Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=45

level: medium

position: 1..3.......2...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1 initial

Autosolve

position: 1..3......32...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1 autosolve

Pair Reduction Variants

Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # B1: 6,9 => CTR => B1: 4,5
* DIS # C1: 8,9 => CTR => C1: 4,5
* DIS # F7: 2,4 => CTR => F7: 3,7
* DIS # F8: 2,4 => CTR => F8: 3,7,9
* DIS # A9: 4,7 => CTR => A9: 2,5,9
* DIS # C5: 4,7 => CTR => C5: 1,5
* DIS # E8: 2,7 => CTR => E8: 1,3
* DIS # F8: 2,7 => CTR => F8: 3,4,9
* DIS # D9: 2,7 => CTR => D9: 4,9
* DIS # A9: 2,7 => CTR => A9: 4,5,9
* DIS # E6: 2,7 => CTR => E6: 3,5,6
* CNT  11 HDP CHAINS /  41 HYP OPENED
```

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

Pair Reduction

The following important HDP chains were detected:

```* DIS # B1: 6,9 => CTR => B1: 4,5
* DIS B1: 4,5 # A2: 5 => CTR => A2: 6,9
* DIS B1: 4,5 + A2: 6,9 # F5: 2,4 => CTR => F5: 3,7
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 # F7: 2,4 => CTR => F7: 3,7
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 # F8: 2,4 => CTR => F8: 9
* PRF B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 # A7: 4,7 => SOL
* STA B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 + A7: 4,7
* CNT   6 HDP CHAINS /  14 HYP OPENED
```

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

Details

Positions

 1..3.......2...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1 initial 1..3......32...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1 autosolve 145362978632978415978145326329854167461297853857631249714583692286719534593426781 solved

level: medium

Pairing Analysis

```--------------------------------------------------
* PAIRS (5)
A3: 6,9
C3: 8,9
F4: 2,4
C7: 4,7
E9: 2,7

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
H2,G3: 1.. / H2 = 1  => 10 pairs (_) / G3 = 1  => 14 pairs (_)
C4,C5: 1.. / C4 = 1  =>  0 pairs (X) / C5 = 1  =>  5 pairs (_)
D8,E8: 1.. / D8 = 1  => 14 pairs (_) / E8 = 1  =>  8 pairs (_)
D3,G3: 1.. / D3 = 1  => 10 pairs (_) / G3 = 1  => 14 pairs (_)
E2,E8: 1.. / E2 = 1  => 14 pairs (_) / E8 = 1  =>  8 pairs (_)
E1,F1: 2.. / E1 = 2  =>  9 pairs (_) / F1 = 2  => 10 pairs (_)
G3,I3: 3.. / G3 = 3  =>  0 pairs (*) / I3 = 3  =>  0 pairs (X)
F5,E6: 3.. / F5 = 3  =>  0 pairs (X) / E6 = 3  =>  7 pairs (_)
E6,E8: 3.. / E6 = 3  =>  7 pairs (_) / E8 = 3  =>  0 pairs (X)
B1,C1: 4.. / B1 = 4  =>  5 pairs (_) / C1 = 4  =>  0 pairs (X)
E4,E6: 5.. / E4 = 5  =>  5 pairs (_) / E6 = 5  =>  0 pairs (X)
A9,B9: 5.. / A9 = 5  => 12 pairs (_) / B9 = 5  =>  0 pairs (X)
G7,I7: 6.. / G7 = 6  =>  7 pairs (_) / I7 = 6  =>  9 pairs (_)
C1,C3: 8.. / C1 = 8  =>  0 pairs (X) / C3 = 8  =>  5 pairs (_)
F1,F2: 8.. / F1 = 8  => 11 pairs (_) / F2 = 8  =>  6 pairs (_)
B8,B9: 8.. / B8 = 8  =>  5 pairs (_) / B9 = 8  =>  0 pairs (X)
G1,G3: 9.. / G1 = 9  =>  6 pairs (_) / G3 = 9  =>  9 pairs (_)
B4,C4: 9.. / B4 = 9  =>  0 pairs (X) / C4 = 9  =>  6 pairs (_)
* DURATION: 0:00:21.500649  START: 04:18:57.768126  END: 04:19:19.268775 2017-05-01
* CP COUNT: (18)
* SOLUTION FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A3,C3,C7,E9,F4)
* 1..3......32...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1
* PAIR A3: 6,9 BLK 1
B1: 6,9,4,5                              # reduction candidate for 6,9
B1: 6,9 => CTR
* 1943.....5329..41.6781459233.985.1.77.169....8.5731..9.1758..9..86.195...53..6..1
B1: 4,5                                  #  6 pairs
A2: 6,9,5                                # reduction candidate for 6,9
A2: 6,9                                  # 12 pairs
* PAIR A3: 6,9 ROW 3
D3: 6,9,1                                # reduction candidate for 6,9
D3: 6,9                                  # 14 pairs
G3: 6,9,1,3,8                            # reduction candidate for 6,9
G3: 6,9                                  # 10 pairs
* PAIR C3: 8,9 BLK 1
C1: 8,9,4,5                              # reduction candidate for 8,9
C1: 8,9 => CTR
* 149328...532.6.418678145923394852167721694...86..312.9.1.58..9..86.1.5...53.76841
C1: 4,5                                  #  6 pairs
* PAIR C3: 8,9 ROW 3
G3: 8,9,1,3,6                            # reduction candidate for 8,9
G3: 8,9                                  # 11 pairs
G3: 1,3,6                                #  6 pairs
* PAIR F4: 2,4 BLK 5
D5: 2,4,6,7                              # reduction candidate for 2,4
D5: 2,4                                  #  9 pairs
F5: 2,4,3,7                              # reduction candidate for 2,4
F5: 2,4                                  # 21 pairs
D6: 2,4,6,7                              # reduction candidate for 2,4
D6: 2,4                                  #  9 pairs
* PAIR F4: 2,4 ROW 4
B4: 2,4,5,6,9                            # reduction candidate for 2,4
B4: 2,4                                  #  9 pairs
B4: 5,6,9                                #  5 pairs
* PAIR F4: 2,4 COL F
F7: 2,4,3,7                              # reduction candidate for 2,4
F7: 2,4 => CTR
* 1..32.76..32.6.41.6781459233..85..47....97.328.7.3165971458.396.86.1357...3.76..1
F7: 3,7                                  #  6 pairs
F8: 2,4,3,7,9                            # reduction candidate for 2,4
F8: 2,4 => CTR
* 14.32.....3276.41.67.145.2.3..854.674...97...8.7.312.9.14583.9...6.1.5.4..3476.81
F8: 3,7,9                                #  5 pairs
* PAIR C7: 4,7 BLK 7
A7: 4,7,2                                # reduction candidate for 4,7
A7: 4,7                                  # 10 pairs
A8: 4,7,2,9                              # reduction candidate for 4,7
A8: 4,7                                  #  9 pairs
A9: 4,7,2,5,9                            # reduction candidate for 4,7
A9: 4,7 => CTR
* 1..328...5327694186781459233..85...77...9.8328...3165921758..9.986.1.5...53976281
A9: 2,5,9                                #  5 pairs
* PAIR C7: 4,7 ROW 7
F7: 4,7,2,3                              # reduction candidate for 4,7
F7: 4,7                                  # 12 pairs
F7: 2,3                                  #  6 pairs
* PAIR C7: 4,7 COL C
C5: 4,7,1,5                              # reduction candidate for 4,7
C5: 4,7 => CTR
* 14.3.....532...41.67.145.2.39185...7....9.1..8.5.31249.1.58..9..86.1.5...53..6..1
C5: 1,5                                  #  6 pairs
C6: 4,7,5                                # reduction candidate for 4,7
C6: 4,7                                  #  9 pairs
* PAIR E9: 2,7 BLK 8
F7: 2,7,3,4                              # reduction candidate for 2,7
F7: 2,7                                  # 16 pairs
D8: 2,7,1,4,9                            # reduction candidate for 2,7
D8: 2,7                                  # 11 pairs
E8: 2,7,1,3                              # reduction candidate for 2,7
E8: 2,7 => CTR
* 1..362978.32.184..6789451233.9854.17..1.97...8.7.31..9.14583.9...61.9534.934.6.81
E8: 1,3                                  #  6 pairs
F8: 2,7,3,4,9                            # reduction candidate for 2,7
F8: 2,7 => CTR
* 14.32.....3276.41.67.145.2.3..854.674...97...8.7.31249.14583.92..6.125.4253.76..1
F8: 3,4,9                                #  5 pairs
D9: 2,7,4,9                              # reduction candidate for 2,7
D9: 2,7 => CTR
* 1..3..76..32...41..7.1459233..8...577...9..848.4.51.39.1758.396.864..57...3..6841
D9: 4,9                                  #  6 pairs
* PAIR E9: 2,7 ROW 9
A9: 2,7,4,5,9                            # reduction candidate for 2,7
A9: 2,7 => CTR
* 1..3.87.55327694186781459233..8....77.5.9.1848.4.31.59417582.9.986.1.572253976841
A9: 4,5,9                                #  5 pairs
G9: 2,7,8                                # reduction candidate for 2,7
G9: 2,7                                  #  8 pairs
* PAIR E9: 2,7 COL E
E1: 2,7,6                                # reduction candidate for 2,7
E1: 2,7                                  #  9 pairs
E6: 2,7,3,5,6                            # reduction candidate for 2,7
E6: 2,7 => CTR
* 1..36297..32.184656789451233.9854.17..1.93.548....1.392145873967.613.5....3..6..1
E6: 3,5,6                                #  5 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0045-base-pr-000.dot
* REASONING
* DIS # B1: 6,9 => CTR => B1: 4,5
* DIS # C1: 8,9 => CTR => C1: 4,5
* DIS # F7: 2,4 => CTR => F7: 3,7
* DIS # F8: 2,4 => CTR => F8: 3,7,9
* DIS # A9: 4,7 => CTR => A9: 2,5,9
* DIS # C5: 4,7 => CTR => C5: 1,5
* DIS # E8: 2,7 => CTR => E8: 1,3
* DIS # F8: 2,7 => CTR => F8: 3,4,9
* DIS # D9: 2,7 => CTR => D9: 4,9
* DIS # A9: 2,7 => CTR => A9: 4,5,9
* DIS # E6: 2,7 => CTR => E6: 3,5,6
* CNT  11 HDP CHAINS /  41 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A3,C3,C7,E9,F4)
* 1..3......32...4...7..45.2.3..8....7....9....8....1..9.1.58..9...6...5....3..6..1
* PAIR A3: 6,9 BLK 1
B1: 6,9,4,5                              # reduction candidate for 6,9
B1: 6,9 => CTR
* 1943.....5329..41.6781459233.985.1.77.169....8.5731..9.1758..9..86.195...53..6..1
A2: 6,9,5                                # reduction candidate for 6,9
A2: 6,9                                  # 12 pairs
* RESTART
* PAIR A3: 6,9 BLK 1
A2: 6,9,5                                # reduction candidate for 6,9
A2: 5 => CTR
* 149328...532.6.418678145923394852167721694...86..312.9.1.58..9..86.1.5...53.76841
* PAIR A3: 6,9 ROW 3
D3: 6,9,1                                # reduction candidate for 6,9
D3: 6,9                                  # 14 pairs
G3: 6,9,1,3                              # reduction candidate for 6,9
G3: 6,9                                  # 18 pairs
* PAIR F4: 2,4 BLK 5
D5: 2,4,6,7                              # reduction candidate for 2,4
D5: 2,4                                  # 16 pairs
F5: 2,4,3,7                              # reduction candidate for 2,4
F5: 2,4 => CTR
* 1..32.9689321684756789451233.985..174.17923..8...31..9.1.583796786.1953.593.76841
D6: 2,4,6,7                              # reduction candidate for 2,4
D6: 2,4                                  # 16 pairs
* PAIR F4: 2,4 ROW 4
B4: 2,4,5,6                              # reduction candidate for 2,4
B4: 2,4                                  # 16 pairs
B4: 5,6                                  # 14 pairs
* PAIR F4: 2,4 COL F
F7: 2,4,3,7                              # reduction candidate for 2,4
F7: 2,4 => CTR
* 1453297689327684156781459233.98..1.7..1.97.3.8.7.31..97145823962.6.135745.3476281
F8: 2,4,9                                # reduction candidate for 2,4
F8: 2,4 => CTR
* 14532.....3276.4156781459233.985.1.74.1.973..8.7.31..9.14583.9...6.1.5345.3476281
* PAIR RESTART
* PAIR C7: 4,7 BLK 7
A7: 4,7,2                                # reduction candidate for 4,7
A7: 4,7 => SOLVED
* 145362978632978415978145326329854167461297853857631249714583692286719534593426781
* DURATION: 0:00:14.577079  START: 04:19:44.717998  END: 04:19:59.295077 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0045-base-pr-001.dot
* REASONING
* DIS # B1: 6,9 => CTR => B1: 4,5
* DIS B1: 4,5 # A2: 5 => CTR => A2: 6,9
* DIS B1: 4,5 + A2: 6,9 # F5: 2,4 => CTR => F5: 3,7
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 # F7: 2,4 => CTR => F7: 3,7
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 # F8: 2,4 => CTR => F8: 9
* PRF B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 # A7: 4,7 => SOL
* STA B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 + A7: 4,7
* CNT   6 HDP CHAINS /  14 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=45
```

Solution

position: 145362978632978415978145326329854167461297853857631249714583692286719534593426781 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 # B1: 6,9 => CTR => B1: 4,5
* INC # B1: 4,5 => UNS
* INC # A2: 6,9 => UNS
* INC # D3: 6,9 => UNS
* INC # G3: 6,9 => UNS
* DIS # C1: 8,9 => CTR => C1: 4,5
* INC # C1: 4,5 => UNS
* INC # G3: 8,9 => UNS
* INC # G3: 1,3,6 => UNS
* INC # D5: 2,4 => UNS
* INC # F5: 2,4 => UNS
* INC # D6: 2,4 => UNS
* INC # B4: 2,4 => UNS
* INC # B4: 5,6,9 => UNS
* DIS # F7: 2,4 => CTR => F7: 3,7
* INC # F7: 3,7 => UNS
* DIS # F8: 2,4 => CTR => F8: 3,7,9
* INC # F8: 3,7,9 => UNS
* INC # A7: 4,7 => UNS
* INC # A8: 4,7 => UNS
* DIS # A9: 4,7 => CTR => A9: 2,5,9
* INC # A9: 2,5,9 => UNS
* INC # F7: 4,7 => UNS
* INC # F7: 2,3 => UNS
* DIS # C5: 4,7 => CTR => C5: 1,5
* INC # C5: 1,5 => UNS
* INC # C6: 4,7 => UNS
* INC # F7: 2,7 => UNS
* INC # D8: 2,7 => UNS
* DIS # E8: 2,7 => CTR => E8: 1,3
* INC # E8: 1,3 => UNS
* DIS # F8: 2,7 => CTR => F8: 3,4,9
* INC # F8: 3,4,9 => UNS
* DIS # D9: 2,7 => CTR => D9: 4,9
* INC # D9: 4,9 => UNS
* DIS # A9: 2,7 => CTR => A9: 4,5,9
* INC # A9: 4,5,9 => UNS
* INC # G9: 2,7 => UNS
* INC # E1: 2,7 => UNS
* DIS # E6: 2,7 => CTR => E6: 3,5,6
* INC # E6: 3,5,6 => UNS
* CNT  41 HDP CHAINS /  41 HYP OPENED
```

A2. Pair Reduction

Full list of HDP chains traversed:

```* DIS # B1: 6,9 => CTR => B1: 4,5
* INC B1: 4,5 # A2: 6,9 => UNS
* INC B1: 4,5 # A2: 6,9 => UNS
* DIS B1: 4,5 # A2: 5 => CTR => A2: 6,9
* INC B1: 4,5 + A2: 6,9 # D3: 6,9 => UNS
* INC B1: 4,5 + A2: 6,9 # G3: 6,9 => UNS
* INC B1: 4,5 + A2: 6,9 # D5: 2,4 => UNS
* DIS B1: 4,5 + A2: 6,9 # F5: 2,4 => CTR => F5: 3,7
* INC B1: 4,5 + A2: 6,9 + F5: 3,7 # D6: 2,4 => UNS
* INC B1: 4,5 + A2: 6,9 + F5: 3,7 # B4: 2,4 => UNS
* INC B1: 4,5 + A2: 6,9 + F5: 3,7 # B4: 5,6 => UNS
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 # F7: 2,4 => CTR => F7: 3,7
* DIS B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 # F8: 2,4 => CTR => F8: 9
* PRF B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 # A7: 4,7 => SOL
* STA B1: 4,5 + A2: 6,9 + F5: 3,7 + F7: 3,7 + F8: 9 + A7: 4,7
* CNT  14 HDP CHAINS /  14 HYP OPENED
```