Dec 25 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 can only be <1>
R1C3 can only be <7>
R1C2 can only be <6>
R3C3 can only be <8>
R2C1 can only be <2>
R4C2 is the only square in row 4 that can be <7>
R9C3 is the only square in column 3 that can be <1>
R3C7 is the only square in column 7 that can be <4>
R2C7 is the only square in column 7 that can be <1>
R2C8 is the only square in row 2 that can be <8>
R4C1 is the only square in block 4 that can be <5>
R9C1 can only be <9>
R4C4 is the only square in row 4 that can be <8>
Intersection of row 6 with block 5. The values <45> only appears in one or more of squares R6C4, R6C5 and R6C6 of row 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R4C5 - removing <4> from <1349> leaving <139>
R4C6 - removing <4> from <1249> leaving <129>
R5C5 - removing <4> from <13469> leaving <1369>
R6C5 is the only square in column 5 that can be <4>
Squares R1C6 and R6C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <59>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C6 - removing <5> from <1256> leaving <126>
R4C6 - removing <9> from <129> leaving <12>
R7C6 - removing <9> from <149> leaving <14>
R8C6 - removing <9> from <469> leaving <46>
Intersection of column 7 with block 9. The values <28> only appears in one or more of squares R7C7, R8C7 and R9C7 of column 7. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain these values.
R7C8 - removing <2> from <2379> leaving <379>
Intersection of block 8 with column 4. The value <9> only appears in one or more of squares R7C4, R8C4 and R9C4 of block 8. These squares are the ones that intersect with column 4. Thus, the other (non-intersecting) squares of column 4 cannot contain this value.
R2C4 - removing <9> from <679> leaving <67>
R5C4 - removing <9> from <12369> leaving <1236>
R6C4 - removing <9> from <359> leaving <35>
Squares R4C9<349>, R5C9<349> and R6C8<39> in block 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <349>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C8 - removing <39> from <1239> leaving <12>
R5C8 - removing <39> from <1239> leaving <12>
Squares R4C6 and R4C8 in row 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <12>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C5 - removing <1> from <139> leaving <39>
Squares R4C5<39>, R6C4<35> and R6C6<59> in block 5 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <359>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R5C4 - removing <3> from <1236> leaving <126>
R5C5 - removing <39> from <1369> leaving <16>
Squares R7C7<2389>, R7C8<379>, R7C9<379>, R8C7<389> and R9C7<23> in block 9 form a comprehensive naked set. These 5 squares can only contain the 5 possibilities <23789>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C8 - removing <39> from <3569> leaving <56>
R9C9 - removing <37> from <3567> leaving <56>
Intersection of row 9 with block 8. The value <7> only appears in one or more of squares R9C4, R9C5 and R9C6 of row 9. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R7C4 - removing <7> from <1379> leaving <139>
Squares R3C6 and R8C6 in column 6 and R3C8 and R8C8 in column 8 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in rows 3 and 8 can be removed.
R3C4 - removing <6> from <12567> leaving <1257>
R8C4 - removing <6> from <369> leaving <39>
R3C5 - removing <6> from <167> leaving <17>
R3C9 - removing <6> from <567> leaving <57>
Squares R2C5 and R2C9 in row 2, R4C3, R4C5 and R4C9 in row 4 and R5C3 and R5C9 in row 5 form a Swordfish pattern on possibility <9>. All other instances of this possibility in columns 3, 5 and 9 can be removed.
R7C9 - removing <9> from <379> leaving <37>
Squares R3C5 (XY), R5C5 (XZ) and R2C4 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R2C5 - removing <6> from <679> leaving <79>
R5C4 - removing <6> from <126> leaving <12>
R5C5 is the only square in row 5 that can be <6>
R3C5 is the only square in column 5 that can be <1>
Squares R3C8 (XYZ), R3C9 (XZ) and R8C8 (YZ) form an XYZ-Wing pattern on <5>. All squares that are buddies of all three squares cannot be <5>.
R1C8 - removing <5> from <359> leaving <39>
R1C6 is the only square in row 1 that can be <5>
R6C6 can only be <9>
R6C8 can only be <3>
R4C5 can only be <3>
R6C4 can only be <5>
R1C8 can only be <9>
R1C7 can only be <3>
R7C8 can only be <7>
R9C5 can only be <7>
R7C9 can only be <3>
R9C7 can only be <2>
R2C5 can only be <9>
R7C2 is the only square in row 7 that can be <2>
Squares R4C3, R4C9, R5C3 and R5C9 form a Type-1 Unique Rectangle on <49>.
R5C3 - removing <49> from <349> leaving <3>
R5C2 can only be <8>
R8C3 can only be <4>
R8C6 can only be <6>
R4C3 can only be <9>
R7C1 can only be <8>
R8C8 can only be <5>
R3C6 can only be <2>
R9C4 can only be <3>
R3C8 can only be <6>
R9C9 can only be <6>
R9C2 can only be <5>
R8C4 can only be <9>
R2C9 can only be <7>
R2C4 can only be <6>
R3C9 can only be <5>
R3C4 can only be <7>
R4C6 can only be <1>
R4C9 can only be <4>
R4C8 can only be <2>
R7C6 can only be <4>
R5C4 can only be <2>
R5C8 can only be <1>
R5C9 can only be <9>
R5C1 can only be <4>
R8C2 can only be <3>
R7C7 can only be <9>
R7C4 can only be <1>
R8C7 can only be <8>
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