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SE121 Some interesting Sudokus
In this page, you will find some interesting Sudokus. "Interesting" in the sense they require rare (and usually difficult) solving techniques. The Sudokus are given in a "computer-friendly" form, which can be cut and pasted in most existing Sudoku solvers, such as the Sudoku Explainer. The Sudokus are classified by the hardest solving techniques required to solve them.
This page is not an explanation of the solving techniques, but rather a collection of Sudokus intended to explore various solving techniques. Because solving techniques are not explained there, it is advised, whenever you get stuck, to solve these Sudokus with a computer solver that explains each solving step, such as the Sudoku Explainer.
About the solving techniques
Hidden Single
Direct Intersections
Direct Hidden Pairs
Naked Single
Intersections
Naked Pair and Naked Triplet
Hidden Pair and Hidden Triplet
X-Wing and Swordfish
XY-Wing and XYZ-Wing
Naked Quad
Jellyfish
Hidden Quad
Unique Rectangles and Loops
Bivalue Universal Grave
Aligned Pair Exclusion
Forcing Chains
Beyond Forcing Chains
Others
Sudoku Explainer - a step-by-step Sudoku solver
Sudoku Explainer FAQ
Equivalence between Naked Sets, Hidden Sets and Fishy patterns
Sudoku on Wikipedia - The free encyclopedia
Sudoku Susser - another Sudoku solver
Sudoku Assistant/Solver - another Sudoku solver (in Javascript)
Sudoku Players' Forums
Sudoku Programmers' Forums
How many Sudoku solving techniques are there? five? six? no! A more correct answer would be more than 20. And nearly no single technique can solve all Sudokus, meaning that they are all potentially useful for some Sudoku. The bad news is that 10% of these solving techniques already allow you to solve 90% of all Sudokus one can find in newspapers and books. In this collection, you will find various Sudokus requiring the remaining 90% of the solving techniques, that is, the really interesting ones.
Apart from the hidden single and naked single solving techniques (the easiest ones), all the other solving techniques can usually only be applied when all possible candidates of every empty cell have been written out in small digits, as with the Sudoku Explainer. (I say "usually" because some techniques can in some simple cases be applied without writing candidates). The reason is that these techniques usually do not allow you to place a value in a cell, but just to exclude some values from one or more cells that are still empty.
The following Sudokus are the two I found that require the largest number of different solving techniques (except forcing chains). They are a good start to get a quick overview of the state of the art in Sudoku solving techniques. The Sudoku Explainer can be used to get a step-by-step explanation of the solving steps. Note that these two Sudokus are difficult Sudokus (more difficult than most "diabolical" Sudokus one can find in newspapers), but still far from the hardest known Sudokus.
..1.7.6..
92...4...
.....3...
..9.....1
37..8.9..
.5....36.
..5.1...6
.....8.5.
...5.7.4.
|
..8....1.
27.84..6.
..6...4.8
...4.6.7.
.8..2..5.
.4.1.3...
7.4...6..
.6..35.42
.2....9..
|
The Sudokus of this collection are classified by the hardest solving technique that is required to solve them. Please note that this classification depends on how the solving techniques are rated. This collection is based on the difficulty ratings of the Sudoku Explainer. When used on other solvers with different ratings, the hardest solving technique that is required may not be the same. Also note that only "logical" techniques, that is, techniques that can be explained and understood by a human player, are covered. Any computer can quickly solve any Sudoku using backtracking, but this technique is not discussed there.
Hidden Single is the name of the simplest solving technique. It can be applied within a 3x3 block, a row or a column.
It is a common mistake to think that Sudokus with fewer clues are harder to solve. The following Sudoku has only 17 clues (the minimum possible), but can be solved using the hidden single method only. It doesn't even require hidden singles in row or column (except for finishing a row or column). Also note that it has only 8 different given values (the value 9 does not appear).
......3.1 86....... ...2..... ....4.76. ..1...... .......8. .7....64. 5..1.3... ...5.....
Of course, there is globally a small tendency for Sudokus with less clues to be more difficult (the above Sudoku for instance is still less easy than most easy Sudokus requiring the Hidden Single technique only). But this tendency is very weak, and rating the difficulty of a Sudoku using the number of clues is usually considered as one of the worst rating schemes. For another, complementary example, see Forcing Chains at the end of this page: a very difficult Sudoku with 35 clues.
The following Sudokus can be solved with the Hidden Single technique only, even if they might seem strange or complicated.
....54683
.....9..1
......2.9
.......78
8.......5
69.......
5.6......
2..8.....
34756....
|
967241583
2.......6
4.......9
8.......5
795832164
1.......2
6.......7
5.......1
321756498
|
.........
4........
2.574391.
5.8....6.
1.9.78.3.
3.2..6.7.
9.3285.4.
8......9.
72649158.
|
.........
..39.67..
.8.257.9.
.56...13.
..9...2..
.12...67.
.2.498.6.
..47.13..
.........
|
476......
1.8..756.
523..8.7.
.....978.
.615.....
.4.1.....
.358..426
......9.3
......857
|
2.9.3.8.4
.4.8.2.1.
6.8.9.3.2
.9.3.8.6.
8.4.7.2.5
.2.4.5.9.
9.5.8.1.6
.7.9.6.8.
1.6.4.9.7
|
...4985..
..3....2.
.1..32..9
.5.3..6.8
.3....2.7
..2..4..3
2..58..7.
.6....8..
..8213...
|
Direct Pointing and Direct Claiming are special cases of the more general techniques Pointing and Claiming in which it is not necessary to write down the candidates in the empty cells. In these techniques, the possible positions of a value in a region exclude some possible positions in another region and reveal a Hidden Single in that region. The following Sudokus can be solved using Direct Pointing:
.1.34..8.
....6.7.1
.6.8..24.
..8.3....
1....4.72
4..7..1.8
..6..8...
......6.9
2..45....
|
.8.9.....
91......7
3.4...8..
...5.49..
5..1.8..3
..93.2...
..7...6.4
4......31
.....3.5.
|
....4....
2.7...1.5
85.....92
..5.6.8..
...378...
..8.1.2..
46.....57
1.3...4.6
....2....
|
..78912..
...4.6...
4...7...8
38.....47
5.4...8.1
71.....53
9...4...2
...1.9...
..53284..
|
6.238....
..9......
..47....9
..35.6..8
.5.....2.
8..9.74..
9....42..
......5..
....938.4
|
It seems that Direct Claimings do not occur in natural ways.
Direct Hidden Pair is a special case of the more general Hidden Pair technique. In a Direct Hidden Pair, the Hidden Pair immediately reveals a Hidden Single in the same region. Direct Hidden Pairs, unlike Hidden Pairs in general, can be found without writing down the candidates in the empty cells.
41.....52
3.9...1.6
.6254139.
.........
.9.3.4.2.
5...7...9
6.......4
.7.....6.
..59.38..
|
.158.967.
2...4...8
8.......1
5.......6
.8.....9.
4.......5
9.......2
1...5...3
.423.185.
|
....4....
..67.91..
..23.64..
2.......1
.379.864.
9.......5
..48.37..
..95.43..
....7....
|
....27.8.
8..9.4.2.
....8...6
.......15
45.....32
68.......
5...4....
.7.1.2..3
.2.39....
|
.3..58.9.
..1.97.4.
6........
.........
.83...4..
.5...6.7.
...4....1
...375..6
.......8.
|
The following Sudokus require the naked single solving technique to be solved, the hidden single alone and the other techniques presented above are not enough.
169......
4.3......
752......
...1..947
......8.3
......612
...896134
...4.25.9
...753268
|
.........
.219..7..
.....485.
.172..9..
.........
..3..864.
.457.....
..9..513.
.........
|
..8...3..
..35...6.
47...38.2
.2..4.1..
...9.7...
..4.6..3.
9.21...83
.6...85..
..7...6..
|
...8..1..
8....345.
3...7...9
.....47..
9..561..4
..13.....
1...3...6
.564....2
..9..6...
|
Intersections are the easiest "indirect" solving techniques. "Indirect" means that these techniques (usually) do not allow you to place a value in a cell. They only allow you to exclude a possible value from one or more empty cells. Usually, a hidden single or naked single shows up after some values have been excluded.
The following Sudokus require various kinds of intersections (Pointing or Claiming) multiple times in order to be solved:
.29.8....
4......3.
7...4.69.
......4.6
5.8.2....
......2.1
..35.47..
.65......
...1.9...
|
..7.1.4..
.4.7.6.3.
3...2...9
.7.....2.
5.2...9.1
.3.....6.
6...8...2
.8.2.5.4.
..5.3.6..
|
......942
..9.6..7.
.....5.6.
.2..16.87
..8.9.4..
57.28..9.
.8.6.....
.9..5.2..
236......
|
..3...6..
.4.136.9.
5.......3
.6.3.2.8.
.2..6..4.
.8.4.7.3.
6.......9
.5.619.2.
..9...8..
|
The naked single technique can be extended to more than one cell. In that case, it does not directly allow the placement of a value in a cell, but only to remove potential candidates from one or more empty cells. Unlike naked singles, naked pairs and naked triplets have three variants: in row, column or block. The following Sudokus require naked pairs and naked triplets in order to be solved:
.1..5..7.
.2.....3.
..86.79..
..51.32..
....6....
..34.58..
..95.43..
.8.....1.
.5..8..2.
|
....7.6.1
...8.37..
.......28
.9.2.4.3.
3.......6
6..3.9.5.
56.......
..9..2...
..175....
|
.....5.96
...8..7.3
....6.25.
.6..1...7
..39.24..
4...3..2.
.17.8....
5.8..1...
64.2.....
|
3..2.....
...8.3...
..9...7..
.6.71.9.4
.7.....6.
......51.
21..4.6..
..4...1..
8..9.....
|
......3..
4.7..5...
9..17...6
....9..4.
5..3.42..
...78.1..
.3.....2.
..4......
8...1.96.
|
Hidden pairs and hidden triplets are the extensions of the hidden single technique to 2 and 3 cells, like naked pairs and naked triplets for the naked single technique. The following sudokus require hidden pairs or hidden triplets:
..68.91..
.1.....9.
4...2...8
8...5...3
..16.27..
9...4...5
6...1...7
.7.....2.
..37.54..
|
...3...56
95.1....8
........7
.8..62.75
.........
63.58..1.
2........
7....5.63
36...4...
|
9..82...7
.......1.
.673....4
37...41..
....1....
4....3.79
..42..59.
.9.......
..5.492..
|
....6.8.7
..7..3.4.
.4...2.5.
........2
4.5.2...8
27..1..9.
..3......
6.......1
7...592..
|
...1.7...
5....394.
......8..
.....4...
8..57....
.63....1.
..241..8.
.........
.169....2
|
These two techniques only involve rows and columns, and a single value at a time.
9...8...5
..1...6..
...9.2...
8.96.37.4
3..824..9
5.41.92.3
...5.7...
..5...9..
6...3...7
|
.7.....4.
5...1...2
.42...96.
72..5..16
...2.3...
49..7..53
.57...62.
2...9...8
.6.....9.
|
..26.71..
....8....
6..1.3..4
7.39.12.6
.1.....9.
9.67.84.3
1..2.9..7
....7....
..95.63..
|
.....9...
.3..52.9.
..51362..
568...1..
.29...84.
..3...569
..65239..
.9.84..5.
...6.....
|
..71.63..
...5.8...
2...3...1
71.....36
..8...1..
53.....24
6...5...2
...8.1...
..53.26..
|
XY-Wings and XYZ-Wings occur quite frequently, although the latter is less frequent than the former. The following Sudokus requires various XY-Wings and XYZ-Wings.
7..345..8
.4.....1.
..91.73..
25.....93
..6.9.5..
19.....87
..18.42..
.2.....3.
9..532..6
|
54..6..23
9.3...4.7
......6..
....37...
3..2.6..1
...48....
..2......
7.5...3.2
61..2..95
|
...3.54.9
......25.
...14..76
6.5.1...4
..86.45..
1...5.6.8
49..71...
.51......
7.65.9...
|
......67.
.59..7..2
.....2.35
.751..3..
..15...4.
.389..7..
.....6.93
.93..1..4
......58.
|
31.......
..6.795.1
9....6.47
4.86.7...
........3
.....4..2
.5.....3.
.9......4
..3.1.6..
|
The naked quad is the extension of the naked single, naked pair and naked triplet to four cells. Note that, while it is theoretically possible to extend the technique to five cells or more, this is useless in practice: with 9 empty cells in a row, column or block, a naked set of 5 cells always occur with a hidden quad (4 cells). The following Sudokus require the Naked Quad solving technique to be solved:
.........
.1769425.
.2.1.8.7.
.35...62.
.9.....1.
.78...94.
.8.9.7.6.
.4231678.
.........
|
.....2.4.
......396
..79..81.
..8.97..4
...5.6...
6..84.2..
.82..96..
975......
.3.7.....
|
....7..83
..6...5..
827..3.61
.1...7..2
..81.534.
2....4...
.62......
58..46.3.
4...1....
|
19.5..3..
34.8...7.
........2
78..2....
...1.3...
....4..26
8........
.5...6.98
..6..5.13
|
1..839..5
.9.4.6.1.
.........
23.....64
9...2...7
45.....81
.........
.2.3.8.7.
7..245..9
|
The jellyfish is a very rare and beautiful pattern, but still more frequent than the hidden quad. The jellyfish is the extension of the X-Wing and swordfish to 4 columns. Like for the naked quad, an extension to 5 columns is possible in theory, but useless in practice: a fish with 5 columns always come with a jellyfish (4 columns) which is orthogonal to it. The following Sudokus require the jellyfish solving technique:
...53....
.5.6.29.3
.......8.
28..9....
6..7...4.
.1......2
.6....4.1
..8.1..9.
.4...96..
|
92...7...
.14.896..
..8..2.3.
..6.7.824
..1..3.6.
.9.......
....1634.
.4.7...96
........7
|
.1..4....
..2..1..5
.......7.
..13....4
.36.52...
.287.....
....68..1
..4.2.7..
8.....43.
|
463.8.1..
2........
8.1.3.5.7
...3.8...
3.7.9.8.5
...1.5...
1.6.5.7.8
........9
..8.1.256
|
.........
..8.3.6..
.1.5.9.4.
.4.....1.
5...2...8
.7.....3.
.5.1.4.9.
..3.6.7..
.........
|
The third one is a "minimal" jellyfish, with only 8 cells. The last one is a "maximal" jellyfish, with 16 cells.
The hidden quad is probably the rarest solving technique, and is also very difficult to spot. The reason hidden quads are so rare is that they can only occur in a block, row or column where all the cells are empty. If only 8 cells are empty, a naked quad (the transposed technique), which is an easier one, can be found whenever a hidden quad is present, and makes the hidden quad unnecessary. With less than 8 empty cells, only lower patterns can be found (hidden triplets and hidden pairs).
The following Sudokus are the only ones I know that require a hidden quad as the hardest solving technique:
...1..2..
826....4.
3.......6
.19..25..
...7.....
..4..8..7
.....71..
..29.5...
.7...1..3
|
..9.2.48.
6....89..
82..9...1
.9.......
7.8...1.2
.......7.
9...7..65
..61....9
.52.6.8..
|
5.26..7..
...9...1.
......385
..4.961..
.........
..527.9..
837......
.6...9...
..9..82.3
|
...5...3.
..9...2.7
..3.64.8.
8..61....
5.2.3....
3..49....
..1.78.2.
..8...6.1
...1...7.
|
....1....
4........
123..8...
.7..923..
2.9.74..6
.4.......
.1..3.2..
..28.91..
8...4.53.
|
....2..1.
1...4.5..
.49..6...
.6..91.8.
29..3...6
..56....1
..8......
97....34.
3.......2
|
.......5.
.3....6..
5.7....23
...5.2..4
6......7.
.1.3..5..
..8....4.
.9.28..6.
.....19.5
|
.2.1.3.9.
.8.4.9.6.
.........
.6..1..3.
..8.3.1..
2.......5
..56.84..
.42...65.
81.7.5.29
|
Unique rectangles and loops are interesting solving techniques: while nearly all other techniques are based on the fact that a valid Sudoku must have a solution, unique rectangles and loops are based on the fact that a valid Sudoku cannot have more than one solution. These techniques cannot be used to solve Sudokus with two or more solutions.
The term "unique" is confusing. In fact, a unique pattern (rectangle or loop) is a pattern that can only occur in a Sudoku having more than one solution. The unique rectangles and loops solving techniques consist in avoiding a unique pattern.
There are four types of Unique Rectangles and Loops, named type 1 to type 4. These four types correspond to four different "ways" of avoiding the unique pattern. It is common to have more than one type at the same time for the same unique pattern. The type 3 is the hardest because it can only be used in combination with a Naked Set or Hidden Set.
..5...82.
1...94...
3..25....
52...1...
...5...48
...4..23.
...61...7
.13...5..
26....4..
|
.1....4.6
4..6.2...
..8...2..
.6.5.4..1
....6.5..
.8.9.7...
2.6.7.98.
......37.
3..4.....
|
53.......
27..83...
...59....
..16...5.
.24...79.
.5...71..
....78...
...36..17
.......48
|
91..2.3..
6.....45.
...94..62
..9..8...
4.6...7.3
...2..8..
36..95...
.45.....1
..2.1..35
|
The unique rectangle technique is quite frequent. Unique loops are less frequent. But long unique loops (with more than 6 cells) are rare. The following Sudokus require unique loops of long length. The theoretical maximum length is 16.
...4..87.
.....3..2
....685.4
........7
.25.4.9.8
.8.......
7...8...3
.94.16...
.3.9...4.
|
.25.1.7..
8...5...6
6..8....1
..2.95..8
98.6.....
...7..9.4
2....84.9
.......5.
.785.96..
|
.6.....4.
.9.2.4..6
..4..978.
......89.
4..3.8..2
.58......
.871..4..
1..9.6.2.
.2.....1.
|
..72..4.3
5......8.
..9..8..6
.5.6..8..
..3......
7..1.2..9
1...4.6.7
2....1...
.749...1.
|
2..4...3.
......56.
..6...1..
4...6...1
...58.2..
.....38.5
.48.79...
53.......
...8.2...
|
The Bivalue Universal Grave (BUG) is a technique similar to unique rectangles and loops: Sudoku that have a BUG have an even number of solutions, that is, no solution, or two solutions, four, six, etc. The solving technique itself again consists in avoiding a BUG pattern.
Like with unique rectangle and loops, there are four types of solving techniques based on the BUG pattern (BUG type 1, 2, 3 and 4), corresponding to the four ways of avoiding the BUG pattern itself. Interesting Sudokus are the ones that require both a BUG type 2 and a BUG type 1, such as the following:
54.6....3
..6.147.5
.8...9.6.
.18.....6
.5.....4.
4.....89.
.2.1...3.
6.943.1..
1....6.29
|
9......45
..2...6.9
..46.91..
6...2..17
.....328.
...5.7..6
.5..8....
2.8...7..
.1.7.....
|
..71.8...
..5.3....
..24.7583
7.6...3.4
.2.....7.
5.9...8.2
4735.69..
....2.4..
...7.91..
|
.5...4.6.
...89.5.7
7.9..6.4.
..4..96.1
.......5.
..32...8.
2..4.57..
.3......8
..1...2..
|
...5.6.4.
.31......
.........
6.42.....
....7.3..
1........
.8.....52
....14...
......7..
|
Because a BUG type 3 nearly always occurs with a BUG type 4, and because the type 3 is rated harder, the BUG type 3 is nearly as rare as the Jellyfish. The following Sudokus require a BUG type 3:
4..1.2..3
98..4..12
....6....
..5...3..
...7.3...
..24.17..
2...3...9
..19.72..
3.......6
|
6....9...
.2..3..6.
.7..1....
.8..2.3.4
.......9.
.3..6..1.
1....4.5.
8....36.2
..25..9.1
|
.91...46.
.4.126.5.
..64.71..
9.......2
5.3.6.9.4
.........
.8..3..9.
6...4...8
...5.2...
|
3...4...1
.5.1.7.2.
1.7...9.5
...8.3...
..34925..
....1....
.4.689.3.
..83.47..
.........
|
..53..4..
..3..2.69
.7..9..2.
.8.......
3..1.7..4
.......3.
.9..1..5.
75.6..3..
..2..96..
|
The following ones require a BUG type 4:
..25.38..
....8....
8..6.2..1
6.48.17.9
.3.....5.
7.59.46.2
1..3.8..7
....9....
..71.69..
|
8...5.1.3
.6.1.....
..3..86.9
.9..1.8..
2..8.3..6
..6.2..7.
6.87..2..
.....1.9.
9.4.8...7
|
.4......2
...46..8.
9..1.3...
..3..5.4.
.28.1.53.
.6.3..9..
...6.8..4
.3..91...
8......7.
|
..135....
.5.......
9.6..1...
...8..7..
248.3...9
.....4..5
6.9.8.5.2
.1..9..7.
..3.4.8..
|
.19.8....
..21..6.9
.345..712
....7.12.
7..2.1..3
.28.5....
947..523.
2.5..74..
....2.56.
|
The BUG type 4 is frequent enough to find Sudokus requiring both a BUG type 4 and a BUG type 1, such as the following ones:
....7....
1.58...7.
.3.6.4...
.....84.7
..69..2..
..9.41...
.5.......
...1..53.
7....29.8
|
4.....178
...7.....
81.5.9.2.
6.41.53..
...36....
..5......
...91...2
.........
.63.289..
|
8...13...
.35.6....
..1..8.2.
...7.5..4
2..19..8.
.......6.
..95..3..
7.2...91.
.........
|
97....15.
.84.12.7.
3..9.....
.9.8....3
6..4.35..
....7.6.1
7....54..
.....8...
.4.......
|
76..123..
..2..8.1.
.3.4..2..
.....6..3
8.3.2.9.5
1..5.....
..7..1.9.
.1.7..8..
..923..71
|
Aligned Pair Exclusions are between what we can consider as logical or brute-force techniques. They require considering all combinations of possible values between two arbitrary cells. Most occurrences of the Aligned Pair Exclusion pattern are already covered by the XY-Wing and XYZ-Wing techniques. The following Sudokus require one or more Aligned Pair Exclusions:
..49.....
.5.8.347.
.9......3
.3..9..62
...2.5...
16..8..4.
4......1.
.197.8.3.
.....15..
|
..9.4.3..
67..3..94
...8.6...
4...5...2
.5.9.3.1.
.9.....6.
....2....
5..4.1..9
.3.....2.
|
.3.4.2...
2...16..4
.1.7....3
..1....78
..984.1..
....2...5
6....5...
145....8.
..7..45..
|
4..6.8..3
.8..9..4.
..34.71..
7.8...4.9
.6.....7.
5.4...2.8
..52.16..
.1..6..5.
6..9.5..1
|
The Aligned Pair Exclusion technique can be extended to any number of cells. The Sudoku Explainer only implements Aligned Pair Exclusions and Aligned Triplet Exclusions (3 cells). Higher versions are usually not really simpler than equivalent Forcing Chains.
The following ones require Aligned Triplet Exclusions:
......9..
..12..87.
..93...51
69..2....
48...1...
.3....26.
......61.
.5.134...
....97...
|
.9.64..1.
.4...736.
.5..1..7.
...2.4...
6...7....
....618..
.2.73..8.
..8...734
..9......
|
56..4....
...13....
......67.
...2..5..
..8..12.9
9.....8..
14..6..5.
..7.1....
38....4..
|
Forcing chains are not pleasant solving techniques to use when solving a Sudoku by hand. While these solving techniques are still more elegant, and more "logical" than a brute-force search, they are less elegant (and much more difficult) than all other solving techniques and are often considered as a form of "trial and error". The Sudoku Explainer classifies all forcing chains into various categories, from the simplest to the hardest. Simpler chains usually exhibit some patterns, and feel "more logical". Harder chains are less elegant and closer to brute-force searching. The simplest form of forcing chain, the XY-Wing, is usually not considered as such.
When considering all possible Sudokus, the ones requiring forcing chains to solve are not very frequent, but still more frequent than Sudokus requiring rare patterns such as hidden quads, naked quads or jellyfishes. But these Sudokus nearly never occur in newspapers and books because they are usually considered to be too hard to solve by hand.
In a forcing chain, each "link" of the chain is based on one of the three following rules: elimination (a value cannot occur more than once in a row, column or block), hidden single (only remaining position for a value in a row, column or block) or naked single (only remaining possible value for an empty cell). The forcing chain solving technique alone (when properly implemented) can solve nearly every possible Sudokus. The following Sudokus require only a few, not too hard forcing chains to be solved (along with other techniques). Note that the last one has 35 clues and is minimal (no clue can be removed):
..4...91.
1..3.....
3..941..7
..12.457.
..6.7.1..
.371.64..
6..897..1
.....5..9
.95...7..
|
5..6....3
...34....
...5.961.
392...76.
.4.....5.
..7.....4
..58..1..
..327....
9....1..5
|
.......3.
.9....5..
5...6...1
.7.4...5.
8..1..4..
..9..6.7.
.56.8...3
.4...1...
....43..2
|
.7...3.8.
..46....7
...7.2..9
.6...183.
1....867.
...2...9.
..6.2...3
5......1.
..2.1....
|
1...5678.
78.12.45.
4..7..1.3
2.1.648..
.6..9...1
8..2..5..
3.2645...
64..7....
.....2...
|
On the other hand, the following Sudokus are very hard ones, and they require nearly every kinds of forcing chains multiple times to be solved:
6...3...1
.5...7.4.
..295....
...5.3.2.
...4..3.6
....724..
52....8..
8.1....6.
.76.....4
|
6.2.5....
.....3.4.
.........
43...8...
.1....2..
......7..
5..27....
.......81
...6.....
|
4..7..8.1
.....2.4.
..384...7
5.79...8.
..8.5.1..
.3...85.9
3...867..
.8.1.....
7.2..4..8
|
.........
.9..3..4.
..26.15..
..4...2..
.3..5..1.
..6...7..
..58.26..
.7..4..9.
.........
|
The first one shows nearly every kind of chains (according to the classification of the Sudoku Explainer). The second one is one of the hardest Sudokus with only 17 clues.
Special cases of forcing chains that exhibit regular patterns are Bidirectional Cycles. A Bidirectional Cycle is a cyclic Forcing Chain that can be reversed. Bidirectional Cycles usually allow the removal of many candidates at once. There are two special cases: A Bidirectional X-Cycles only involves a single value, and a Bidirectional Y-Cycle only involves cells with two candidates. The following Sudokus can be solved using Bidirectional cycles.
.......4.
89.2..5..
.2..35..9
..1.7.6..
.4...6..7
..8.1.3..
.1..54..3
25.9..4..
.......7.
|
..26.....
63..24...
97..5..4.
5.......7
.64...18.
.8..1.36.
.....17..
...93..14
.4.86..5.
|
.4..8..5.
..9...71.
...27.4..
..1..25..
.3..5..4.
..46..9..
..7.63...
.25...6..
.8..2..3.
|
There are a few Sudokus that still cannot be solved using Forcing Chains and all the techniques depicted above. These Sudokus can be solved in various other ways. Let's have a look at the two schemes implemented in the Sudoku Explainer.
In a standard Forcing Chain, only the two basic rules of Sudoku are used:
- A value can occur only once in a region
- A cell can contain only one value
These two rules do in fact correspond to the Hidden Single and Naked Single solving techniques. The Forcing Chain technique can be extended by also allowing more advanced techniques within the chains, such as Pointing, Hidden Pair, Naked Pair, X-Wing, etc. Using such "Advanced" Forcing Chains, it is possible to solve the following Sudokus that cannot be solved by standard Forcing Chains alone:
7.8...3..
...2.1...
5........
.4.....26
3...8....
...1...9.
.9.6....4
....7.5..
.........
|
7.....4..
.2..7..8.
..3..8..9
...5..3..
.6..2..9.
..1..7..6
...3..9..
.3..4..6.
..9..1.35
|
..1.....2
.3..4....
5..6..3..
..7..8..1
.6..1..3.
1..9..8..
..5..9..4
.8..5..7.
2..1..9..
|
..1...2..
.3..4..5.
6.......7
...1.3...
.8.....3.
...6.4...
2.......6
.4..5..8.
..7...1..
|
..1...2..
.3..4..5.
6.......7
...1.3...
.8..7..3.
...5.6...
7.......6
.5..3..8.
..9...1..
|
.....1..2
.1..2..3.
4..5.....
..4.....6
.7..3..1.
8.....9..
5....8...
....1..7.
..64..5..
|
Following the idea of using advanced techniques within a Forcing Chain to the extreme, one can use Forcing Chains within Forcing Chains to get a powerfull solving technique. In fact, if the nesting depth is not limited, this technique of "Nested" Forcing Chains can actually solve any Sudoku. The Sudoku Explainer only implements one level of nesting. The following Sudokus (from here) can be solved using Nested Forcing Chains, but not by any of the other techniques:
.6.9....3
5....4.2.
....8.4..
8......5.
..3...7..
.9......1
..1.5....
.7.3....9
9....2.4.
|
1....7.9.
.3..2...8
..96..5..
..53..9..
.1..8...2
6....4...
3......1.
.4......7
..7...3..
|
9....18..
.6..3....
..25....7
..9.....6
.5.....2.
4.....3..
3....81..
....4..8.
..79....5
|
..1..2...
.3..4..5.
6..7..8..
..6.....7
.1.....3.
9.....6..
..7..1..8
.4..3..2.
...5..9..
|
..1..2...
.3..4..5.
5..6..7..
..7.....6
.1.....3.
8.....9..
..9..3..2
.2..1..4.
...7..8..
|
As this page was last updated (November 2006), these Sudokus were considered to be among the hardest known Sudokus. Note that they may require from some minutes to some hours to be solved by the Sudoku Explainer.
Here's some interesting Sudokus which require combinations of rare solving techniques.
The first one requires both a Hidden Quad and a Unique Rectangle. The second one requires both a Jellyfish and two BUGs. The 3rd one requires 5 Turbot Fishes (Forcing X-Chains of length 5), an incredible long skewed unique loop (14 cells) and a BUG. The 4th one has a Naked Quad on a block with 9 empty cells, and a BUG type 2. The last one requires a BUG type 2 two times.
6.214..7.
8.......4
.4..8..1.
...85....
1..2.4...
...96....
.8..2..6.
7.......9
2.673..4.
|
.9.81....
.....3.5.
..4...81.
...7.9...
3.8...5..
.4.......
.....63..
9....8.7.
71.43....
|
...9..45.
6..5.....
.1..8.2.7
...7.....
.3..927..
..8.4....
5.6....49
...6...7.
.....4...
|
.94.2.6..
.....7..1
6..91...8
.1....8..
8.6...4.2
..5....9.
4...76..9
7..3.....
..3.4.72.
|
6.79..2..
....5.34.
3...82.91
1.....8..
.56...41.
..8.....6
56.31...4
.81.4....
..4..91.5
|
| Last update: 2006-11-27 |