# Original Sudoku

level: medium

position: ....5..6.7.9..1....4....8...5...3.91.........21.7...3...5....7....2..4.3.8..6.... initial

# Autosolve

position: ....5..6.7.9..1...54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6.... autosolve

# Pair Reduction Variants

## Pair Reduction Analysis

The following important HDP chains were detected:

```* DIS # B2: 2,3 => CTR => B2: 6
* DIS # G1: 2,3 => CTR => G1: 1,7,9
* DIS # B7: 9 => CTR => B7: 2,3
* DIS # C3: 3,6 => CTR => C3: 1,2
* CNT   4 HDP CHAINS /  19 HYP OPENED
```

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

## Pair Reduction

The following important HDP chains were detected:

```* DIS # B2: 2,3 => CTR => B2: 6
* DIS B2: 6 # G1: 2,3 => CTR => G1: 1,7,9
* DIS B2: 6 + G1: 1,7,9 # B7: 9 => CTR => B7: 2,3
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 # C3: 3 => CTR => C3: 1,2
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C9: 1,2 => CTR => C9: 3,4,7
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # E2: 4,8 => CTR => E2: 2
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 # H9: 5 => CTR => H9: 1,2
* PRF B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 # G5: 5,6 => SOL
* STA B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 + G5: 5,6
* CNT   8 HDP CHAINS /  36 HYP OPENED
```

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

# Details

## Positions

 ....5..6.7.9..1....4....8...5...3.91.........21.7...3...5....7....2..4.3.8..6.... initial ....5..6.7.9..1...54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6.... autosolve 138954762769821354542376819457683291893512647216749538925438176671295483384167925 solved

level: medium

## Pairing Analysis

```--------------------------------------------------
* PAIRS (3)
B1: 2,3
H3: 1,2
G6: 5,6

--------------------------------------------------
* CONSTRAINT PAIRS (AUTO SOLVE)
G1,H3: 1.. / G1 = 1  => 18 pairs (_) / H3 = 1  =>  6 pairs (_)
D5,E5: 1.. / D5 = 1  =>  0 pairs (X) / E5 = 1  =>  3 pairs (_)
C3,H3: 1.. / C3 = 1  => 18 pairs (_) / H3 = 1  =>  6 pairs (_)
B7,C9: 2.. / B7 = 2  =>  9 pairs (_) / C9 = 2  =>  0 pairs (X)
E4,G4: 2.. / E4 = 2  =>  4 pairs (_) / G4 = 2  =>  7 pairs (_)
G1,G2: 3.. / G1 = 3  =>  0 pairs (X) / G2 = 3  =>  4 pairs (_)
H2,H5: 4.. / H2 = 4  =>  5 pairs (_) / H5 = 4  =>  5 pairs (_)
F8,H8: 5.. / F8 = 5  =>  4 pairs (_) / H8 = 5  =>  0 pairs (X)
D5,D9: 5.. / D5 = 5  =>  3 pairs (_) / D9 = 5  =>  0 pairs (X)
B2,C3: 6.. / B2 = 6  =>  4 pairs (_) / C3 = 6  =>  0 pairs (X)
G7,I7: 6.. / G7 = 6  =>  8 pairs (_) / I7 = 6  =>  5 pairs (_)
B2,D2: 6.. / B2 = 6  =>  4 pairs (_) / D2 = 6  =>  0 pairs (X)
C4,G4: 7.. / C4 = 7  =>  6 pairs (_) / G4 = 7  =>  5 pairs (_)
C9,F9: 7.. / C9 = 7  =>  0 pairs (X) / F9 = 7  =>  4 pairs (_)
B5,B8: 7.. / B5 = 7  =>  7 pairs (_) / B8 = 7  =>  5 pairs (_)
E3,E8: 7.. / E3 = 7  =>  4 pairs (_) / E8 = 7  =>  0 pairs (X)
A1,C1: 8.. / A1 = 8  =>  5 pairs (_) / C1 = 8  =>  5 pairs (_)
D2,E2: 8.. / D2 = 8  =>  5 pairs (_) / E2 = 8  =>  7 pairs (_)
I7,H8: 8.. / I7 = 8  =>  0 pairs (X) / H8 = 8  =>  5 pairs (_)
H5,H8: 8.. / H5 = 8  =>  0 pairs (X) / H8 = 8  =>  5 pairs (_)
A5,B5: 9.. / A5 = 9  =>  4 pairs (_) / B5 = 9  =>  6 pairs (_)
E6,F6: 9.. / E6 = 9  =>  3 pairs (_) / F6 = 9  =>  5 pairs (_)
* DURATION: 0:00:25.286557  START: 06:25:11.341799  END: 06:25:36.628356 2017-05-01
* CP COUNT: (22)
* CLUE FOUND

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (B1,G6,H3)
* ....5..6.7.9..1...54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6....
* PAIR B1: 2,3 BLK 1
C1: 2,3,1,8                              # reduction candidate for 2,3
C1: 2,3                                  # 18 pairs
B2: 2,3,6                                # reduction candidate for 2,3
B2: 2,3 => CTR
* ....5.9677.96813.55469..81..5...3.919.3......21.7...3..95....7....29.4.3.82.67.59
B2: 6                                    #  4 pairs
C3: 2,3,1,6                              # reduction candidate for 2,3
C3: 2,3                                  #  8 pairs
* PAIR B1: 2,3 ROW 1
G1: 2,3,1,7,9                            # reduction candidate for 2,3
G1: 2,3 => CTR
* .3..5.26.769.213..542...81..5...3.91...512.4821.7..53..25....76...2.5483.8..6....
G1: 1,7,9                                #  4 pairs
* PAIR B1: 2,3 COL B
B7: 2,3,9                                # reduction candidate for 2,3
B7: 9 => CTR
* .2..54.677698213..54..768...5.6.329193.51274.21.7.953.495138672671295483382467...
B7: 2,3                                  #  6 pairs
* PAIR H3: 1,2 BLK 3
G1: 1,2,3,7,9                            # reduction candidate for 1,2
G1: 1,2                                  # 13 pairs
G1: 3,7,9                                #  6 pairs
* PAIR H3: 1,2 ROW 3
C3: 1,2,3,6                              # reduction candidate for 1,2
C3: 3,6 => CTR
* ....5.9677.9..13.554..7.812.5...3.91...51..2.21.7..53...5.3.276...2.5483382467159
C3: 1,2                                  #  8 pairs
* PAIR H3: 1,2 COL H
H9: 1,2,5                                # reduction candidate for 1,2
H9: 5                                    #  6 pairs
H9: 1,2                                  #  6 pairs
* PAIR G6: 5,6 BLK 6
G5: 5,6,2,7                              # reduction candidate for 5,6
G5: 5,6                                  #  7 pairs
I5: 5,6,2,4,7,8                          # reduction candidate for 5,6
I5: 5,6                                  #  9 pairs
I6: 5,6,4,8                              # reduction candidate for 5,6
I6: 5,6                                  # 11 pairs
* PAIR G6: 5,6 ROW 6
F6: 5,6,4,8,9                            # reduction candidate for 5,6
F6: 5,6                                  #  6 pairs
F6: 4,8,9                                #  3 pairs
* INCONCLUSIVE
* SAVE PR GRAPH zz-handy-001-base-pr-000.dot
* REASONING
* DIS # B2: 2,3 => CTR => B2: 6
* DIS # G1: 2,3 => CTR => G1: 1,7,9
* DIS # B7: 9 => CTR => B7: 2,3
* DIS # C3: 3,6 => CTR => C3: 1,2
* CNT   4 HDP CHAINS /  19 HYP OPENED

--------------------------------------------------
* PREPARE PR GRAPH
* PAIR REDUCTION ..
* LEVEL 0 PASS 1 ROUND 1 (AUTO SOLVE) (B1,G6,H3)
* ....5..6.7.9..1...54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6....
* PAIR B1: 2,3 BLK 1
C1: 2,3,1,8                              # reduction candidate for 2,3
C1: 2,3                                  # 18 pairs
B2: 2,3,6                                # reduction candidate for 2,3
B2: 2,3 => CTR
* ....5.9677.96813.55469..81..5...3.919.3......21.7...3..95....7....29.4.3.82.67.59
C3: 2,3,1                                # reduction candidate for 2,3
C3: 2,3                                  #  8 pairs
* PAIR B1: 2,3 ROW 1
G1: 2,3,1,7,9                            # reduction candidate for 2,3
G1: 2,3 => CTR
* .3..5.26.769.213..542...81..5...3.91...512.4821.7..53..25....76...2.5483.8..6....
* PAIR B1: 2,3 COL B
B7: 2,3,9                                # reduction candidate for 2,3
B7: 9 => CTR
* 12835..6.7698.13..54397681285.6.3.9193.512.4821.789.3.49513827667129548338246715.
* PAIR RESTART
* PAIR B1: 2,3 BLK 1
C1: 2,3,1,8                              # reduction candidate for 2,3
C1: 2,3                                  # 18 pairs
C3: 2,3,1                                # reduction candidate for 2,3
C3: 2,3                                  # 12 pairs
* PAIR H3: 1,2 ROW 3
C3: 1,2,3                                # reduction candidate for 1,2
C3: 3 => CTR
* ....5..6.769..13..543...81..5...3.9139.......21.7...3..35....7..7.2..4.3.82367.59
* PAIR H3: 1,2 COL H
H9: 1,2,5                                # reduction candidate for 1,2
H9: 1,2                                  # 13 pairs
H9: 5                                    # 12 pairs
* PAIR RESTART
* PAIR H3: 1,2 COL H
H9: 1,2,5                                # reduction candidate for 1,2
H9: 1,2                                  # 13 pairs
H9: 5                                    # 12 pairs
* PAIR G6: 5,6 BLK 6
G5: 5,6,2,7                              # reduction candidate for 5,6
G5: 5,6                                  # 15 pairs
I5: 5,6,2,4,7,8                          # reduction candidate for 5,6
I5: 5,6                                  # 16 pairs
I6: 5,6,4,8                              # reduction candidate for 5,6
I6: 5,6                                  # 18 pairs
* PAIR G6: 5,6 ROW 6
F6: 5,6,4,8,9                            # reduction candidate for 5,6
F6: 5,6                                  # 14 pairs
F6: 4,8,9                                # 10 pairs

* LEVEL 0 PASS 1 ROUND 2 (AUTO SOLVE) (B1,B5,B7,B8,C3,D1,D2,G6,H3,I3)
* ....5..6.769..13..54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6....
* PAIR B1: 2,3 BLK 1
C1: 2,3,1,8                              # reduction candidate for 2,3
C1: 2,3                                  # 18 pairs
C1: 1,8                                  # 11 pairs
* PAIR C3: 1,2 BLK 1
C1: 1,2,3,8                              # reduction candidate for 1,2
C1: 3,8                                  # 11 pairs
C1: 1,2                                  # 11 pairs
* PAIR C3: 1,2 COL C
C9: 1,2,3,4,7                            # reduction candidate for 1,2
C9: 1,2 => CTR
* .3.95..6.7698.13..54.3768.9.5.6.3.91..351....21.7..53.425139678...2854133814679..
* PAIR RESTART
* PAIR C3: 1,2 BLK 1
C1: 1,2,8                                # reduction candidate for 1,2
C1: 8                                    # 12 pairs
C1: 1,2                                  # 11 pairs
* PAIR D1: 4,9 BLK 2
F1: 4,9,2,7                              # reduction candidate for 4,9
F1: 4,9                                  # 17 pairs
F1: 2,7                                  # 10 pairs
* PAIR D1: 4,9 ROW 1
I1: 4,9,2,7                              # reduction candidate for 4,9
I1: 2,7                                  # 17 pairs
I1: 4,9                                  # 14 pairs
* PAIR D1: 4,9 COL D
D7: 4,9,1,3,8                            # reduction candidate for 4,9
D7: 4,9                                  # 19 pairs
D9: 4,9,1,3,5                            # reduction candidate for 4,9
D9: 4,9                                  # 15 pairs
* PAIR D2: 4,8 BLK 2
E2: 4,8,2                                # reduction candidate for 4,8
E2: 4,8 => CTR
* .389527647694813..5423768198576.3.91.9.51824721.7..538.2583..76.7.295483.8.1679..
* PAIR H3: 1,2 COL H
H9: 1,2,5                                # reduction candidate for 1,2
H9: 5 => CTR
* .32.5.167769821345541376829.576.3291.9.51278.2187.9536.25.3.678.762..413.8..67.52
* PAIR RESTART
* PAIR G6: 5,6 BLK 6
G5: 5,6,7                                # reduction candidate for 5,6
G5: 5,6 => SOLVED
* 138954762769821354542376819457683291893512647216749538925438176671295483384167925
* DURATION: 0:00:33.478204  START: 06:25:49.184918  END: 06:26:22.663122 2017-05-01
* SOLUTION FOUND
* SAVE PR GRAPH zz-handy-001-base-pr-001.dot
* REASONING
* DIS # B2: 2,3 => CTR => B2: 6
* DIS B2: 6 # G1: 2,3 => CTR => G1: 1,7,9
* DIS B2: 6 + G1: 1,7,9 # B7: 9 => CTR => B7: 2,3
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 # C3: 3 => CTR => C3: 1,2
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C9: 1,2 => CTR => C9: 3,4,7
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # E2: 4,8 => CTR => E2: 2
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 # H9: 5 => CTR => H9: 1,2
* PRF B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 # G5: 5,6 => SOL
* STA B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 + G5: 5,6
* CNT   8 HDP CHAINS /  36 HYP OPENED
```

```|:info:| zz-handy-001-step-0

strictly solvable by pair-reduction, 2 iterations

* PAIR REDUCTION ..
* ROUND 1: ....5..6.7.9..1...54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6....
B1: 2,3
C1: 1,2,3,8                              # reduction candidate for 2,3
B2: 2,3,6                                # reduction candidate for 2,3
B2: 2,3 => CTR
* ....5.9677.96813.55469..81..5...3.919.3......21.7...3..95....7....29.4.3.82.67.59
C3: 1,2,3                                # reduction candidate for 2,3
G1: 1,2,3,7,9                            # reduction candidate for 2,3
G1: 2,3 => CTR
* .3..5.26.769.213..542...81..5...3.91...512.4821.7..53..25....76...2.5483.8..6....
B7: 2,3,9                                # reduction candidate for 2,3
B7: 9 => CTR
* 12835..6.7698.13..54397681285.6.3.9193.512.4821.789.3.49513827667129548338246715.
H3: 1,2
C3: 1,2,3                                # reduction candidate for 1,2
C3: 3 => CTR
* ....5..6.769..13..543...81..5...3.9139.......21.7...3..35....7..7.2..4.3.82367.59
H9: 1,2,5                                # reduction candidate for 1,2
G6: 5,6
G5: 2,5,6,7                              # reduction candidate for 5,6
I5: 2,4,5,6,7,8                          # reduction candidate for 5,6
I6: 4,5,6,8                              # reduction candidate for 5,6
F6: 4,5,6,8,9                            # reduction candidate for 5,6
* ROUND 2: ....5..6.769..13..54....8...5...3.91.........21.7...3...5....7....2..4.3.8..6....
B1: 2,3
C1: 1,2,3,8                              # reduction candidate for 2,3
C3: 1,2
C1: 1,2,3,8                              # reduction candidate for 1,2
C9: 1,2,3,4,7                            # reduction candidate for 1,2
C9: 1,2 => CTR
* .3.95..6.7698.13..54.3768.9.5.6.3.91..351....21.7..53.425139678...2854133814679..
D1: 4,9
F1: 2,4,7,9                              # reduction candidate for 4,9
I1: 2,4,7,9                              # reduction candidate for 4,9
D7: 1,3,4,8,9                            # reduction candidate for 4,9
D9: 1,3,4,5,9                            # reduction candidate for 4,9
D2: 4,8
E2: 2,4,8                                # reduction candidate for 4,8
E2: 4,8 => CTR
* .389527647694813..5423768198576.3.91.9.51824721.7..538.2583..76.7.295483.8.1679..
H3: 1,2
H9: 1,2,5                                # reduction candidate for 1,2
H9: 5 => CTR
* .32.5.167769821345541376829.576.3291.9.51278.2187.9536.25.3.678.762..413.8..67.52
G6: 5,6
G5: 5,6,7                                # reduction candidate for 5,6
G5: 5,6 => SOLVED
* 138954762769821354542376819457683291893512647216749538925438176671295483384167925
* SOLVED!

* PAIRS (3)
B1: 2,3
H3: 1,2
G6: 5,6

* AUTO ..
A3 = 5                # set value
Q1: 8.. = A1,C1: 8.. => D1,F1 != 8
Q4: 9.. = A5,B5: 9.. => D5,E5,F5 != 9
Q9: 6.. = G7,I7: 6.. => A7,B7 != 6

2,3
1,2
4,8
4,8
4,9
1,9

|:step:| 00
--------------------------------------------------

B1: 2,3
B2: 2,3,6
* DISABLE VALUE:: B2 != 6
* AUTO ..
E3 = 3,7 => CTR
=> B2 = 6

|:step:| 01
--------------------------------------------------

B1: 2,3
G1: 1,2,3,7,9
* DISABLE VALUE:: G1 != 1
* DISABLE VALUE:: G1 != 7
* DISABLE VALUE:: G1 != 9

* AUTO ..
I6 = 0 => CTR

=> G1 = 1,7,9

|:step:| 02
--------------------------------------------------

C3 != 1,2 => C3 = 3 => CTR
=> C3 != 3

E2 !=2 => CTR => E2 = 2

* DISABLE:: A5 != 3
* DISABLE:: A5 != 6
* DISABLE:: A5 != 9
AUTO ..
* SOLVED!

|:step:| 03
--------------------------------------------------

??? |:check:|

D2: 4,8
E2: 2,4,8
* DISABLE VALUE:: E2 != 2
=> D2,E4: 4,8

* DISABLE VALUE:: D1 != 4
* DISABLE VALUE:: F1 != 4
* DISABLE VALUE:: H2 != 4
* DISABLE VALUE:: I2 != 4

H2: 2,5
I2: 2,5
* DISABLE VALUE:: I3 != 2
* DISABLE VALUE:: H3 != 2
H3: 1                 # naked single
* DISABLE VALUE:: I1 != 2

* AUTO ..
H3 = 1                # set value
I1: 4..               # hidden single
H5: 4..               # hidden single
I1 = 4                # set value
H5 = 4                # set value
H8: 8..               # hidden single
H8 = 8                # set value
F8: 5..               # hidden single
F8 = 5                # set value
D5: 5..               # hidden single
D5 = 5                # set value
E5: 1..               # hidden single
E5 = 1                # set value
Q8,Q9: 1.. => A7,A9,C9 != 1

H2: 2,5
I2: 2,5
H9: 2,5
I9: 2,5,9
* FORCE VALUE:: I9 = 9
I9 = 9                # set value
I3: 7                 # naked single
G1: 9..               # hidden single

* AUTO ..
G1 = 9                # set value
D1: 3                 # naked single
I3 = 7                # set value
F1: 7..               # hidden single
E8: 7..               # hidden single
D1 = 3                # set value
B1: 2                 # naked single
F1 = 7                # set value
F9: 4                 # naked single
E8 = 7                # set value
B8: 9                 # naked single
F9 = 4                # set value
D9: 1                 # naked single
A9: 3                 # naked single
C3: 3..               # hidden single
A7: 4..               # hidden single
C9: 7..               # hidden single
E7: 3..               # hidden single
B5: 7..               # hidden single
B1 = 2                # set value
C3: 3                 # naked single
C3 = 3                # set value
B5 = 7                # set value
A7 = 4                # set value
E7 = 3                # set value
B7: 9                 # naked single
B8 = 9                # set value
B7 = 0 => CTR

=> E2 != 4,8
* DISABLE VALUE:: E2 != 4
* DISABLE VALUE:: E2 != 8
=> E2 = 2
E2: 2                 # naked single
* AUTO ..
E2 = 2                # set value
D2: 8..               # hidden single
F5: 2..               # hidden single
G4: 2..               # hidden single
D2 = 8                # set value
G4 = 2                # set value
F5 = 2                # set value
C4: 7..               # hidden single
C4 = 7                # set value
B8: 7..               # hidden single
F9: 7..               # hidden single
B8 = 7                # set value
F9 = 7                # set value
E3: 7..               # hidden single
E3 = 7                # set value
E7: 3..               # hidden single
E7 = 3                # set value
Q2: 4.. = D1,F1: 4.. => I1 != 4
D4: 6                 # naked single
Q2,Q8: 4.. => D4,D5,F6 != 4
D4 = 6                # set value
F3: 6..               # hidden single
F3 = 6                # set value

|:step:| 03
--------------------------------------------------

I3: 2,9
I1: 2,7,9
* DISABLE VALUE:: I1 != 7
* DISABLE VALUE:: I9 != 2
* DISABLE VALUE:: I9 != 9
I9: 5                 # naked single
* DISABLE VALUE:: I7 != 9
* DISABLE VALUE:: I7 != 2
* DISABLE VALUE:: G1 != 9
* AUTO ..
* SOLVED!

I9 = 5                # set value
I2: 4                 # naked single
H2: 5..               # hidden single
G1: 7..               # hidden single
F8: 5..               # hidden single
H9: 2..               # hidden single
B7: 2..               # hidden single
D5: 5..               # hidden single
I5: 7..               # hidden single
G1 = 7                # set value
H2 = 5                # set value
I2 = 4                # set value
D5 = 5                # set value
G5: 6                 # naked single
G5 = 6                # set value
G6: 5                 # naked single
I6: 8                 # naked single
I5 = 7                # set value
G6 = 5                # set value
I6 = 8                # set value
H5: 4                 # naked single
I7: 6                 # naked single
F6: 9                 # naked single
B7 = 2                # set value
B1: 3                 # naked single
I7 = 6                # set value
F8 = 5                # set value
H9 = 2                # set value
H3: 1                 # naked single
C6: 6..               # hidden single
E5: 1..               # hidden single
H8: 8..               # hidden single
F7: 8..               # hidden single
A8: 6..               # hidden single
B5: 9..               # hidden single
B1 = 3                # set value
B5: 9                 # naked single
H3 = 1                # set value
H8: 8                 # naked single
C3: 2                 # naked single
B5 = 9                # set value
E5 = 1                # set value
H5 = 4                # set value
C6 = 6                # set value
C8: 1                 # naked single
F6 = 9                # set value
E6: 4                 # naked single
F1: 4                 # naked single
F7 = 8                # set value
E8: 9                 # naked single
A8 = 6                # set value
C8 = 1                # set value
E8 = 9                # set value
H8 = 8                # set value
A1: 1..               # hidden single
D3: 3..               # hidden single
A4: 4..               # hidden single
E4: 8..               # hidden single
C9: 4..               # hidden single
A1 = 1                # set value
F1 = 4                # set value
D1: 9                 # naked single
C3 = 2                # set value
C1: 8                 # naked single
I3: 9                 # naked single
D3 = 3                # set value
I3 = 9                # set value
I1: 2                 # naked single
A4 = 4                # set value
A7: 9                 # naked single
E4: 8                 # naked single
E4 = 8                # set value
E6 = 4                # set value
A7 = 9                # set value
A9: 3                 # naked single
G7: 1                 # naked single
G7 = 1                # set value
G9: 9                 # naked single
D7: 4                 # naked single
A9 = 3                # set value
C9: 4                 # naked single
A5: 8                 # naked single
C9 = 4                # set value
D9: 1                 # naked single
D9 = 1                # set value
G9 = 9                # set value
C5: 3..               # hidden single
C1 = 8                # set value
C5: 3                 # naked single
D1 = 9                # set value
I1 = 2                # set value
A5 = 8                # set value
C5 = 3                # set value
D7 = 4                # set value

|:step:| 04
--------------------------------------------------

alternate solution

G6: 5,6
G5: 5,6,7
* DISABLE VALUE:: G5 != 7
=> G5,G6: 5,6

* DISABLE VALUE:: H5 != 5
* DISABLE VALUE:: I5 != 5
* DISABLE VALUE:: I5 != 6
* DISABLE VALUE:: I6 != 5
* DISABLE VALUE:: I6 != 6
* DISABLE VALUE:: G7 != 6
* DISABLE VALUE:: G9 != 5

* NEW PAIR
G7: 1,9
G9: 1,9

* DISABLE VALUE:: G1 != 1
* DISABLE VALUE:: G1 != 9
G1: 7                 # naked single

* AUTO ..
* SOLVED!

G1 = 7                # set value
H3: 1..               # hidden single
I5: 7..               # hidden single
I7: 6..               # hidden single
H3 = 1                # set value
I5 = 7                # set value
I7 = 6                # set value
H8: 8..               # hidden single
B7: 2..               # hidden single
F7: 8..               # hidden single
H9: 2..               # hidden single
I6: 8..               # hidden single
I6 = 8                # set value
H5: 4                 # naked single
B7 = 2                # set value
B1: 3                 # naked single
F7 = 8                # set value
H8 = 8                # set value
H9 = 2                # set value
I9: 5..               # hidden single
F8: 5..               # hidden single
B5: 9..               # hidden single
F1: 4..               # hidden single
H2: 5..               # hidden single
I2: 4..               # hidden single
B1 = 3                # set value
C3: 2                 # naked single
B5: 9                 # naked single
F1 = 4                # set value
D1: 9                 # naked single
H2 = 5                # set value
I2: 4                 # naked single
I2 = 4                # set value
C3 = 2                # set value
I3: 9                 # naked single
I3 = 9                # set value
I1: 2                 # naked single
I9: 5                 # naked single
D3: 3                 # naked single
B5 = 9                # set value
H5 = 4                # set value
F8 = 5                # set value
F6: 9                 # naked single
I9 = 5                # set value
D5: 5..               # hidden single
G6: 5..               # hidden single
D1 = 9                # set value
I1 = 2                # set value
D3 = 3                # set value
D5 = 5                # set value
G5: 6                 # naked single
G5 = 6                # set value
G6: 5                 # naked single
F6 = 9                # set value
E6: 4                 # naked single
G6 = 5                # set value
C6: 6..               # hidden single
E5: 1..               # hidden single
E8: 9..               # hidden single
A8: 6..               # hidden single
E5 = 1                # set value
E8: 9                 # naked single
C6 = 6                # set value
C8: 1                 # naked single
E6 = 4                # set value
E4: 8                 # naked single
A8 = 6                # set value
C8 = 1                # set value
C1: 8                 # naked single
E8 = 9                # set value
A1: 1..               # hidden single
A4: 4..               # hidden single
C9: 4..               # hidden single
A1 = 1                # set value
C1 = 8                # set value
C5: 3                 # naked single
A4 = 4                # set value
A7: 9                 # naked single
E4 = 8                # set value
C5 = 3                # set value
A5: 8                 # naked single
C9: 4                 # naked single
A7 = 9                # set value
A9: 3                 # naked single
G7: 1                 # naked single
G7 = 1                # set value
G9: 9                 # naked single
D7: 4                 # naked single
A9 = 3                # set value
C9 = 4                # set value
D9: 1                 # naked single
D9 = 1                # set value
G9 = 9                # set value
A5 = 8                # set value
D7 = 4                # set value

|:step:| 10
--------------------------------------------------

|:info:| old ..

Q9: 6.. = G7,I7: 6.. => A7,B7 != 6
Q1: 8.. = A1,C1: 8.. => D1,F1 != 8
Q4: 9.. = A5,B5: 9.. => D5,E5,F5 != 9

F8 != 5 => CTR => F8 = 5
B2 != 6 => CTR => B2 = 6
C3 = 1 => CTR => C3 != 1
F5 != 2 => CTR => F5 = 2

G2 != 3 => CTR => G2 = 3
C3 = 3 => CTR => C3 != 3
D1 != 9 => CTR => D1 = 9
I9,H9: 2,5 => ...

? C4 != 7 => A4,C4,C6 = 4,6,8 + A4,C4,D4 = 4,6,8 => Q4\A4,C4,C6 != 4,6,8 + E4,G4 != 4,6,8 => E4 = 2
? B5 != 3 =>
```

# Solution

position: 138954762769821354542376819457683291893512647216749538925438176671295483384167925 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 # C1: 2,3 => UNS
* DIS # B2: 2,3 => CTR => B2: 6
* INC # B2: 6 => UNS
* INC # C3: 2,3 => UNS
* DIS # G1: 2,3 => CTR => G1: 1,7,9
* INC # G1: 1,7,9 => UNS
* INC # B7: 2,3 => UNS
* DIS # B7: 9 => CTR => B7: 2,3
* INC # G1: 1,2 => UNS
* INC # G1: 3,7,9 => UNS
* INC # C3: 1,2 => UNS
* DIS # C3: 3,6 => CTR => C3: 1,2
* INC # H9: 1,2 => UNS
* INC # H9: 5 => UNS
* INC # G5: 5,6 => UNS
* INC # I5: 5,6 => UNS
* INC # I6: 5,6 => UNS
* INC # F6: 5,6 => UNS
* INC # F6: 4,8,9 => UNS
* CNT  19 HDP CHAINS /  19 HYP OPENED
```

## A2. Pair Reduction

Full list of HDP chains traversed:

```* INC # C1: 2,3 => UNS
* DIS # B2: 2,3 => CTR => B2: 6
* INC B2: 6 # C3: 2,3 => UNS
* DIS B2: 6 # G1: 2,3 => CTR => G1: 1,7,9
* INC B2: 6 + G1: 1,7,9 # B7: 2,3 => UNS
* DIS B2: 6 + G1: 1,7,9 # B7: 9 => CTR => B7: 2,3
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 # C1: 2,3 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 # C3: 2,3 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 # C3: 1,2 => UNS
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 # C3: 3 => CTR => C3: 1,2
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # H9: 1,2 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # H9: 5 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # H9: 1,2 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # H9: 5 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # G5: 5,6 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # I5: 5,6 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # I6: 5,6 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # F6: 5,6 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # F6: 4,8,9 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C1: 2,3 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C1: 1,8 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C1: 1,2 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C1: 3,8 => UNS
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 # C9: 1,2 => CTR => C9: 3,4,7
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # C1: 1,2 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # C1: 8 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # F1: 4,9 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # F1: 2,7 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # I1: 4,9 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # I1: 2,7 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # D7: 4,9 => UNS
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # D9: 4,9 => UNS
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 # E2: 4,8 => CTR => E2: 2
* INC B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 # H9: 1,2 => UNS
* DIS B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 # H9: 5 => CTR => H9: 1,2
* PRF B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 # G5: 5,6 => SOL
* STA B2: 6 + G1: 1,7,9 + B7: 2,3 + C3: 1,2 + C9: 3,4,7 + E2: 2 + H9: 1,2 + G5: 5,6
* CNT  36 HDP CHAINS /  36 HYP OPENED
```