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A General Logic for Sudoku Monster #224 from list 1465 |
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Solution to Monster #224
from List 1465 This is a 17-step set-logic solution to
monster puzzle #224 from the 1465 list.
The original solution to the puzzle by (ttt) was presented here. The
elimination by sets is of the form: rank N(n sets+ m linksets), [ cover set list
]x[ cover linkset list ] reason for elimination
=> results For more information on notation and set logic eliminations, see this quick reference. For a full description of sets and Sudoku set logic, see the SudokuOne.com website. Elimination 1, ALS Chain This elimination works like a simple chain
where one link is replaced by an ALS made of the two cell sets. Column
set 9c3 overlaps 2 cell linksets 7n3 and 8n3 to eliminate the 2
candidates. The set logic notation is: Rank 1(5+6), [9r3 2r7 6c3 16n2]x[129c2 78n3
9c3] 9c2*(78n3)
=> r78c3<>9
Elimination 2, ALS Chain This elimination works like a simple chain
where one link is replaced by an ALS made of the two cell sets. Column
set 2c3 overlaps cell linkset 3n3 to eliminate the candidate. The set
logic notation is: Rank 1(4+5), [9r3 2r7 16n2]x[129c2 2c3 3n3] 2c2*3n3 => r3c3<>2
Elimination 3, 3D Kraken
+ ALS Chain, Dual Overlap
This logic works like a chain where 2 links
are made of ALSs. The ends of the chain then overlap to eliminate the 2
candidates. Alternatively row sets 46r4 can
be considered the body of a kraken structure. The set logic notation
for the elimination is: Rank 1(6+7), [12r5 46r6 59c4]x[5n46 6n479 8n4
2b6] 2b6*5n46
=> r6c7<>2,
r6c9<>2
Elimination 4, Finned
X-Wing + ALS Chain for 2nd Fin
This logic would be a dual finned X-wing in
row sets, 3r36c146, with one fin in box set 6b2. The other fin that
would be in 6b5 is covered by a short ALS chain, 6r6c(6-4)=4r6c4-(als)456r78c4-6r36c4.
The set logic notation for the elimination is: Rank 1(5+6), [6r36 78n7 4b6]x[6c4 456c7 6n9
6b3] 6c7*6b3
=> r2c7<>6
Elimination 5, Multi-loops
This rank 1 logic works by the overlap of 2
linksets 5c7
and 5r5. The looped structure is made of 3 ALS and 1 cell set 5n5. This might be expressed as 5r5c4-als(5=46r78c4)-als(46=46r6c146)-(59=r568c1)-46r5c5.
The set logic notation for the elimination is: Rank 1(8+7), [46r6 59c4 78n7 5n8]x[59r5 456c7
6n49 8n4] 5c7*5r5
=> r5c7<>5
Elimination 6, Multi-loops
This rank 2 group of multi-loops can
eliminate candidates anywhere that 3 linksets overlap.
Candidate 5r6c7 is therefore eliminated by the overlap of 5r6, 6n7, and 5b6. Candidate 5r7c8 is eliminated by the
overlap of 2 linksets 5c8 and 7c8 augmented by a triplet at 5r4c8,
which requires set 5c8 be a linkset thus the two eliminations use two
different groups of cover sets for the same candidates. The set logic notation for the elimination is: Rank 2(12+14), [8r1 46r6 6c1 5c3 3c7 356c8
23n4 6b2]x [6r1 36r2 5r6 3r9 168c4 1n3 4n8 6n79 7n8 5b6] 5r6*6n7*5b6 => r6c7<>5, (L)5c8*7c8 =>
r7c8<>5
Elimination 7, Complex
Structure
This is a complex
rank 2 elimination where the candidate is overlapped by 2 linksets
(highlight below) augmented by the linkset triplet A at 1r1c1. The set
logic notation for the elimination is: Rank 2(16+18), [13568r1 6r2 1r6 5c3 9n1
1236n4 1b3 9b4 6b5]x [1r3 59r6 169c1 1c2 368c4 6c6 1n139 2n8 6n3
13b2], (T1r1c1)1r3*1b2
=> r3c5<>1
Elimination 8, Locked
Candidates The set logic notation for the elimination is: Rank 0(1+1), [1c5]x[1b8] 1b8 =>
r7c6<>1, r8c4<>1, r8c6<>1
Elimination 9,
Intertwined ALSs Produce Rank 0 Sets
This logic is made of a column ALS,
4r2378c256, an AALS, 14569r78c678, and a branching chain that
interleaves through both. Although the overall rank is 2, linksets 4r7 and 8n5, highlighed black, are both rank 0 and can
eliminate candidates without overlap. The rank of both sets is lowered
by the presence of 2 linkset triplets each. All 3 triplets are in
column set 4r5. The set logic notation is: Rank 2(8+10), [5r7 4c256 7n89 8n7 1b8]x[4r3
149r7 4r8 78n5 4b2 56b9] (TT)4r7 => r7c3<>4,
r7c7<>4, (TT)8n5 => r8c5<>3, r8c5<>5
Elimination 10,
Intertwined ALSs Produce Rank 0 Sets This logic is almost the same as elimination
9. The logic is made of a cell ALS,
1249r`36c2, a column AALS, 19r1269c1, and a branching chain that
interleaves through both. The overall rank is 2 and linkset 2c3, highlighted black, is rank 0 because of 2
linkset triplets. The set logic notation is: Rank 2 = 10-8, [2r6 19c1 136n2 3n5 2b7], [4r3
129c2 2c3 2c5 69n1 1b1 9b4] (TT)2c3 => r1c3<>2, r2c3<>2
Elimination 11, Complex
Structure
This is a complex piece of rank 4 logic where
2 overlap linksets eliminate the candidate at 2r3c7. The logic must
therefore have at least 3 linkset triplets to augment the rank of the
overlap linksets. The most obvious is the nearby triplet A. Four other
triplets are marked with red arrows. The set logic notation for the
elimination is: Rank 4 = 19-15, [4r269 6r36 5c48 7c9 3n5 5n7
7n7 8n357 6b6]x [2r3 57r5 156r8 4c3 6c4 4c5 2456c7 6c9 4n8
6n49 9n9 4b2] (TTT)2r3*2c7
=> r3c7<>2
Elimination 12, 3D
Multi-loop with Triple Overlap
This logic forms a rank 4 multiloop that has
3 short chain branches. All three branches converge to eliminate the
candidate at 6r7c7, which assigns 5r7c7 triggering more eliminations.
The rank of the 3 branch chains is augmented by two triplets A and B.
The set logic notation for the elimination is: Rank 4 = 14-10, [5r147 6c1 8c3 14c5 6c8 3n7
4b1], [6r2 48r3 6r7 5c15 6c7 1n13 2n3 4n8 7n57 8n5] (TT)6r7*6c7*7n7 => r7c7<>6 5r7c7 =>
r7c57<>5 => 5r8c5 => r56c4<>5
r8c4<>39 => 5r5c8 => r4c8<>5 r5c8<>9.
Elimination 13, Hidden
Pair The set logic notation for the elimination is: Rank 0(2+2), [39r8]x[8n26] 8n26 => r8c2<>1, r8c2<>4, r8c6<>4
Elimination 14, Box Ended
Chain This logic forms a simple chain with a box
set end. The set logic notation for the elimination is: Rank 1(2+3), [4c2 4b8]x[4r37 4c5] 4r3*4c5 =>
r3c5<>4 => 4r2c6 => r2c3<>4
r2c6<>1236 r7c6<>4 => 2r3c5 =>
r3c29<>2r46c5<>2 r1c6<>2
Elimination 15, Simple
Chain
This logic is a simple chain or discontinuous
nice loop that eliminates the candidate 4r7c2. This leads to the
assignment of 4r3c2 and several subsequent eliminations. The set logic
notation for the elimination is: Rank 1(4+5), [2r7 6c3 4c5 8n7], [4r7 46r8
7n23] 4r7*7n2 =>
r7c2<>4 => 4r3c2 => r3c2<>19,
r3c3<>4 => 9r3c3 => r3c3<>18 r79c3<>9,
r5c7<>2 => 8r3c7 => r3c7<>6,
r3c4<>8=> 7r5c7 => r5c9<>7,
r9c7<>7 => 7r9c9 => r9c9<>1349
Elimination 16,
Multi-loop + Converging Chains
Columns 1 and 2 contain a rank 2 AALS like
structure with 1 box and 2 column sets. This then has 2 branches from
2r2 and 9r6 that converge at triplet 2r5c6, which continue as a single
rank 1 chain. The end of the chain 6r1 overlaps cell linkset 1n1 to
eliminate candidate 6r1c1. at The set logic notation for the
elimination is: Rank 2(8+10), [1r5 2c167 5c1 6c6 9c4
9b4]x[2r25 6r1 9r6 146n1 5n4 45n6] (T)6r1*1n1 => r1c1<>6 => 6r2c1 => r2c1<>12,
r2c48<>6, => 2r2c7, r2c7<>38, r1c9<>2,
Elimination 17, Discontinuous Nice Multi-loop with ALS
Rows 1 2 and 3 contain part of a
discontinuous nice loop with 2 strong links that meet at 6r3c6 to
assign the candidate. However, part of the loop in rows 4 and 5 is made
of a rank 1 dual loop that connects to the DNL through linkset triplet
6r4c3. The set logic notation for the
elimination is: Rank 1(8+9), [6r3 19r5 26c6 4n8 36b3]x [6r4
3c8 1n9 3n9 4n6 5n46 6b2 9b6] 6r3*6b3 => 6r3c9, r1c9<>6,
r3c4<>6 (singles to the end)
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