Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=126

level: medium

position: .....2..4.7.1...9.6.1.5.2...86...7......9......9...36...2.6.5.1.6...9.2.5..7..... initial

Autosolve

position: .....2..427.1...9.6.1.5.2...862..7.9....9......9...36...2.6.5.1.6.5.9.2.5..72.9.6 autosolve

Pair Reduction Variants

Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # D5: 4,8 => CTR => D5: 3,6
* DIS # F5: 4,8 => CTR => F5: 1,3,5,6,7
* DIS # D3: 4,8 => CTR => D3: 3,9
* DIS # H7: 4,8 => CTR => H7: 3,7
* PRF # C8: 4,8 => SOL
* DIS # E8: 4,8 => CTR => E8: 1,3
* CNT   6 HDP CHAINS /  23 HYP OPENED
```

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

Pair Reduction

The following important HDP chains were detected:

```* DIS # D5: 4,8 => CTR => D5: 3,6
* DIS D5: 3,6 # F5: 4,8 => CTR => F5: 1,3,5,6,7
* DIS D5: 3,6 + F5: 1,3,5,6,7 # D3: 4,8 => CTR => D3: 3,9
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 # D7: 3 => CTR => D7: 4,8
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 # H7: 4,8 => CTR => H7: 3,7
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 # H9: 3 => CTR => H9: 4,8
* PRF D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 # C8: 4,8 => SOL
* STA D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 + C8: 4,8
* CNT   7 HDP CHAINS /  20 HYP OPENED
```

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

Details

Positions

 .....2..4.7.1...9.6.1.5.2...86...7......9......9...36...2.6.5.1.6...9.2.5..7..... initial .....2..427.1...9.6.1.5.2...862..7.9....9......9...36...2.6.5.1.6.5.9.2.5..72.9.6 autosolve 895372614273146895641958237486231759327695148159487362932864571768519423514723986 solved

level: medium

Pairing Analysis

```--------------------------------------------------
* PAIRS (3)
G2: 6,8
D6: 4,8
G8: 4,8

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
G1,H1: 1.. / G1 = 1  =>  5 pairs (_) / H1 = 1  => 14 pairs (_)
A8,B9: 1.. / A8 = 1  =>  0 pairs (X) / B9 = 1  =>  4 pairs (_)
E8,F9: 1.. / E8 = 1  =>  4 pairs (_) / F9 = 1  =>  0 pairs (X)
A8,E8: 1.. / A8 = 1  =>  0 pairs (X) / E8 = 1  =>  4 pairs (_)
B9,F9: 1.. / B9 = 1  =>  4 pairs (_) / F9 = 1  =>  0 pairs (X)
G1,G5: 1.. / G1 = 1  =>  5 pairs (_) / G5 = 1  => 14 pairs (_)
B5,B6: 2.. / B5 = 2  =>  4 pairs (_) / B6 = 2  =>  4 pairs (_)
I5,I6: 2.. / I5 = 2  =>  4 pairs (_) / I6 = 2  =>  4 pairs (_)
B5,I5: 2.. / B5 = 2  =>  4 pairs (_) / I5 = 2  =>  4 pairs (_)
B6,I6: 2.. / B6 = 2  =>  4 pairs (_) / I6 = 2  =>  4 pairs (_)
C2,B3: 4.. / C2 = 4  => 15 pairs (_) / B3 = 4  =>  5 pairs (_)
G5,G8: 4.. / G5 = 4  =>  0 pairs (X) / G8 = 4  =>  5 pairs (_)
H1,I2: 5.. / H1 = 5  =>  0 pairs (X) / I2 = 5  =>  8 pairs (_)
C2,I2: 5.. / C2 = 5  =>  0 pairs (X) / I2 = 5  =>  8 pairs (_)
F4,H4: 5.. / F4 = 5  =>  6 pairs (_) / H4 = 5  => 10 pairs (_)
D1,F2: 6.. / D1 = 6  =>  5 pairs (_) / F2 = 6  => 14 pairs (_)
G1,G2: 6.. / G1 = 6  => 14 pairs (_) / G2 = 6  =>  5 pairs (_)
D5,F5: 6.. / D5 = 6  => 14 pairs (_) / F5 = 6  =>  5 pairs (_)
D1,G1: 6.. / D1 = 6  =>  5 pairs (_) / G1 = 6  => 14 pairs (_)
F2,G2: 6.. / F2 = 6  => 14 pairs (_) / G2 = 6  =>  5 pairs (_)
D1,D5: 6.. / D1 = 6  =>  5 pairs (_) / D5 = 6  => 14 pairs (_)
F2,F5: 6.. / F2 = 6  => 14 pairs (_) / F5 = 6  =>  5 pairs (_)
E1,F3: 7.. / E1 = 7  =>  3 pairs (_) / F3 = 7  =>  0 pairs (X)
H7,I8: 7.. / H7 = 7  =>  5 pairs (_) / I8 = 7  =>  0 pairs (X)
E1,H1: 7.. / E1 = 7  =>  3 pairs (_) / H1 = 7  =>  0 pairs (X)
A7,H7: 7.. / A7 = 7  =>  0 pairs (X) / H7 = 7  =>  5 pairs (_)
C5,C8: 7.. / C5 = 7  => 10 pairs (_) / C8 = 7  =>  5 pairs (_)
E1,E6: 7.. / E1 = 7  =>  3 pairs (_) / E6 = 7  =>  0 pairs (X)
I3,I8: 7.. / I3 = 7  =>  5 pairs (_) / I8 = 7  =>  0 pairs (X)
D1,D3: 9.. / D1 = 9  =>  0 pairs (X) / D3 = 9  =>  4 pairs (_)
A7,B7: 9.. / A7 = 9  =>  7 pairs (_) / B7 = 9  =>  6 pairs (_)
B3,D3: 9.. / B3 = 9  =>  0 pairs (X) / D3 = 9  =>  4 pairs (_)
A1,A7: 9.. / A1 = 9  =>  6 pairs (_) / A7 = 9  =>  7 pairs (_)
* DURATION: 0:00:41.799690  START: 05:50:16.717043  END: 05:50:58.516733 2017-05-01
* CP COUNT: (33)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (D6,G2,G8)
* .....2..427.1...9.6.1.5.2...862..7.9....9......9...36...2.6.5.1.6.5.9.2.5..72.9.6
* PAIR G2: 6,8 BLK 3
G1: 6,8,1                                # reduction candidate for 6,8
G1: 6,8                                  # 14 pairs
G1: 1                                    #  5 pairs
* PAIR G2: 6,8 ROW 2
F2: 6,8,3,4                              # reduction candidate for 6,8
F2: 6,8                                  #  8 pairs
F2: 3,4                                  #  6 pairs
* PAIR D6: 4,8 BLK 5
D5: 4,8,3,6                              # reduction candidate for 4,8
D5: 4,8 => CTR
* 9..6821742781436956419572...8623.7.9....96.1...9.7.36.79236.5.186.519427514728936
D5: 3,6                                  #  4 pairs
F5: 4,8,1,3,5,6,7                        # reduction candidate for 4,8
F5: 4,8 => CTR
* ....7261427.1.68956.1.5.2...862.5749...69.15..59..7362..2.6.5.1.6.5.942.51.72.9.6
F5: 1,3,5,6,7                            #  3 pairs
E6: 4,8,1,7                              # reduction candidate for 4,8
E6: 4,8                                  #  8 pairs
F6: 4,8,1,5,7                            # reduction candidate for 4,8
F6: 4,8                                  # 10 pairs
* PAIR D6: 4,8 COL D
D3: 4,8,3,9                              # reduction candidate for 4,8
D3: 4,8 => CTR
* ...972614274136895691.5.237.862..7.9...69.1....9...36.942368571.6.5194235137249.6
D3: 3,9                                  #  4 pairs
D7: 4,8,3                                # reduction candidate for 4,8
D7: 4,8                                  #  6 pairs
* PAIR G8: 4,8 BLK 9
H7: 4,8,3,7                              # reduction candidate for 4,8
H7: 4,8 => CTR
* 93..72154275184693641953278.862..7.9..7.9.8....9..736.792.6.581863519427514728936
H7: 3,7                                  #  4 pairs
H9: 4,8,3                                # reduction candidate for 4,8
H9: 4,8                                  #  6 pairs
* PAIR G8: 4,8 ROW 8
A8: 4,8,1,3,7                            # reduction candidate for 4,8
A8: 4,8                                  #  7 pairs
C8: 4,8,3,7                              # reduction candidate for 4,8
C8: 4,8 => SOLVED
* 895372614273146895641958237486231759327695148159487362932864571768519423514723986
C8: 3,7                                  #  4 pairs
E8: 4,8,1,3                              # reduction candidate for 4,8
E8: 4,8 => CTR
* .5..72..427.1.68956.1.5.2374862137593.5697...719..5362..236.571167589423534721986
E8: 1,3                                  #  4 pairs
* PAIR G8: 4,8 COL G
G5: 4,8,1                                # reduction candidate for 4,8
G5: 1                                    # 14 pairs
G5: 4,8                                  #  5 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0126-base-pr-000.dot
* REASONING
* DIS # D5: 4,8 => CTR => D5: 3,6
* DIS # F5: 4,8 => CTR => F5: 1,3,5,6,7
* DIS # D3: 4,8 => CTR => D3: 3,9
* DIS # H7: 4,8 => CTR => H7: 3,7
* PRF # C8: 4,8 => SOL
* DIS # E8: 4,8 => CTR => E8: 1,3
* CNT   6 HDP CHAINS /  23 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (D6,G2,G8)
* .....2..427.1...9.6.1.5.2...862..7.9....9......9...36...2.6.5.1.6.5.9.2.5..72.9.6
* PAIR G2: 6,8 BLK 3
G1: 6,8,1                                # reduction candidate for 6,8
G1: 6,8                                  # 14 pairs
G1: 1                                    #  5 pairs
* PAIR G2: 6,8 ROW 2
F2: 6,8,3,4                              # reduction candidate for 6,8
F2: 6,8                                  #  8 pairs
F2: 3,4                                  #  6 pairs
* PAIR D6: 4,8 BLK 5
D5: 4,8,3,6                              # reduction candidate for 4,8
D5: 4,8 => CTR
* 9..6821742781436956419572...8623.7.9....96.1...9.7.36.79236.5.186.519427514728936
F5: 4,8,1,3,5,6,7                        # reduction candidate for 4,8
F5: 4,8 => CTR
* ....7261427.1.68956.1.5.2...862.5749...69.15..59..7362..2.6.5.1.6.5.942.51.72.9.6
E6: 4,8,1,7                              # reduction candidate for 4,8
E6: 4,8                                  #  8 pairs
F6: 4,8,1,5,7                            # reduction candidate for 4,8
F6: 4,8                                  # 10 pairs
* PAIR D6: 4,8 COL D
D3: 4,8,3,9                              # reduction candidate for 4,8
D3: 4,8 => CTR
* ...972614274136895691.5.237.862..7.9.2.69.1.8..9...362942368571.6.5194235137249.6
D7: 4,8,3                                # reduction candidate for 4,8
D7: 4,8                                  #  7 pairs
* RESTART
* PAIR D6: 4,8 COL D
D7: 4,8,3                                # reduction candidate for 4,8
D7: 3 => CTR
* ...8.26142781...9.6.1.5.2...862..7.9....9......94..36...236.5.1.6.5.9.2.5..72.9.6
* PAIR RESTART
* PAIR D6: 4,8 BLK 5
E6: 4,8,1,7                              # reduction candidate for 4,8
E6: 4,8                                  # 10 pairs
F6: 4,8,1,5,7                            # reduction candidate for 4,8
F6: 4,8                                  # 12 pairs
* PAIR G8: 4,8 BLK 9
H7: 4,8,3,7                              # reduction candidate for 4,8
H7: 4,8 => CTR
* 9....2..427.1...9.64195728..862..7.9..7.9......9...36.792.635.1863519427514728936
H9: 4,8,3                                # reduction candidate for 4,8
H9: 4,8                                  # 10 pairs
* RESTART
* PAIR G8: 4,8 BLK 9
H9: 4,8,3                                # reduction candidate for 4,8
H9: 3 => CTR
* ....721542751..693641953287.862.57.9...3968....9..736...2.6.571.6.53942851.72.936
* PAIR G8: 4,8 ROW 8
A8: 4,8,1,3,7                            # reduction candidate for 4,8
A8: 4,8                                  # 13 pairs
C8: 4,8,3,7                              # reduction candidate for 4,8
C8: 4,8 => SOLVED
* 895372614273146895641958237486231759327695148159487362932864571768519423514723986
* DURATION: 0:00:16.570022  START: 05:51:15.677942  END: 05:51:32.247964 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0126-base-pr-001.dot
* REASONING
* DIS # D5: 4,8 => CTR => D5: 3,6
* DIS D5: 3,6 # F5: 4,8 => CTR => F5: 1,3,5,6,7
* DIS D5: 3,6 + F5: 1,3,5,6,7 # D3: 4,8 => CTR => D3: 3,9
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 # D7: 3 => CTR => D7: 4,8
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 # H7: 4,8 => CTR => H7: 3,7
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 # H9: 3 => CTR => H9: 4,8
* PRF D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 # C8: 4,8 => SOL
* STA D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 + C8: 4,8
* CNT   7 HDP CHAINS /  20 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=126
```

Solution

position: 895372614273146895641958237486231759327695148159487362932864571768519423514723986 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:

```* INC # G1: 6,8 => UNS
* INC # G1: 1 => UNS
* INC # F2: 6,8 => UNS
* INC # F2: 3,4 => UNS
* DIS # D5: 4,8 => CTR => D5: 3,6
* INC # D5: 3,6 => UNS
* DIS # F5: 4,8 => CTR => F5: 1,3,5,6,7
* INC # F5: 1,3,5,6,7 => UNS
* INC # E6: 4,8 => UNS
* INC # F6: 4,8 => UNS
* DIS # D3: 4,8 => CTR => D3: 3,9
* INC # D3: 3,9 => UNS
* INC # D7: 4,8 => UNS
* DIS # H7: 4,8 => CTR => H7: 3,7
* INC # H7: 3,7 => UNS
* INC # H9: 4,8 => UNS
* INC # A8: 4,8 => UNS
* PRF # C8: 4,8 => SOL
* INC # C8: 3,7 => UNS
* DIS # E8: 4,8 => CTR => E8: 1,3
* INC # E8: 1,3 => UNS
* INC # G5: 4,8 => UNS
* INC # G5: 1 => UNS
* CNT  23 HDP CHAINS /  23 HYP OPENED
```

A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # G1: 6,8 => UNS
* INC # G1: 1 => UNS
* INC # F2: 6,8 => UNS
* INC # F2: 3,4 => UNS
* DIS # D5: 4,8 => CTR => D5: 3,6
* DIS D5: 3,6 # F5: 4,8 => CTR => F5: 1,3,5,6,7
* INC D5: 3,6 + F5: 1,3,5,6,7 # E6: 4,8 => UNS
* INC D5: 3,6 + F5: 1,3,5,6,7 # F6: 4,8 => UNS
* DIS D5: 3,6 + F5: 1,3,5,6,7 # D3: 4,8 => CTR => D3: 3,9
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 # D7: 4,8 => UNS
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 # D7: 4,8 => UNS
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 # D7: 3 => CTR => D7: 4,8
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 # E6: 4,8 => UNS
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 # F6: 4,8 => UNS
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 # H7: 4,8 => CTR => H7: 3,7
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 # H9: 4,8 => UNS
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 # H9: 4,8 => UNS
* DIS D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 # H9: 3 => CTR => H9: 4,8
* INC D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 # A8: 4,8 => UNS
* PRF D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 # C8: 4,8 => SOL
* STA D5: 3,6 + F5: 1,3,5,6,7 + D3: 3,9 + D7: 4,8 + H7: 3,7 + H9: 4,8 + C8: 4,8
* CNT  20 HDP CHAINS /  20 HYP OPENED
```