# Sudoku from http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=281

level: medium

position: .2.34....4...5......6..17..5..2......3.....4...1..68....9....7....1..9.......9.68 initial

# Autosolve

position: .2.34....4..65......6..17.45..2......3.....4...1..68....9....7....1..9.......9.68 autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # F2: 2 => CTR => F2: 7,8
* DIS # C1: 5 => CTR => C1: 7,8
* DIS # E3: 8,9 => CTR => E3: 2
* DIS # B4: 4,8 => CTR => B4: 6,7,9
* PRF # C8: 4,8 => SOL
* DIS # A5: 2,8 => CTR => A5: 6,7,9
* DIS # C8: 2,8 => CTR => C8: 3,4,5
* PRF # D5: 5,8 => SOL
* DIS # F7: 5,8 => CTR => F7: 2,3,4
* DIS # F8: 5,8 => CTR => F8: 2,3,4
* CNT  10 HDP CHAINS /  40 HYP OPENED
```

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

## Pair Reduction

The following important HDP chains were detected:

```* DIS # F2: 2 => CTR => F2: 7,8
* DIS F2: 7,8 # C1: 5 => CTR => C1: 7,8
* DIS F2: 7,8 + C1: 7,8 # B4: 4,8 => CTR => B4: 6,7,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 # A5: 2,8 => CTR => A5: 6,7,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 # I1: 1,9 => CTR => I1: 5,6
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 # H1: 5 => CTR => H1: 1,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 9 => CTR => I2: 1,2
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 # G7: 1,2 => CTR => G7: 3,4,5
* PRF F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 # G9: 1,2 => SOL
* STA F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 + G9: 1,2
* CNT   9 HDP CHAINS /  32 HYP OPENED
```

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

# Details

## Positions

 .2.34....4...5......6..17..5..2......3.....4...1..68....9....7....1..9.......9.68 initial .2.34....4..65......6..17.45..2......3.....4...1..68....9....7....1..9.......9.68 autosolve 127348596493657182856921734568294317732815649941736825289563471674182953315479268 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (9)
A1: 1,9
B2: 1,9
A3: 3,8
B3: 5,8
F1: 7,8
D3: 8,9
C4: 4,8
C5: 2,8
F5: 5,8

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
A1,B2: 1.. / A1 = 1  =>  9 pairs (_) / B2 = 1  =>  9 pairs (_)
E4,E5: 1.. / E4 = 1  => 13 pairs (_) / E5 = 1  =>  9 pairs (_)
F2,E3: 2.. / F2 = 2  =>  0 pairs (X) / E3 = 2  => 12 pairs (_)
E3,H3: 2.. / E3 = 2  => 12 pairs (_) / H3 = 2  =>  0 pairs (X)
C2,A3: 3.. / C2 = 3  => 11 pairs (_) / A3 = 3  =>  9 pairs (_)
A3,H3: 3.. / A3 = 3  =>  9 pairs (_) / H3 = 3  => 11 pairs (_)
F4,D6: 4.. / F4 = 4  =>  0 pairs (*) / D6 = 4  =>  0 pairs (X)
G7,G9: 4.. / G7 = 4  => 10 pairs (_) / G9 = 4  => 10 pairs (_)
B6,D6: 4.. / B6 = 4  =>  0 pairs (*) / D6 = 4  =>  0 pairs (X)
C1,B3: 5.. / C1 = 5  =>  0 pairs (X) / B3 = 5  =>  9 pairs (_)
B3,H3: 5.. / B3 = 5  =>  9 pairs (_) / H3 = 5  =>  0 pairs (X)
G1,I1: 6.. / G1 = 6  => 10 pairs (_) / I1 = 6  => 10 pairs (_)
B4,A5: 6.. / B4 = 6  => 10 pairs (_) / A5 = 6  =>  9 pairs (_)
E7,E8: 6.. / E7 = 6  =>  9 pairs (_) / E8 = 6  =>  9 pairs (_)
C1,C2: 7.. / C1 = 7  => 11 pairs (_) / C2 = 7  =>  9 pairs (_)
F1,F2: 7.. / F1 = 7  =>  9 pairs (_) / F2 = 7  => 11 pairs (_)
C1,F1: 7.. / C1 = 7  => 11 pairs (_) / F1 = 7  =>  9 pairs (_)
C2,F2: 7.. / C2 = 7  =>  9 pairs (_) / F2 = 7  => 11 pairs (_)
A1,B2: 9.. / A1 = 9  =>  9 pairs (_) / B2 = 9  =>  9 pairs (_)
D3,E3: 9.. / D3 = 9  =>  9 pairs (_) / E3 = 9  =>  0 pairs (X)
* DURATION: 0:00:16.818960  START: 14:46:26.769017  END: 14:46:43.587977 2019-04-28
* CP COUNT: (20)
* SOLUTION FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A1,A3,B2,B3,C4,C5,D3,F1,F5)
* .2.34....4..65......6..17.45..2......3.....4...1..68....9....7....1..9.......9.68
* PAIR A1: 1,9 ROW 1
H1: 1,9,5,8                              # reduction candidate for 1,9
H1: 1,9                                  # 12 pairs
I1: 1,9,5,6                              # reduction candidate for 1,9
I1: 1,9                                  # 13 pairs
* PAIR B2: 1,9 ROW 2
H2: 1,9,2,3,8                            # reduction candidate for 1,9
H2: 1,9                                  # 16 pairs
I2: 1,9,2,3                              # reduction candidate for 1,9
I2: 1,9                                  # 11 pairs
* PAIR A3: 3,8 BLK 1
C2: 3,8,7                                # reduction candidate for 3,8
C2: 3,8                                  # 11 pairs
C2: 7                                    #  9 pairs
* PAIR A3: 3,8 ROW 3
H3: 3,8,2,5                              # reduction candidate for 3,8
H3: 3,8                                  # 13 pairs
H3: 2,5                                  # 10 pairs
* PAIR A3: 3,8 COL A
A7: 3,8,1,2,6                            # reduction candidate for 3,8
A7: 3,8                                  # 10 pairs
A8: 3,8,2,6,7                            # reduction candidate for 3,8
A8: 3,8                                  # 12 pairs
* PAIR B3: 5,8 BLK 1
C1: 5,8,7                                # reduction candidate for 5,8
C1: 7                                    # 11 pairs
C1: 5,8                                  #  9 pairs
* PAIR B3: 5,8 ROW 3
H3: 5,8,2,3                              # reduction candidate for 5,8
H3: 5,8                                  # 13 pairs
H3: 2,3                                  # 12 pairs
* PAIR B3: 5,8 COL B
B7: 5,8,1,4,6                            # reduction candidate for 5,8
B7: 5,8                                  # 10 pairs
B8: 5,8,4,6,7                            # reduction candidate for 5,8
B8: 5,8                                  # 12 pairs
* PAIR F1: 7,8 BLK 2
F2: 7,8,2                                # reduction candidate for 7,8
F2: 2 => CTR
* .28347...4.7652.8.356..17245.42......32....4...14.6852..9....7....1..935.....9.68
F2: 7,8                                  # 12 pairs
* PAIR F1: 7,8 ROW 1
C1: 7,8,5                                # reduction candidate for 7,8
C1: 5 => CTR
* .25347.8.4.7658...3869217545.4283...83271564...1..68.5..9..2.71.581.49....3.79.68
C1: 7,8                                  #  9 pairs
* PAIR D3: 8,9 BLK 2
E3: 8,9,2                                # reduction candidate for 8,9
E3: 8,9 => CTR
* .28347...4.7652.8.356..17245.42......32....4...14.6852..9....7....1..935.....9.68
E3: 2                                    # 12 pairs
* PAIR D3: 8,9 COL D
D5: 8,9,5,7                              # reduction candidate for 8,9
D5: 8,9                                  # 11 pairs
D5: 5,7                                  # 10 pairs
* PAIR C4: 4,8 BLK 4
B4: 4,8,6,7,9                            # reduction candidate for 4,8
B4: 4,8 => CTR
* .2.34....4..65......6..17.45..2.....63.....4...1.368....9....7....1..9.......9.68
B4: 6,7,9                                #  9 pairs
* PAIR C4: 4,8 ROW 4
F4: 4,8,3                                # reduction candidate for 4,8
F4: 3                                    # 29 pairs
F4: 4,8                                  # 11 pairs
* PAIR C4: 4,8 COL C
C8: 4,8,2,3,5                            # reduction candidate for 4,8
C8: 4,8 => SOLVED
* 127348596493657182856921734568294317732815649941736825289563471674182953315479268
C8: 2,3,5                                # 10 pairs
* PAIR C5: 2,8 BLK 4
A5: 2,8,6,7,9                            # reduction candidate for 2,8
A5: 2,8 => CTR
* .2.347...4.7652.8.3.6..17245642.8....3...5.4...1436852..9....7....1..935..3..9.68
A5: 6,7,9                                #  9 pairs
* PAIR C5: 2,8 COL C
C8: 2,8,3,4,5                            # reduction candidate for 2,8
C8: 2,8 => CTR
* 92.34....41.65......69217.45.42.3....3...5.4.2.14.68....9....7...21..9.......9.68
C8: 3,4,5                                #  9 pairs
* PAIR F5: 5,8 BLK 5
D5: 5,8,7,9                              # reduction candidate for 5,8
D5: 5,8 => SOLVED
* 127348596493657182856921734568294317732815649941736825289563471674182953315479268
D5: 7,9                                  # 10 pairs
* PAIR F5: 5,8 COL F
F7: 5,8,2,3,4                            # reduction candidate for 5,8
F7: 5,8 => CTR
* .28347...4.7652.8.356..17245.42.3....32....4...14.68.2..9....7....1.49.......9.68
F7: 2,3,4                                #  9 pairs
F8: 5,8,2,3,4                            # reduction candidate for 5,8
F8: 5,8 => CTR
* .28347...4.7652.8.356..17245.42.3....32....4...14.68.2..9..4.7..4.1..9.......9468
F8: 2,3,4                                #  9 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-www.sudokuwiki.org-0281-base-pr-000.dot
* REASONING
* DIS # F2: 2 => CTR => F2: 7,8
* DIS # C1: 5 => CTR => C1: 7,8
* DIS # E3: 8,9 => CTR => E3: 2
* DIS # B4: 4,8 => CTR => B4: 6,7,9
* PRF # C8: 4,8 => SOL
* DIS # A5: 2,8 => CTR => A5: 6,7,9
* DIS # C8: 2,8 => CTR => C8: 3,4,5
* PRF # D5: 5,8 => SOL
* DIS # F7: 5,8 => CTR => F7: 2,3,4
* DIS # F8: 5,8 => CTR => F8: 2,3,4
* CNT  10 HDP CHAINS /  40 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (A1,A3,B2,B3,C4,C5,D3,F1,F5)
* .2.34....4..65......6..17.45..2......3.....4...1..68....9....7....1..9.......9.68
* PAIR A1: 1,9 ROW 1
H1: 1,9,5,8                              # reduction candidate for 1,9
H1: 1,9                                  # 12 pairs
I1: 1,9,5,6                              # reduction candidate for 1,9
I1: 1,9                                  # 13 pairs
* PAIR B2: 1,9 ROW 2
H2: 1,9,2,3,8                            # reduction candidate for 1,9
H2: 1,9                                  # 16 pairs
I2: 1,9,2,3                              # reduction candidate for 1,9
I2: 1,9                                  # 11 pairs
* PAIR A3: 3,8 BLK 1
C2: 3,8,7                                # reduction candidate for 3,8
C2: 3,8                                  # 11 pairs
C2: 7                                    #  9 pairs
* PAIR A3: 3,8 ROW 3
H3: 3,8,2,5                              # reduction candidate for 3,8
H3: 3,8                                  # 13 pairs
H3: 2,5                                  # 10 pairs
* PAIR A3: 3,8 COL A
A7: 3,8,1,2,6                            # reduction candidate for 3,8
A7: 3,8                                  # 10 pairs
A8: 3,8,2,6,7                            # reduction candidate for 3,8
A8: 3,8                                  # 12 pairs
* PAIR B3: 5,8 BLK 1
C1: 5,8,7                                # reduction candidate for 5,8
C1: 7                                    # 11 pairs
C1: 5,8                                  #  9 pairs
* PAIR B3: 5,8 ROW 3
H3: 5,8,2,3                              # reduction candidate for 5,8
H3: 5,8                                  # 13 pairs
H3: 2,3                                  # 12 pairs
* PAIR B3: 5,8 COL B
B7: 5,8,1,4,6                            # reduction candidate for 5,8
B7: 5,8                                  # 10 pairs
B8: 5,8,4,6,7                            # reduction candidate for 5,8
B8: 5,8                                  # 12 pairs
* PAIR F1: 7,8 BLK 2
F2: 7,8,2                                # reduction candidate for 7,8
F2: 2 => CTR
* .28347...4.7652.8.356..17245.42......32....4...14.6852..9....7....1..935.....9.68
* PAIR F1: 7,8 ROW 1
C1: 7,8,5                                # reduction candidate for 7,8
C1: 5 => CTR
* .25347.8.4.7658...3869217545.4283...83271564...1..68.5..9..2.71.581.49....3.79.68
* PAIR RESTART
* PAIR C4: 4,8 BLK 4
B4: 4,8,6,7,9                            # reduction candidate for 4,8
B4: 4,8 => CTR
* .2.34....4..65.....569217.45..2.....63...5.4...14368....9....7....1..9.......9.68
* PAIR C5: 2,8 BLK 4
A5: 2,8,6,7,9                            # reduction candidate for 2,8
A5: 2,8 => CTR
* .2.34....4..65.....569217.456.2......3...5.4...14368....9....7....1..9.......9.68

* LEVEL 0 PASS 1 ROUND 2 (AUTO SOLVE) (A1,A8,B2,C4,C5,D5,D6,E9,F4,G2,H4,H8)
* .27348...4.3657.8.8569217345..2......3...5.4...1..68....9....7....1..9.......9.68
* PAIR A1: 1,9 ROW 1
H1: 1,9,5                                # reduction candidate for 1,9
H1: 1,9                                  # 16 pairs
I1: 1,9,5,6                              # reduction candidate for 1,9
I1: 1,9 => CTR
* .2734865.4.3657.8.8569217345..2...1..3..1524.2.1..6895..9....7...51.4923..2..9.68
* RESTART
* PAIR A1: 1,9 ROW 1
H1: 1,9,5                                # reduction candidate for 1,9
H1: 5 => CTR
* 9273481564136572898569217345..2...1..3...5.4...1..68....9....7....1..9.......9.68
* PAIR RESTART
* PAIR B2: 1,9 ROW 2
I2: 1,9,2                                # reduction candidate for 1,9
I2: 2                                    # 18 pairs
I2: 1,9                                  # 18 pairs
* PAIR G2: 1,2 BLK 3
I2: 1,2,9                                # reduction candidate for 1,2
I2: 9 => CTR
* 927348.1.4136572898569217345.4283.9..3...5.4..914.68....9....7....1..9.......9.68
* PAIR G2: 1,2 COL G
G7: 1,2,3,4,5                            # reduction candidate for 1,2
G7: 1,2 => CTR
* 127348596493657.8.85692173456.294317.38.1564...1.368....9....7....1..9..312579468
G9: 1,2,3,4,5                            # reduction candidate for 1,2
G9: 1,2 => SOLVED
* 127348596493657182856921734568294317732815649941736825289563471674182953315479268
* DURATION: 0:00:20.348645  START: 14:47:03.666438  END: 14:47:24.015083 2019-04-28
* SOLUTION FOUND
* SAVE PR GRAPH zz-www.sudokuwiki.org-0281-base-pr-001.dot
* REASONING
* DIS # F2: 2 => CTR => F2: 7,8
* DIS F2: 7,8 # C1: 5 => CTR => C1: 7,8
* DIS F2: 7,8 + C1: 7,8 # B4: 4,8 => CTR => B4: 6,7,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 # A5: 2,8 => CTR => A5: 6,7,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 # I1: 1,9 => CTR => I1: 5,6
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 # H1: 5 => CTR => H1: 1,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 9 => CTR => I2: 1,2
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 # G7: 1,2 => CTR => G7: 3,4,5
* PRF F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 # G9: 1,2 => SOL
* STA F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 + G9: 1,2
* CNT   9 HDP CHAINS /  32 HYP OPENED
```

```http://www.sudokuwiki.org/Print_Weekly_Sudoku.asp?unsolvable=281
```

# Solution

position: 127348596493657182856921734568294317732815649941736825289563471674182953315479268 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 # H1: 1,9 => UNS
* INC # I1: 1,9 => UNS
* INC # H2: 1,9 => UNS
* INC # I2: 1,9 => UNS
* INC # C2: 3,8 => UNS
* INC # C2: 7 => UNS
* INC # H3: 3,8 => UNS
* INC # H3: 2,5 => UNS
* INC # A7: 3,8 => UNS
* INC # A8: 3,8 => UNS
* INC # C1: 5,8 => UNS
* INC # C1: 7 => UNS
* INC # H3: 5,8 => UNS
* INC # H3: 2,3 => UNS
* INC # B7: 5,8 => UNS
* INC # B8: 5,8 => UNS
* INC # F2: 7,8 => UNS
* DIS # F2: 2 => CTR => F2: 7,8
* INC # C1: 7,8 => UNS
* DIS # C1: 5 => CTR => C1: 7,8
* DIS # E3: 8,9 => CTR => E3: 2
* INC # E3: 2 => UNS
* INC # D5: 8,9 => UNS
* INC # D5: 5,7 => UNS
* DIS # B4: 4,8 => CTR => B4: 6,7,9
* INC # B4: 6,7,9 => UNS
* INC # F4: 4,8 => UNS
* INC # F4: 3 => UNS
* PRF # C8: 4,8 => SOL
* INC # C8: 2,3,5 => UNS
* DIS # A5: 2,8 => CTR => A5: 6,7,9
* INC # A5: 6,7,9 => UNS
* DIS # C8: 2,8 => CTR => C8: 3,4,5
* INC # C8: 3,4,5 => UNS
* PRF # D5: 5,8 => SOL
* INC # D5: 7,9 => UNS
* DIS # F7: 5,8 => CTR => F7: 2,3,4
* INC # F7: 2,3,4 => UNS
* DIS # F8: 5,8 => CTR => F8: 2,3,4
* INC # F8: 2,3,4 => UNS
* CNT  40 HDP CHAINS /  40 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # H1: 1,9 => UNS
* INC # I1: 1,9 => UNS
* INC # H2: 1,9 => UNS
* INC # I2: 1,9 => UNS
* INC # C2: 3,8 => UNS
* INC # C2: 7 => UNS
* INC # H3: 3,8 => UNS
* INC # H3: 2,5 => UNS
* INC # A7: 3,8 => UNS
* INC # A8: 3,8 => UNS
* INC # C1: 5,8 => UNS
* INC # C1: 7 => UNS
* INC # H3: 5,8 => UNS
* INC # H3: 2,3 => UNS
* INC # B7: 5,8 => UNS
* INC # B8: 5,8 => UNS
* INC # F2: 7,8 => UNS
* DIS # F2: 2 => CTR => F2: 7,8
* INC F2: 7,8 # C1: 7,8 => UNS
* DIS F2: 7,8 # C1: 5 => CTR => C1: 7,8
* DIS F2: 7,8 + C1: 7,8 # B4: 4,8 => CTR => B4: 6,7,9
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 # A5: 2,8 => CTR => A5: 6,7,9
* INC F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 # H1: 1,9 => UNS
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 # I1: 1,9 => CTR => I1: 5,6
* INC F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 # H1: 1,9 => UNS
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 # H1: 5 => CTR => H1: 1,9
* INC F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 1,9 => UNS
* INC F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 2 => UNS
* INC F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 1,2 => UNS
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 # I2: 9 => CTR => I2: 1,2
* DIS F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 # G7: 1,2 => CTR => G7: 3,4,5
* PRF F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 # G9: 1,2 => SOL
* STA F2: 7,8 + C1: 7,8 + B4: 6,7,9 + A5: 6,7,9 + I1: 5,6 + H1: 1,9 + I2: 1,2 + G7: 3,4,5 + G9: 1,2
* CNT  32 HDP CHAINS /  32 HYP OPENED
```