# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=118

level: medium

position: .5..3..4...82..6..9....5..1...8...36....2....48.......7..4....8..5..97...9..6..2. initial

# Autosolve

position: .5..3..4...82..6..9....5..1...8...36....2....48.......7..4.2..8..5..97...9..6..2. autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # D1: 6,7 => CTR => D1: 1,9
* DIS # F1: 6,7 => CTR => F1: 1,8
* DIS # B3: 6,7 => CTR => B3: 2,3,4
* DIS # C3: 6,7 => CTR => C3: 2,3,4
* DIS # D9: 1,5 => CTR => D9: 3,7
* DIS # D9: 1,3 => CTR => D9: 5,7
* DIS # A8: 1,6 => CTR => A8: 2,3,8
* DIS # B8: 1,6 => CTR => B8: 2,3,4
* CNT   8 HDP CHAINS /  46 HYP OPENED
```

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

## Pair Reduction

The following important HDP chains were detected:

```* DIS # D1: 6,7 => CTR => D1: 1,9
* DIS D1: 1,9 # F1: 6,7 => CTR => F1: 1,8
* DIS D1: 1,9 + F1: 1,8 # D9: 1,5 => CTR => D9: 3,7
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 # A8: 1,8 => CTR => A8: 2,3,6
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 # B8: 1,6 => CTR => B8: 2,3,4
* PRF D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 # H7: 9 => SOL
* STA D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 + H7: 9
* CNT   6 HDP CHAINS /  26 HYP OPENED
```

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

# Details

## Positions

 .5..3..4...82..6..9....5..1...8...36....2....48.......7..4....8..5..97...9..6..2. initial .5..3..4...82..6..9....5..1...8...36....2....48.......7..4.2..8..5..97...9..6..2. autosolve 156938247378214659924675381512847936639521874487396512763452198245189763891763425 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (8)
A2: 1,3
D3: 6,7
H3: 7,8
E7: 1,5
D8: 1,3
E8: 1,8
H8: 1,6
I8: 3,4

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
A8,B8: 2.. / A8 = 2  => 11 pairs (_) / B8 = 2  =>  9 pairs (_)
I1,I6: 2.. / I1 = 2  => 11 pairs (_) / I6 = 2  =>  9 pairs (_)
I2,G3: 3.. / I2 = 3  => 24 pairs (_) / G3 = 3  => 11 pairs (_)
B8,C9: 4.. / B8 = 4  => 17 pairs (_) / C9 = 4  =>  8 pairs (_)
B8,I8: 4.. / B8 = 4  => 17 pairs (_) / I8 = 4  =>  8 pairs (_)
C3,C9: 4.. / C3 = 4  => 17 pairs (_) / C9 = 4  =>  8 pairs (_)
H2,I2: 5.. / H2 = 5  =>  8 pairs (_) / I2 = 5  => 15 pairs (_)
A4,A5: 5.. / A4 = 5  =>  8 pairs (_) / A5 = 5  =>  9 pairs (_)
E7,D9: 5.. / E7 = 5  =>  7 pairs (_) / D9 = 5  =>  0 pairs (X)
H7,H8: 6.. / H7 = 6  =>  0 pairs (X) / H8 = 6  =>  7 pairs (_)
D9,F9: 7.. / D9 = 7  =>  7 pairs (_) / F9 = 7  => 14 pairs (_)
F1,E3: 8.. / F1 = 8  =>  9 pairs (_) / E3 = 8  => 18 pairs (_)
G5,H5: 8.. / G5 = 8  =>  9 pairs (_) / H5 = 8  =>  9 pairs (_)
A8,A9: 8.. / A8 = 8  => 18 pairs (_) / A9 = 8  =>  9 pairs (_)
E8,F9: 8.. / E8 = 8  =>  9 pairs (_) / F9 = 8  => 18 pairs (_)
F1,G1: 8.. / F1 = 8  =>  9 pairs (_) / G1 = 8  => 18 pairs (_)
A8,E8: 8.. / A8 = 8  => 18 pairs (_) / E8 = 8  =>  9 pairs (_)
A9,F9: 8.. / A9 = 8  =>  9 pairs (_) / F9 = 8  => 18 pairs (_)
E3,E8: 8.. / E3 = 8  => 18 pairs (_) / E8 = 8  =>  9 pairs (_)
F1,F9: 8.. / F1 = 8  =>  9 pairs (_) / F9 = 8  => 18 pairs (_)
H3,H5: 8.. / H3 = 8  =>  9 pairs (_) / H5 = 8  =>  9 pairs (_)
D1,E2: 9.. / D1 = 9  => 15 pairs (_) / E2 = 9  =>  9 pairs (_)
G7,H7: 9.. / G7 = 9  => 11 pairs (_) / H7 = 9  =>  8 pairs (_)
* DURATION: 0:00:25.106729  START: 05:38:35.551006  END: 05:39:00.657735 2017-05-01
* CP COUNT: (23)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A2,D3,D8,E7,E8,H3,H8,I8)
* .5..3..4...82..6..9....5..1...8...36....2....48.......7..4.2..8..5..97...9..6..2.
* PAIR A2: 1,3 BLK 1
B2: 1,3,4,7                              # reduction candidate for 1,3
B2: 1,3                                  # 17 pairs
B2: 4,7                                  #  9 pairs
* PAIR A2: 1,3 COL A
A5: 1,3,5,6                              # reduction candidate for 1,3
A5: 1,3                                  # 12 pairs
A8: 1,3,2,6,8                            # reduction candidate for 1,3
A8: 1,3                                  # 13 pairs
A9: 1,3,8                                # reduction candidate for 1,3
A9: 1,3                                  # 18 pairs
* PAIR D3: 6,7 BLK 2
D1: 6,7,1,9                              # reduction candidate for 6,7
D1: 6,7 => CTR
* .5..38.4.3482916..9...45381...8.4.36....2....48.......7..4.2..8..5.897..894.67.2.
D1: 1,9                                  #  9 pairs
F1: 6,7,1,8                              # reduction candidate for 6,7
F1: 6,7 => CTR
* .5.93784237824165994268537121.8.45365.912.48748.5.....7..452..8825319764.94768.2.
F1: 1,8                                  #  9 pairs
* PAIR D3: 6,7 ROW 3
B3: 6,7,2,3,4                            # reduction candidate for 6,7
B3: 6,7 => CTR
* .5..38947148297653973645281...8..436....248..48.3.65.27..452398..5189764894763125
B3: 2,3,4                                #  8 pairs
C3: 6,7,2,3,4                            # reduction candidate for 6,7
C3: 6,7 => CTR
* .5..38947148297653937645281...8..436....248..48...65.27..452398325189764894763125
C3: 2,3,4                                #  8 pairs
* PAIR D3: 6,7 COL D
D5: 6,7,1,3,5,9                          # reduction candidate for 6,7
D5: 6,7                                  # 14 pairs
D6: 6,7,1,3,5,9                          # reduction candidate for 6,7
D6: 6,7                                  # 18 pairs
* PAIR H3: 7,8 ROW 3
E3: 7,8,4                                # reduction candidate for 7,8
E3: 7,8                                  #  9 pairs
E3: 4                                    #  9 pairs
* PAIR H3: 7,8 COL H
H5: 7,8,1,5,9                            # reduction candidate for 7,8
H5: 7,8                                  # 10 pairs
H5: 1,5,9                                #  9 pairs
* PAIR E7: 1,5 BLK 8
D9: 1,5,3,7                              # reduction candidate for 1,5
D9: 1,5 => CTR
* .5..38.4...82..6539..645871...8.4.363..52648.486..35.27..452..8..53897.4894167325
D9: 3,7                                  #  8 pairs
* PAIR E7: 1,5 ROW 7
G7: 1,5,3,9                              # reduction candidate for 1,5
G7: 1,5                                  # 13 pairs
H7: 1,5,6,9                              # reduction candidate for 1,5
H7: 1,5                                  # 13 pairs
* PAIR E7: 1,5 COL E
E4: 1,5,4,7,9                            # reduction candidate for 1,5
E4: 1,5                                  # 12 pairs
E6: 1,5,7,9                              # reduction candidate for 1,5
E6: 1,5                                  # 10 pairs
* PAIR D8: 1,3 BLK 8
D9: 1,3,5,7                              # reduction candidate for 1,3
D9: 1,3 => CTR
* 15..38.4.3782..6599.4.75381...8...36....2.8.448.......7..452..8.45.897.3891367425
D9: 5,7                                  #  9 pairs
F9: 1,3,7,8                              # reduction candidate for 1,3
F9: 1,3                                  #  9 pairs
* PAIR D8: 1,3 ROW 8
A8: 1,3,2,6,8                            # reduction candidate for 1,3
A8: 1,3                                  # 13 pairs
B8: 1,3,2,4,6                            # reduction candidate for 1,3
B8: 1,3                                  # 11 pairs
* PAIR D8: 1,3 COL D
D5: 1,3,5,6,7,9                          # reduction candidate for 1,3
D5: 1,3                                  # 10 pairs
D6: 1,3,5,6,7,9                          # reduction candidate for 1,3
D6: 1,3                                  # 10 pairs
* PAIR E8: 1,8 BLK 8
F9: 1,8,3,7                              # reduction candidate for 1,8
F9: 1,8                                  # 13 pairs
F9: 3,7                                  # 10 pairs
* PAIR E8: 1,8 ROW 8
A8: 1,8,2,3,6                            # reduction candidate for 1,8
A8: 1,8                                  # 11 pairs
A8: 2,3,6                                #  9 pairs
* PAIR H8: 1,6 BLK 9
H7: 1,6,5,9                              # reduction candidate for 1,6
H7: 1,6                                  # 11 pairs
H7: 5,9                                  #  8 pairs
* PAIR H8: 1,6 ROW 8
A8: 1,6,2,3,8                            # reduction candidate for 1,6
A8: 1,6 => CTR
* .5..38.4...82..6539.26458712..8.4.36....2.48.48.......7..4.2..8.253897.4894.6.325
A8: 2,3,8                                #  8 pairs
B8: 1,6,2,3,4                            # reduction candidate for 1,6
B8: 1,6 => CTR
* 65..382491482976539..6458715..8.4.363.95264874869735127..4521982153897.4894.6.325
B8: 2,3,4                                #  8 pairs
* PAIR I8: 3,4 BLK 9
G9: 3,4,1,5                              # reduction candidate for 3,4
G9: 3,4                                  # 11 pairs
I9: 3,4,5                                # reduction candidate for 3,4
I9: 3,4                                  # 13 pairs
* PAIR I8: 3,4 ROW 8
B8: 3,4,1,2,6                            # reduction candidate for 3,4
B8: 3,4                                  # 17 pairs
B8: 1,2,6                                #  8 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0118-base-pr-000.dot
* REASONING
* DIS # D1: 6,7 => CTR => D1: 1,9
* DIS # F1: 6,7 => CTR => F1: 1,8
* DIS # B3: 6,7 => CTR => B3: 2,3,4
* DIS # C3: 6,7 => CTR => C3: 2,3,4
* DIS # D9: 1,5 => CTR => D9: 3,7
* DIS # D9: 1,3 => CTR => D9: 5,7
* DIS # A8: 1,6 => CTR => A8: 2,3,8
* DIS # B8: 1,6 => CTR => B8: 2,3,4
* CNT   8 HDP CHAINS /  46 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A2,D3,D8,E7,E8,H3,H8,I8)
* .5..3..4...82..6..9....5..1...8...36....2....48.......7..4.2..8..5..97...9..6..2.
* PAIR A2: 1,3 BLK 1
B2: 1,3,4,7                              # reduction candidate for 1,3
B2: 1,3                                  # 17 pairs
B2: 4,7                                  #  9 pairs
* PAIR A2: 1,3 COL A
A5: 1,3,5,6                              # reduction candidate for 1,3
A5: 1,3                                  # 12 pairs
A8: 1,3,2,6,8                            # reduction candidate for 1,3
A8: 1,3                                  # 13 pairs
A9: 1,3,8                                # reduction candidate for 1,3
A9: 1,3                                  # 18 pairs
* PAIR D3: 6,7 BLK 2
D1: 6,7,1,9                              # reduction candidate for 6,7
D1: 6,7 => CTR
* .5..38.4.3482916..9...45381...8.4.36....2....48.......7..4.2..8..5.897..894.67.2.
F1: 6,7,1,8                              # reduction candidate for 6,7
F1: 6,7 => CTR
* .5.93784237824165994268537121.8.45365.912.48748.5.....7..452..8825319764.94768.2.
* PAIR H3: 7,8 ROW 3
E3: 7,8,4                                # reduction candidate for 7,8
E3: 7,8                                  # 11 pairs
E3: 4                                    # 10 pairs
* PAIR H3: 7,8 COL H
H5: 7,8,1,5,9                            # reduction candidate for 7,8
H5: 7,8                                  # 11 pairs
H5: 1,5,9                                #  9 pairs
* PAIR E7: 1,5 BLK 8
D9: 1,5,3,7                              # reduction candidate for 1,5
D9: 1,5 => CTR
* .57.38.4...827.65.9..6.5871...8...36....2.48.48.......7..4.2..8..53897.4894.67325
* PAIR D8: 1,3 BLK 8
F9: 1,3,7,8                              # reduction candidate for 1,3
F9: 7,8                                  # 13 pairs
F9: 1,3                                  #  9 pairs
* PAIR D8: 1,3 ROW 8
A8: 1,3,2,6,8                            # reduction candidate for 1,3
A8: 1,3                                  # 14 pairs
B8: 1,3,2,4,6                            # reduction candidate for 1,3
B8: 1,3                                  # 12 pairs
* PAIR D8: 1,3 COL D
D5: 1,3,5,7,9                            # reduction candidate for 1,3
D5: 1,3                                  # 19 pairs
D6: 1,3,5,7,9                            # reduction candidate for 1,3
D6: 1,3                                  # 16 pairs
* PAIR E8: 1,8 BLK 8
F9: 1,8,3,7                              # reduction candidate for 1,8
F9: 3,7                                  # 17 pairs
F9: 1,8                                  # 17 pairs
* PAIR E8: 1,8 ROW 8
A8: 1,8,2,3,6                            # reduction candidate for 1,8
A8: 1,8 => CTR
* 657.3..4...82..6539326458712..8.45365...2.48.48.5.....7..452..8.25389764.9476..25
* PAIR RESTART
* PAIR H8: 1,6 BLK 9
H7: 1,6,9                                # reduction candidate for 1,6
H7: 1,6                                  # 15 pairs
H7: 9                                    # 11 pairs
* PAIR H8: 1,6 ROW 8
B8: 1,6,2,3,4                            # reduction candidate for 1,6
B8: 1,6 => CTR
* 6571382491482976539..6458715..8.4.363.95264874869735127..4521982153897.4894761325
* PAIR RESTART
* PAIR H8: 1,6 BLK 9
H7: 1,6,9                                # reduction candidate for 1,6
H7: 9 => SOLVED
* 156938247378214659924675381512847936639521874487396512763452198245189763891763425
* DURATION: 0:00:22.261383  START: 05:39:29.225063  END: 05:39:51.486446 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0118-base-pr-001.dot
* REASONING
* DIS # D1: 6,7 => CTR => D1: 1,9
* DIS D1: 1,9 # F1: 6,7 => CTR => F1: 1,8
* DIS D1: 1,9 + F1: 1,8 # D9: 1,5 => CTR => D9: 3,7
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 # A8: 1,8 => CTR => A8: 2,3,6
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 # B8: 1,6 => CTR => B8: 2,3,4
* PRF D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 # H7: 9 => SOL
* STA D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 + H7: 9
* CNT   6 HDP CHAINS /  26 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=118
```

# Solution

position: 156938247378214659924675381512847936639521874487396512763452198245189763891763425 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 # B2: 1,3 => UNS
* INC # B2: 4,7 => UNS
* INC # A5: 1,3 => UNS
* INC # A8: 1,3 => UNS
* INC # A9: 1,3 => UNS
* DIS # D1: 6,7 => CTR => D1: 1,9
* INC # D1: 1,9 => UNS
* DIS # F1: 6,7 => CTR => F1: 1,8
* INC # F1: 1,8 => UNS
* DIS # B3: 6,7 => CTR => B3: 2,3,4
* INC # B3: 2,3,4 => UNS
* DIS # C3: 6,7 => CTR => C3: 2,3,4
* INC # C3: 2,3,4 => UNS
* INC # D5: 6,7 => UNS
* INC # D6: 6,7 => UNS
* INC # E3: 7,8 => UNS
* INC # E3: 4 => UNS
* INC # H5: 7,8 => UNS
* INC # H5: 1,5,9 => UNS
* DIS # D9: 1,5 => CTR => D9: 3,7
* INC # D9: 3,7 => UNS
* INC # G7: 1,5 => UNS
* INC # H7: 1,5 => UNS
* INC # E4: 1,5 => UNS
* INC # E6: 1,5 => UNS
* DIS # D9: 1,3 => CTR => D9: 5,7
* INC # D9: 5,7 => UNS
* INC # F9: 1,3 => UNS
* INC # A8: 1,3 => UNS
* INC # B8: 1,3 => UNS
* INC # D5: 1,3 => UNS
* INC # D6: 1,3 => UNS
* INC # F9: 1,8 => UNS
* INC # F9: 3,7 => UNS
* INC # A8: 1,8 => UNS
* INC # A8: 2,3,6 => UNS
* INC # H7: 1,6 => UNS
* INC # H7: 5,9 => UNS
* DIS # A8: 1,6 => CTR => A8: 2,3,8
* INC # A8: 2,3,8 => UNS
* DIS # B8: 1,6 => CTR => B8: 2,3,4
* INC # B8: 2,3,4 => UNS
* INC # G9: 3,4 => UNS
* INC # I9: 3,4 => UNS
* INC # B8: 3,4 => UNS
* INC # B8: 1,2,6 => UNS
* CNT  46 HDP CHAINS /  46 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # B2: 1,3 => UNS
* INC # B2: 4,7 => UNS
* INC # A5: 1,3 => UNS
* INC # A8: 1,3 => UNS
* INC # A9: 1,3 => UNS
* DIS # D1: 6,7 => CTR => D1: 1,9
* DIS D1: 1,9 # F1: 6,7 => CTR => F1: 1,8
* INC D1: 1,9 + F1: 1,8 # E3: 7,8 => UNS
* INC D1: 1,9 + F1: 1,8 # E3: 4 => UNS
* INC D1: 1,9 + F1: 1,8 # H5: 7,8 => UNS
* INC D1: 1,9 + F1: 1,8 # H5: 1,5,9 => UNS
* DIS D1: 1,9 + F1: 1,8 # D9: 1,5 => CTR => D9: 3,7
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # F9: 1,3 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # F9: 7,8 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # A8: 1,3 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # B8: 1,3 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # D5: 1,3 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # D6: 1,3 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # F9: 1,8 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 # F9: 3,7 => UNS
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 # A8: 1,8 => CTR => A8: 2,3,6
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 # H7: 1,6 => UNS
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 # H7: 9 => UNS
* DIS D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 # B8: 1,6 => CTR => B8: 2,3,4
* INC D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 # H7: 1,6 => UNS
* PRF D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 # H7: 9 => SOL
* STA D1: 1,9 + F1: 1,8 + D9: 3,7 + A8: 2,3,6 + B8: 2,3,4 + H7: 9
* CNT  26 HDP CHAINS /  26 HYP OPENED
```