Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=148

level: medium

position: .....3..1...1..2...57........4.3....9...6..74....5.3...8....96...9..2...3..4....5 initial

Autosolve

position: .....3..1..31..2..157.2......4.3....93..6..74....543..48....96...9..2...3..4....5 autosolve

Pair Reduction Variants

Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # F2: 6,8 => CTR => F2: 5,7,9
* DIS # E8: 1,7 => CTR => E8: 8
* DIS # E9: 1,7 => CTR => E9: 8,9
* DIS # D7: 5,7 => CTR => D7: 3
* DIS # F2: 6,8,9 => CTR => F2: 5,7
* CNT   5 HDP CHAINS /  28 HYP OPENED
```

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

Pair Reduction

The following important HDP chains were detected:

```* DIS # F2: 6,8 => CTR => F2: 5,7,9
* DIS F2: 5,7,9 # E8: 1,7 => CTR => E8: 8
* DIS F2: 5,7,9 + E8: 8 # E9: 1,7 => CTR => E9: 9
* DIS F2: 5,7,9 + E8: 8 + E9: 9 # D7: 5,7 => CTR => D7: 3
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 # D8: 6 => CTR => D8: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 # A1: 6,8 => CTR => A1: 2
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 # I2: 7,9 => CTR => I2: 6,8
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 # A4: 6,8 => CTR => A4: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 # A6: 6,8 => CTR => A6: 7
* PRF F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7 => SOL
* STA F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7
* CNT  10 HDP CHAINS /  37 HYP OPENED
```

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

Details

Positions

 .....3..1...1..2...57........4.3....9...6..74....5.3...8....96...9..2...3..4....5 initial .....3..1..31..2..157.2......4.3....93..6..74....543..48....96...9..2...3..4....5 autosolve 296843751843175296157629843524738619938261574761954328485317962619582437372496185 solved

level: medium

Pairing Analysis

```--------------------------------------------------
* PAIRS (9)
B1: 4,9
A2: 6,8
B2: 4,9
H3: 3,4
D5: 2,8
F5: 1,8
E7: 1,7
F7: 5,7
H8: 3,4

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
F4,F5: 1.. / F4 = 1  => 14 pairs (_) / F5 = 1  =>  9 pairs (_)
C7,E7: 1.. / C7 = 1  =>  0 pairs (X) / E7 = 1  => 10 pairs (_)
A1,C1: 2.. / A1 = 2  => 10 pairs (_) / C1 = 2  => 20 pairs (_)
I7,H9: 2.. / I7 = 2  => 12 pairs (_) / H9 = 2  =>  0 pairs (X)
C5,D5: 2.. / C5 = 2  =>  0 pairs (X) / D5 = 2  =>  8 pairs (_)
C7,I7: 2.. / C7 = 2  =>  0 pairs (X) / I7 = 2  => 12 pairs (_)
H3,I3: 3.. / H3 = 3  =>  7 pairs (_) / I3 = 3  => 11 pairs (_)
D7,D8: 3.. / D7 = 3  => 10 pairs (_) / D8 = 3  =>  0 pairs (X)
D7,I7: 3.. / D7 = 3  => 10 pairs (_) / I7 = 3  =>  0 pairs (X)
H3,H8: 3.. / H3 = 3  =>  7 pairs (_) / H8 = 3  => 11 pairs (_)
B1,B2: 4.. / B1 = 4  =>  8 pairs (_) / B2 = 4  =>  8 pairs (_)
E1,E2: 4.. / E1 = 4  =>  8 pairs (_) / E2 = 4  =>  8 pairs (_)
G3,H3: 4.. / G3 = 4  =>  7 pairs (_) / H3 = 4  => 11 pairs (_)
G8,H8: 4.. / G8 = 4  => 11 pairs (_) / H8 = 4  =>  7 pairs (_)
B1,E1: 4.. / B1 = 4  =>  8 pairs (_) / E1 = 4  =>  8 pairs (_)
B2,E2: 4.. / B2 = 4  =>  8 pairs (_) / E2 = 4  =>  8 pairs (_)
G3,G8: 4.. / G3 = 4  =>  7 pairs (_) / G8 = 4  => 11 pairs (_)
H3,H8: 4.. / H3 = 4  => 11 pairs (_) / H8 = 4  =>  7 pairs (_)
D1,F2: 5.. / D1 = 5  => 11 pairs (_) / F2 = 5  => 17 pairs (_)
A4,C5: 5.. / A4 = 5  => 15 pairs (_) / C5 = 5  => 10 pairs (_)
C7,A8: 5.. / C7 = 5  => 15 pairs (_) / A8 = 5  => 10 pairs (_)
F2,H2: 5.. / F2 = 5  => 17 pairs (_) / H2 = 5  => 11 pairs (_)
C5,G5: 5.. / C5 = 5  => 10 pairs (_) / G5 = 5  => 15 pairs (_)
A8,D8: 5.. / A8 = 5  => 10 pairs (_) / D8 = 5  => 15 pairs (_)
A4,A8: 5.. / A4 = 5  => 15 pairs (_) / A8 = 5  => 10 pairs (_)
C5,C7: 5.. / C5 = 5  => 10 pairs (_) / C7 = 5  => 15 pairs (_)
F2,F7: 5.. / F2 = 5  => 17 pairs (_) / F7 = 5  => 11 pairs (_)
D8,F9: 6.. / D8 = 6  => 13 pairs (_) / F9 = 6  => 12 pairs (_)
G1,I2: 7.. / G1 = 7  => 10 pairs (_) / I2 = 7  => 12 pairs (_)
B1,B2: 9.. / B1 = 9  =>  8 pairs (_) / B2 = 9  =>  8 pairs (_)
E9,F9: 9.. / E9 = 9  =>  9 pairs (_) / F9 = 9  =>  0 pairs (X)
* DURATION: 0:00:44.578695  START: 06:12:17.282304  END: 06:13:01.860999 2017-05-01
* CP COUNT: (31)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A2,B1,B2,D5,E7,F5,F7,H3,H8)
* .....3..1..31..2..157.2......4.3....93..6..74....543..48....96...9..2...3..4....5
* PAIR B1: 4,9 ROW 1
E1: 4,9,7,8                              # reduction candidate for 4,9
E1: 4,9                                  # 11 pairs
E1: 7,8                                  #  9 pairs
* PAIR A2: 6,8 BLK 1
A1: 6,8,2                                # reduction candidate for 6,8
A1: 6,8                                  # 20 pairs
C1: 6,8,2                                # reduction candidate for 6,8
C1: 6,8                                  # 10 pairs
* PAIR A2: 6,8 ROW 2
F2: 6,8,5,7,9                            # reduction candidate for 6,8
F2: 6,8 => CTR
* 2..5.3..1..31..25.157.2......4.3.5..935268174....543..482.1596.519682...37.49.825
F2: 5,7,9                                #  9 pairs
I2: 6,8,7,9                              # reduction candidate for 6,8
I2: 6,8                                  # 12 pairs
* PAIR A2: 6,8 COL A
A4: 6,8,2,5,7                            # reduction candidate for 6,8
A4: 6,8                                  # 13 pairs
A6: 6,8,2,7                              # reduction candidate for 6,8
A6: 6,8                                  # 13 pairs
* PAIR B2: 4,9 ROW 2
E2: 4,9,7,8                              # reduction candidate for 4,9
E2: 4,9                                  # 11 pairs
E2: 7,8                                  #  9 pairs
* PAIR D5: 2,8 BLK 5
D4: 2,8,7,9                              # reduction candidate for 2,8
D4: 2,8                                  # 13 pairs
D6: 2,8,7,9                              # reduction candidate for 2,8
D6: 2,8                                  # 13 pairs
* PAIR D5: 2,8 ROW 5
C5: 2,8,1,5                              # reduction candidate for 2,8
C5: 2,8                                  # 16 pairs
C5: 1,5                                  #  9 pairs
* PAIR F5: 1,8 BLK 5
F4: 1,8,7,9                              # reduction candidate for 1,8
F4: 1,8                                  # 14 pairs
F4: 7,9                                  # 11 pairs
* PAIR F5: 1,8 ROW 5
C5: 1,8,2,5                              # reduction candidate for 1,8
C5: 1,8                                  # 15 pairs
G5: 1,8,5                                # reduction candidate for 1,8
G5: 1,8                                  # 10 pairs
* PAIR E7: 1,7 BLK 8
E8: 1,7,8                                # reduction candidate for 1,7
E8: 1,7 => CTR
* ...5.37.1..31.725.157.2.643..4.3.5..93526..74....543..48.3.596.5.96.24383764..125
E8: 8                                    # 10 pairs
E9: 1,7,8,9                              # reduction candidate for 1,7
E9: 1,7 => CTR
* ...5.37.1..31.725.157826439..4.3.5..93526..74....543..48.3.596.5.96821433..4.98.5
E9: 8,9                                  # 10 pairs
* PAIR F7: 5,7 BLK 8
D7: 5,7,3                                # reduction candidate for 5,7
D7: 5,7 => CTR
* 2....37.1..31..2..157.2.43...4.3.65.93..6..74....543..482.1.9635693.214.371496825
D7: 3                                    # 10 pairs
D8: 5,7,3,6,8                            # reduction candidate for 5,7
D8: 5,7                                  # 16 pairs
* PAIR F7: 5,7 COL F
F2: 5,7,6,8,9                            # reduction candidate for 5,7
F2: 6,8,9 => CTR
* 2..5.3..1..31..25.157.2......4.3.5..935268174....543..482.1596.519682...37.49.825
F2: 5,7                                  # 11 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0148-base-pr-000.dot
* REASONING
* DIS # F2: 6,8 => CTR => F2: 5,7,9
* DIS # E8: 1,7 => CTR => E8: 8
* DIS # E9: 1,7 => CTR => E9: 8,9
* DIS # D7: 5,7 => CTR => D7: 3
* DIS # F2: 6,8,9 => CTR => F2: 5,7
* CNT   5 HDP CHAINS /  28 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A2,B1,B2,D5,E7,F5,F7,H3,H8)
* .....3..1..31..2..157.2......4.3....93..6..74....543..48....96...9..2...3..4....5
* PAIR B1: 4,9 ROW 1
E1: 4,9,7,8                              # reduction candidate for 4,9
E1: 4,9                                  # 11 pairs
E1: 7,8                                  #  9 pairs
* PAIR A2: 6,8 BLK 1
A1: 6,8,2                                # reduction candidate for 6,8
A1: 6,8                                  # 20 pairs
C1: 6,8,2                                # reduction candidate for 6,8
C1: 6,8                                  # 10 pairs
* PAIR A2: 6,8 ROW 2
F2: 6,8,5,7,9                            # reduction candidate for 6,8
F2: 6,8 => CTR
* 2..5.3..1..31..25.157.2......4.3.5..935268174....543..482.1596.519682...37.49.825
I2: 6,8,7,9                              # reduction candidate for 6,8
I2: 6,8                                  # 12 pairs
* RESTART
* PAIR A2: 6,8 ROW 2
I2: 6,8,7,9                              # reduction candidate for 6,8
I2: 7,9                                  # 13 pairs
I2: 6,8                                  # 12 pairs
* PAIR A2: 6,8 COL A
A4: 6,8,2,5,7                            # reduction candidate for 6,8
A4: 6,8                                  # 13 pairs
A6: 6,8,2,7                              # reduction candidate for 6,8
A6: 6,8                                  # 13 pairs
* PAIR RESTART
* PAIR A2: 6,8 BLK 1
A1: 6,8,2                                # reduction candidate for 6,8
A1: 6,8                                  # 21 pairs
C1: 6,8,2                                # reduction candidate for 6,8
C1: 6,8                                  # 10 pairs
* PAIR A2: 6,8 ROW 2
I2: 6,8,7,9                              # reduction candidate for 6,8
I2: 7,9                                  # 13 pairs
I2: 6,8                                  # 12 pairs
* PAIR A2: 6,8 COL A
A4: 6,8,2,5,7                            # reduction candidate for 6,8
A4: 6,8                                  # 13 pairs
A6: 6,8,2,7                              # reduction candidate for 6,8
A6: 6,8                                  # 13 pairs
* PAIR B2: 4,9 ROW 2
E2: 4,9,7,8                              # reduction candidate for 4,9
E2: 4,9                                  # 13 pairs
E2: 7,8                                  #  9 pairs
* PAIR D5: 2,8 BLK 5
D4: 2,8,7,9                              # reduction candidate for 2,8
D4: 2,8                                  # 15 pairs
D6: 2,8,7,9                              # reduction candidate for 2,8
D6: 2,8                                  # 15 pairs
* PAIR D5: 2,8 ROW 5
C5: 2,8,1,5                              # reduction candidate for 2,8
C5: 2,8                                  # 16 pairs
C5: 1,5                                  #  9 pairs
* PAIR F5: 1,8 BLK 5
F4: 1,8,7,9                              # reduction candidate for 1,8
F4: 1,8                                  # 14 pairs
F4: 7,9                                  # 13 pairs
* PAIR F5: 1,8 ROW 5
C5: 1,8,2,5                              # reduction candidate for 1,8
C5: 1,8                                  # 15 pairs
G5: 1,8,5                                # reduction candidate for 1,8
G5: 1,8                                  # 10 pairs
* PAIR E7: 1,7 BLK 8
E8: 1,7,8                                # reduction candidate for 1,7
E8: 1,7 => CTR
* ...5.37.1..31.725.157.2.643..4.3.5..93526..74....543..48.3.596.5.96.24383764..125
E9: 1,7,9                                # reduction candidate for 1,7
E9: 1,7 => CTR
* ...5.3..1..31.725.157826439..4.3.5..93526..74....543..48.3.596.5.9682.433..4.9..5
* PAIR F7: 5,7 BLK 8
D7: 5,7,3                                # reduction candidate for 5,7
D7: 5,7 => CTR
* .....37.1..317.2..157.2.43...4.3.6..93..6.574....543..48..1.963..9382.4737.496.25
D8: 5,7,6                                # reduction candidate for 5,7
D8: 5,7                                  # 17 pairs
* RESTART
* PAIR F7: 5,7 BLK 8
D8: 5,7,6                                # reduction candidate for 5,7
D8: 6 => CTR
* .9.5437818431792561578264.9..4.3....93..6..74....543..48.3159675796821433..497825

* LEVEL 0 PASS 1 ROUND 2 (AUTO SOLVE) (A2,B1,B2,C7,C9,D5,D8,E1,E2,F2,F3,F5,F7,H3,H8,I7,I8)
* .....3..1..31..2..157.2......4.3....93..6..74....543..48.31.96...9.82...3..496..5
* PAIR A2: 6,8 BLK 1
A1: 6,8,2                                # reduction candidate for 6,8
A1: 6,8 => CTR
* ..2.73..1..314529.157.2......4.3....93.26..74..6.543..485317962.69582143321496785
* RESTART
* PAIR A2: 6,8 ROW 2
I2: 6,8,7,9                              # reduction candidate for 6,8
I2: 7,9 => CTR
* 2.8..37516.3175289157.2.436..4.3.6..93..6.574..6.543..48.31.96..69.821433..496825
* PAIR A2: 6,8 COL A
A4: 6,8,5,7                              # reduction candidate for 6,8
A4: 6,8 => CTR
* 29..437.1.4317.2..157.2......4.3.5..93526..747...543..4823159675697821433714968.5
A6: 6,8,7                                # reduction candidate for 6,8
A6: 6,8 => CTR
* 29..437.1.4317.2..157.2....7.4.3.5..93526..74...7543..48231.96...9.82...37.496..5
A6: 7 => SOLVED
* 296843751843175296157629843524738619938261574761954328485317962619582437372496185
* DURATION: 0:00:32.729451  START: 06:13:24.219337  END: 06:13:56.948788 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0148-base-pr-001.dot
* REASONING
* DIS # F2: 6,8 => CTR => F2: 5,7,9
* DIS F2: 5,7,9 # E8: 1,7 => CTR => E8: 8
* DIS F2: 5,7,9 + E8: 8 # E9: 1,7 => CTR => E9: 9
* DIS F2: 5,7,9 + E8: 8 + E9: 9 # D7: 5,7 => CTR => D7: 3
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 # D8: 6 => CTR => D8: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 # A1: 6,8 => CTR => A1: 2
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 # I2: 7,9 => CTR => I2: 6,8
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 # A4: 6,8 => CTR => A4: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 # A6: 6,8 => CTR => A6: 7
* PRF F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7 => SOL
* STA F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7
* CNT  10 HDP CHAINS /  37 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=148
```

Solution

position: 296843751843175296157629843524738619938261574761954328485317962619582437372496185 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 # E1: 4,9 => UNS
* INC # E1: 7,8 => UNS
* INC # A1: 6,8 => UNS
* INC # C1: 6,8 => UNS
* DIS # F2: 6,8 => CTR => F2: 5,7,9
* INC # F2: 5,7,9 => UNS
* INC # I2: 6,8 => UNS
* INC # A4: 6,8 => UNS
* INC # A6: 6,8 => UNS
* INC # E2: 4,9 => UNS
* INC # E2: 7,8 => UNS
* INC # D4: 2,8 => UNS
* INC # D6: 2,8 => UNS
* INC # C5: 2,8 => UNS
* INC # C5: 1,5 => UNS
* INC # F4: 1,8 => UNS
* INC # F4: 7,9 => UNS
* INC # C5: 1,8 => UNS
* INC # G5: 1,8 => UNS
* DIS # E8: 1,7 => CTR => E8: 8
* INC # E8: 8 => UNS
* DIS # E9: 1,7 => CTR => E9: 8,9
* INC # E9: 8,9 => UNS
* DIS # D7: 5,7 => CTR => D7: 3
* INC # D7: 3 => UNS
* INC # D8: 5,7 => UNS
* INC # F2: 5,7 => UNS
* DIS # F2: 6,8,9 => CTR => F2: 5,7
* CNT  28 HDP CHAINS /  28 HYP OPENED
```

A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # E1: 4,9 => UNS
* INC # E1: 7,8 => UNS
* INC # A1: 6,8 => UNS
* INC # C1: 6,8 => UNS
* DIS # F2: 6,8 => CTR => F2: 5,7,9
* INC F2: 5,7,9 # I2: 6,8 => UNS
* INC F2: 5,7,9 # I2: 6,8 => UNS
* INC F2: 5,7,9 # I2: 7,9 => UNS
* INC F2: 5,7,9 # A4: 6,8 => UNS
* INC F2: 5,7,9 # A6: 6,8 => UNS
* INC F2: 5,7,9 # A1: 6,8 => UNS
* INC F2: 5,7,9 # C1: 6,8 => UNS
* INC F2: 5,7,9 # I2: 6,8 => UNS
* INC F2: 5,7,9 # I2: 7,9 => UNS
* INC F2: 5,7,9 # A4: 6,8 => UNS
* INC F2: 5,7,9 # A6: 6,8 => UNS
* INC F2: 5,7,9 # E2: 4,9 => UNS
* INC F2: 5,7,9 # E2: 7,8 => UNS
* INC F2: 5,7,9 # D4: 2,8 => UNS
* INC F2: 5,7,9 # D6: 2,8 => UNS
* INC F2: 5,7,9 # C5: 2,8 => UNS
* INC F2: 5,7,9 # C5: 1,5 => UNS
* INC F2: 5,7,9 # F4: 1,8 => UNS
* INC F2: 5,7,9 # F4: 7,9 => UNS
* INC F2: 5,7,9 # C5: 1,8 => UNS
* INC F2: 5,7,9 # G5: 1,8 => UNS
* DIS F2: 5,7,9 # E8: 1,7 => CTR => E8: 8
* DIS F2: 5,7,9 + E8: 8 # E9: 1,7 => CTR => E9: 9
* DIS F2: 5,7,9 + E8: 8 + E9: 9 # D7: 5,7 => CTR => D7: 3
* INC F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 # D8: 5,7 => UNS
* INC F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 # D8: 5,7 => UNS
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 # D8: 6 => CTR => D8: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 # A1: 6,8 => CTR => A1: 2
* INC F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 # I2: 6,8 => UNS
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 # I2: 7,9 => CTR => I2: 6,8
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 # A4: 6,8 => CTR => A4: 5,7
* DIS F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 # A6: 6,8 => CTR => A6: 7
* PRF F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7 => SOL
* STA F2: 5,7,9 + E8: 8 + E9: 9 + D7: 3 + D8: 5,7 + A1: 2 + I2: 6,8 + A4: 5,7 + A6: 7
* CNT  38 HDP CHAINS /  37 HYP OPENED
```