numbers; we can extend the same and say if three different numbers can only occur in three cells of a row or column or block, then other candidates from those cells can be elimiated.

Here are some free online web sudoku puzzles than can be solved using Hidden Subset strategy. Click on the link to load the example puzzles in the free online sudoku puzzle solver.

Hidden Subset Strategy Free Online Sudoku Puzzle 1
Hidden Subset Strategy Free Online Sudoku Puzzle 2
Hidden Subset Strategy Free Online Sudoku Puzzle 3
Hidden Subset Strategy Free Online Sudoku Puzzle 4
Hidden Subset Strategy Free Online Sudoku Puzzle 5

X-Wing

Consider a puzzle where a particular candidate (say Can1) can only occur in two columns (say Col1, Col2) in a row, and if there are two such rows then that particular number can be elimiated as a candidate from other cells in Col1 and Col2. Let me expalin it better with an example. In the partial Sudoku puzzle below candidate 4 can only occur in cells marked as * (Column 2 and Column 4) in row 3 and row 6.

7 *	* 8	4
3 *	* 2	4
4	4

Thus candidate 4 can be eliminated from other cells of column 2 and column 4.

Here are some free online web sudoku puzzles than can be solved using X-Wing strategy. Click on the link to load the example puzzles in the free online sudoku puzzle solver.

X-Wing Strategy Free Online Sudoku Puzzle 1
X-Wing Strategy Free Online Sudoku Puzzle 2
X-Wing Strategy Free Online Sudoku Puzzle 3
X-Wing Strategy Free Online Sudoku Puzzle 4

Swordfish

Swordfish is similar to X-Wing, actually an extension of X-Wing to three rows or columns. If a particular number can only occur in two cells in a column, and if there are three such columns and also if those cells fall in the exactly three common rows, then we can eliminate that number from all other cells in those three rows. Let me explain it better with an example. Consider the unsolved Sudoku puzzle below.

*       1	3 *    4 5	9
*       6	* 6    5 4	9
5       4	* *    6 7	9

Number 9 can only occur in cells marked as * in column 1, 5 and 6. The cells marked as * fall in three common rows namely row 2, 5, and 8. Thus we can eliminated number 9 from all other cells in row 2, 5 and 8. It is not neccessary that the given candidate has to exactly come only in two rows in column. It can occur in three rows as well.

Swordfish Strategy Free Online Sudoku Puzzle 1
Swordfish Strategy Free Online Sudoku Puzzle 2
Swordfish Strategy Free Online Sudoku Puzzle 3
Swordfish Strategy Free Online Sudoku Puzzle 4
Swordfish Strategy Free Online Sudoku Puzzle 5
Swordfish Strategy Free Online Sudoku Puzzle 6
Swordfish Strategy Free Online Sudoku Puzzle 7

XY-Wing

This is like a mix of both X-Wing and Swordfish. Let me explain it with a general case. Consider the partial Sudoku puzzle below.

9 1 1 1 1 1 1 1 1	1 2 1 1 1 1 1 1 1	1 1 3 1 1 1 1 1 1
9 1 1 1 1 1 1 1 1	1 2 1 1 1 1 1 1 1	1 1 3 1 1 1 1 1 1
9 1 1 1 1 1 1 1 1	1 2 1 1 1 1 1 1 1	1 1 3 1 1 1 1 1 1

Generating Sudoku puzzles

Total number of Sudoku grids

Mathematics of Sudoku puzzles

History of Sudoku puzzles

Free Sudoku solvers for download