Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=266

level: medium

position: 4..8.5..9..6.7.......4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2 initial

Autosolve

position: 4..8.5..9..6.7..5....4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2 autosolve

Pair Reduction Variants

Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # E1: 2,6 => CTR => E1: 1,3
* DIS # B5: 2,9 => CTR => B5: 4,5
* CNT   2 HDP CHAINS /  20 HYP OPENED
```

See Appendix: Full HDP Chains for full list of HDP chains.

Pair Reduction

The following important HDP chains were detected:

```* DIS # E1: 2,6 => CTR => E1: 1,3
* DIS E1: 1,3 # B5: 2,9 => CTR => B5: 4,5
* DIS E1: 1,3 + B5: 4,5 # D2: 1,2 => CTR => D2: 9
* DIS E1: 1,3 + B5: 4,5 + D2: 9 # B1: 2,7 => CTR => B1: 1,3
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # F8: 8 => CTR => F8: 1,3
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # G2: 2 => CTR => G2: 4,8
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 # I8: 1,3,6 => CTR => I8: 4,8
* PRF E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 # G7: 8 => SOL
* STA E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 + G7: 8
* CNT   8 HDP CHAINS /  43 HYP OPENED
```

See Appendix: Full HDP Chains for full list of HDP chains.

Details

Positions

 4..8.5..9..6.7.......4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2 initial 4..8.5..9..6.7..5....4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2 autosolve 417835629326971458598426137683142795241759386759368241965214873872693514134587962 solved

level: medium

Pairing Analysis

```--------------------------------------------------
* PAIRS (5)
E3: 2,6
F3: 6,9
I2: 4,8
C6: 2,9
D7: 1,2

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
B1,B2: 1.. / B1 = 1  =>  5 pairs (_) / B2 = 1  =>  8 pairs (_)
H4,I6: 1.. / H4 = 1  => 14 pairs (_) / I6 = 1  =>  8 pairs (_)
B1,E1: 1.. / B1 = 1  =>  5 pairs (_) / E1 = 1  =>  8 pairs (_)
H5,G6: 2.. / H5 = 2  =>  7 pairs (_) / G6 = 2  =>  0 pairs (*)
H1,H5: 2.. / H1 = 2  =>  0 pairs (*) / H5 = 2  =>  0 pairs (X)
E1,F2: 3.. / E1 = 3  => 10 pairs (_) / F2 = 3  =>  5 pairs (_)
A4,C4: 3.. / A4 = 3  => 14 pairs (_) / C4 = 3  => 11 pairs (_)
I7,I8: 3.. / I7 = 3  =>  5 pairs (_) / I8 = 3  => 11 pairs (_)
F2,F8: 3.. / F2 = 3  =>  5 pairs (_) / F8 = 3  => 10 pairs (_)
G2,I2: 4.. / G2 = 4  => 22 pairs (_) / I2 = 4  =>  8 pairs (_)
C4,B5: 4.. / C4 = 4  => 14 pairs (_) / B5 = 4  =>  7 pairs (_)
E4,E5: 4.. / E4 = 4  =>  7 pairs (_) / E5 = 4  => 14 pairs (_)
I8,G9: 4.. / I8 = 4  => 22 pairs (_) / G9 = 4  =>  8 pairs (_)
C4,E4: 4.. / C4 = 4  => 14 pairs (_) / E4 = 4  =>  7 pairs (_)
B5,E5: 4.. / B5 = 4  =>  7 pairs (_) / E5 = 4  => 14 pairs (_)
G2,G9: 4.. / G2 = 4  => 22 pairs (_) / G9 = 4  =>  8 pairs (_)
I2,I8: 4.. / I2 = 4  =>  8 pairs (_) / I8 = 4  => 22 pairs (_)
A3,B3: 5.. / A3 = 5  =>  7 pairs (_) / B3 = 5  => 19 pairs (_)
E5,E6: 5.. / E5 = 5  =>  7 pairs (_) / E6 = 5  => 13 pairs (_)
B6,E6: 5.. / B6 = 5  =>  7 pairs (_) / E6 = 5  => 13 pairs (_)
A3,A5: 5.. / A3 = 5  =>  7 pairs (_) / A5 = 5  => 19 pairs (_)
A4,A5: 6.. / A4 = 6  => 11 pairs (_) / A5 = 6  => 13 pairs (_)
D4,D5: 7.. / D4 = 7  =>  7 pairs (_) / D5 = 7  =>  6 pairs (_)
B8,C8: 7.. / B8 = 7  =>  5 pairs (_) / C8 = 7  =>  6 pairs (_)
* DURATION: 0:00:19.521316  START: 14:13:13.540045  END: 14:13:33.061361 2019-04-28
* CP COUNT: (24)
* SOLUTION FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (C6,D7,E3,F3,I2)
* 4..8.5..9..6.7..5....4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2
* PAIR E3: 2,6 BLK 2
E1: 2,6,1,3                              # reduction candidate for 2,6
E1: 2,6 => CTR
* 413865279.26173.5.57842913638.7.2..56.19.83277.23.6.4.8652347912.7691583139587462
E1: 1,3                                  #  7 pairs
* PAIR F3: 6,9 COL F
F5: 6,9,8                                # reduction candidate for 6,9
F5: 6,9                                  #  8 pairs
F6: 6,9,1,8                              # reduction candidate for 6,9
F6: 6,9                                  # 10 pairs
* PAIR I2: 4,8 BLK 3
G2: 4,8,2                                # reduction candidate for 4,8
G2: 2                                    # 20 pairs
G2: 4,8                                  #  8 pairs
* PAIR I2: 4,8 COL I
I8: 4,8,1,3,6                            # reduction candidate for 4,8
I8: 4,8                                  # 25 pairs
I8: 1,3,6                                #  8 pairs
* PAIR C6: 2,9 BLK 4
A5: 2,9,5,6                              # reduction candidate for 2,9
A5: 2,9                                  # 12 pairs
B5: 2,9,4,5                              # reduction candidate for 2,9
B5: 2,9 => CTR
* 4..8.5..9..6.7..5..5.4..13.384..2..5..1.4.3..7..35..4..65..4.......9.5..1..5.7..2
B5: 4,5                                  #  6 pairs
B6: 2,9,5                                # reduction candidate for 2,9
B6: 2,9                                  # 13 pairs
* PAIR C6: 2,9 ROW 6
G6: 2,9,6,8                              # reduction candidate for 2,9
G6: 6,8                                  # 10 pairs
G6: 2,9                                  #  8 pairs
* PAIR C6: 2,9 COL C
C3: 2,9,7,8                              # reduction candidate for 2,9
C3: 2,9                                  # 12 pairs
C3: 7,8                                  #  6 pairs
* PAIR D7: 1,2 BLK 8
E7: 1,2,3,8                              # reduction candidate for 1,2
E7: 1,2                                  # 10 pairs
D8: 1,2,6                                # reduction candidate for 1,2
D8: 1,2                                  # 23 pairs
* PAIR D7: 1,2 COL D
D2: 1,2,9                                # reduction candidate for 1,2
D2: 1,2                                  # 11 pairs
D2: 9                                    #  8 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0266-base-pr-000.dot
* REASONING
* DIS # E1: 2,6 => CTR => E1: 1,3
* DIS # B5: 2,9 => CTR => B5: 4,5
* CNT   2 HDP CHAINS /  20 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (C6,D7,E3,F3,I2)
* 4..8.5..9..6.7..5....4..13..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2
* PAIR E3: 2,6 BLK 2
E1: 2,6,1,3                              # reduction candidate for 2,6
E1: 2,6 => CTR
* 413865279.26173.5.57842913638.7.2..56.19.83277.23.6.4.8652347912.7691583139587462
* PAIR F3: 6,9 COL F
F5: 6,9,8                                # reduction candidate for 6,9
F5: 6,9                                  # 11 pairs
F6: 6,9,1,8                              # reduction candidate for 6,9
F6: 6,9                                  # 13 pairs
* PAIR I2: 4,8 BLK 3
G2: 4,8,2                                # reduction candidate for 4,8
G2: 2                                    # 23 pairs
G2: 4,8                                  # 11 pairs
* PAIR I2: 4,8 COL I
I8: 4,8,1,3,6                            # reduction candidate for 4,8
I8: 4,8                                  # 18 pairs
I8: 1,3,6                                # 10 pairs
* PAIR C6: 2,9 BLK 4
A5: 2,9,5,6                              # reduction candidate for 2,9
A5: 2,9                                  # 12 pairs
B5: 2,9,4,5                              # reduction candidate for 2,9
B5: 2,9 => CTR
* 4..8.5..9..6.7..5..5.4..13.384..2..5..1.4.3..7..35..4..65..4.......9.5..1..5.7..2
B6: 2,9,5                                # reduction candidate for 2,9
B6: 2,9                                  # 15 pairs
* PAIR C6: 2,9 ROW 6
G6: 2,9,6,8                              # reduction candidate for 2,9
G6: 6,8                                  # 13 pairs
G6: 2,9                                  # 10 pairs
* PAIR C6: 2,9 COL C
C3: 2,9,7,8                              # reduction candidate for 2,9
C3: 2,9                                  # 14 pairs
C3: 7,8                                  #  9 pairs
* PAIR RESTART
* PAIR C6: 2,9 BLK 4
A5: 2,9,5,6                              # reduction candidate for 2,9
A5: 2,9                                  # 12 pairs
B6: 2,9,5                                # reduction candidate for 2,9
B6: 2,9                                  # 15 pairs
* PAIR C6: 2,9 ROW 6
G6: 2,9,6,8                              # reduction candidate for 2,9
G6: 6,8                                  # 13 pairs
G6: 2,9                                  # 10 pairs
* PAIR C6: 2,9 COL C
C3: 2,9,7,8                              # reduction candidate for 2,9
C3: 2,9                                  # 14 pairs
C3: 7,8                                  #  9 pairs
* PAIR D7: 1,2 BLK 8
E7: 1,2,3,8                              # reduction candidate for 1,2
E7: 1,2                                  # 17 pairs
D8: 1,2,6                                # reduction candidate for 1,2
D8: 1,2                                  # 30 pairs
* PAIR D7: 1,2 COL D
D2: 1,2,9                                # reduction candidate for 1,2
D2: 1,2 => CTR
* 4..8.56798.6.7.254...426138.8...2915..19.83277..3.18463652.4.9.2..69.58.1..5.7462
* PAIR RESTART
* PAIR D7: 1,2 BLK 8
D8: 1,2,6                                # reduction candidate for 1,2
D8: 1,2                                  # 30 pairs
D8: 6                                    # 19 pairs

* LEVEL 0 PASS 1 ROUND 2 (AUTO SOLVE) (B5,C6,D5,D7,E1,F2,F5,I2,I3)
* 4..8.5..9..697..5....42613..8...2..5..1...3..7..3...4..65..4.......9.5..1..5.7..2
* PAIR E1: 1,3 ROW 1
B1: 1,3,2,7                              # reduction candidate for 1,3
B1: 2,7 => CTR
* 4.3815..9.169738545.842613738...2..56.17.93287..3...4.9652.47..8.7.915..1..5.7..2
* PAIR E1: 1,3 COL E
E7: 1,3,8                                # reduction candidate for 1,3
E7: 8                                    # 15 pairs
E7: 1,3                                  # 15 pairs
* PAIR RESTART
* PAIR E1: 1,3 COL E
E7: 1,3,8                                # reduction candidate for 1,3
E7: 8                                    # 15 pairs
E7: 1,3                                  # 15 pairs
* PAIR F2: 1,3 ROW 2
B2: 1,3,2                                # reduction candidate for 1,3
B2: 2                                    # 18 pairs
B2: 1,3                                  # 14 pairs
* PAIR F2: 1,3 COL F
F8: 1,3,8                                # reduction candidate for 1,3
F8: 8 => CTR
* 43.8156.9.16973.54..842613738.64271564175932.7..3.1.4..65.34.......985631.3567482
* PAIR RESTART
* PAIR F2: 1,3 ROW 2
B2: 1,3,2                                # reduction candidate for 1,3
B2: 2                                    # 18 pairs
B2: 1,3                                  # 16 pairs
* PAIR I2: 4,8 BLK 3
G2: 4,8,2                                # reduction candidate for 4,8
G2: 2 => CTR
* 4.28.5..98.697.254...426138.8...2..5..1..832.7293..84..65..4.......9.5..1..5.74.2
* PAIR I2: 4,8 COL I
I8: 4,8,1,3,6                            # reduction candidate for 4,8
I8: 1,3,6 => CTR
* 417835269326971854...426137.83.42..5.4175.32875231..4..65..4....74.935.61..5674.2
* PAIR RESTART

* LEVEL 0 PASS 1 ROUND 3 (AUTO SOLVE) (A5,A7,A8,C6,C8,C9,F5,F6,G1,G2,G4,G6,H1,H4,H5,H9,I2,I8)
* 417835..9326971.5.598426137683142..5.4175.3.675.36..41.65214..3.7.69351.13.587..2
* PAIR G4: 7,9 COL G
G7: 7,9,8                                # reduction candidate for 7,9
G7: 8 => SOLVED
* 417835629326971458598426137683142795241759386759368241965214873872693514134587962
* DURATION: 0:00:25.006702  START: 14:13:42.154626  END: 14:14:07.161328 2019-04-28
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0266-base-pr-001.dot
* REASONING
* DIS # E1: 2,6 => CTR => E1: 1,3
* DIS E1: 1,3 # B5: 2,9 => CTR => B5: 4,5
* DIS E1: 1,3 + B5: 4,5 # D2: 1,2 => CTR => D2: 9
* DIS E1: 1,3 + B5: 4,5 + D2: 9 # B1: 2,7 => CTR => B1: 1,3
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # F8: 8 => CTR => F8: 1,3
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # G2: 2 => CTR => G2: 4,8
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 # I8: 1,3,6 => CTR => I8: 4,8
* PRF E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 # G7: 8 => SOL
* STA E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 + G7: 8
* CNT   8 HDP CHAINS /  43 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=266
```

Solution

position: 417835629326971458598426137683142795241759386759368241965214873872693514134587962 solved

See section Pair Reduction for the HDP chains leading to this result.

Appendix: Full HDP Chains

A1. Pair Reduction Analysis

Full list of HDP chains traversed:

```* DIS # E1: 2,6 => CTR => E1: 1,3
* INC # E1: 1,3 => UNS
* INC # F5: 6,9 => UNS
* INC # F6: 6,9 => UNS
* INC # G2: 4,8 => UNS
* INC # G2: 2 => UNS
* INC # I8: 4,8 => UNS
* INC # I8: 1,3,6 => UNS
* INC # A5: 2,9 => UNS
* DIS # B5: 2,9 => CTR => B5: 4,5
* INC # B5: 4,5 => UNS
* INC # B6: 2,9 => UNS
* INC # G6: 2,9 => UNS
* INC # G6: 6,8 => UNS
* INC # C3: 2,9 => UNS
* INC # C3: 7,8 => UNS
* INC # E7: 1,2 => UNS
* INC # D8: 1,2 => UNS
* INC # D2: 1,2 => UNS
* INC # D2: 9 => UNS
* CNT  20 HDP CHAINS /  20 HYP OPENED
```

A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # E1: 2,6 => CTR => E1: 1,3
* INC E1: 1,3 # F5: 6,9 => UNS
* INC E1: 1,3 # F6: 6,9 => UNS
* INC E1: 1,3 # G2: 4,8 => UNS
* INC E1: 1,3 # G2: 2 => UNS
* INC E1: 1,3 # I8: 4,8 => UNS
* INC E1: 1,3 # I8: 1,3,6 => UNS
* INC E1: 1,3 # A5: 2,9 => UNS
* DIS E1: 1,3 # B5: 2,9 => CTR => B5: 4,5
* INC E1: 1,3 + B5: 4,5 # B6: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # G6: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # G6: 6,8 => UNS
* INC E1: 1,3 + B5: 4,5 # C3: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # C3: 7,8 => UNS
* INC E1: 1,3 + B5: 4,5 # A5: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # B6: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # G6: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # G6: 6,8 => UNS
* INC E1: 1,3 + B5: 4,5 # C3: 2,9 => UNS
* INC E1: 1,3 + B5: 4,5 # C3: 7,8 => UNS
* INC E1: 1,3 + B5: 4,5 # E7: 1,2 => UNS
* INC E1: 1,3 + B5: 4,5 # D8: 1,2 => UNS
* DIS E1: 1,3 + B5: 4,5 # D2: 1,2 => CTR => D2: 9
* INC E1: 1,3 + B5: 4,5 + D2: 9 # D8: 1,2 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 # D8: 6 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 # B1: 1,3 => UNS
* DIS E1: 1,3 + B5: 4,5 + D2: 9 # B1: 2,7 => CTR => B1: 1,3
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # E7: 1,3 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # E7: 8 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # E7: 1,3 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # E7: 8 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # B2: 1,3 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # B2: 2 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # F8: 1,3 => UNS
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 # F8: 8 => CTR => F8: 1,3
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # B2: 1,3 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # B2: 2 => UNS
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # G2: 4,8 => UNS
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 # G2: 2 => CTR => G2: 4,8
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 # I8: 4,8 => UNS
* DIS E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 # I8: 1,3,6 => CTR => I8: 4,8
* INC E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 # G7: 7,9 => UNS
* PRF E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 # G7: 8 => SOL
* STA E1: 1,3 + B5: 4,5 + D2: 9 + B1: 1,3 + F8: 1,3 + G2: 4,8 + I8: 4,8 + G7: 8
* CNT  43 HDP CHAINS /  43 HYP OPENED
```