- Code: Select all
`21 07`

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30| | | |

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11| | | |12

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| | | |38

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30 07 27

Step 1:

The two 7(?) cages must be of length 5 (length 3 must have singleton, length 4 must be even-total)

The two 7(5) cages must be in the form [{12}1{12}]

Their starting ends must be R19C5, their stopping ends must be R5C19

(stopping ends must be different, otherwise two 1s on the same column)

--> R19C5={12} (C5), R5C19={12} (R5)

--> The two 7(5) cages together force D\37=[11] or D/37=[11]

Step 2:

The 11(?) cage also must be of length 5, from R5C1 to R1C5 or R9C5, with the middle cell in R3C3 or R7C3

The 11(5) cage must be in the form [{ab}a{ab}] --> 11=3a+2b --> a must be odd and a<5

Since one of R37C3 is [1] and belongs to a 7(5) cage, a cannot be 1 --> a must be 3

11(5)=[{13}3{13}]=[13331] (Step 1: R5C1 and R19C5 must be from {12})

Step 3:

Summarising steps 1 & 2, we have the following 2 possible cases:

R1C5 -7(5)- R5C1 -11(5)- R9C5 -7(5)- R5C9: [2112133312112]

R9C5 -7(5)- R5C1 -11(5)- R1C5 -7(5)- R5C9: [2112133312112]

--> R5C19=[12], R46C2={23} (C2,N4), R37C3={13} (C3), R28C4={13} (C4)

--> [2] of C6 locked in R28C6, [1] of C7 locked in R37C7, [1] of C8,N6 locked in R46C8

Step 4:

The 21(?) cage must be of length 7/8/9 (length 3/4/5 must have singleton, length 6 must be even-total)

--> 21(?) must include D\456 and all of these 3 cell values must repeat at least once

--> D\46 must contain at least one of {12}

(otherwise min D\456={345} and when these cell values repeat the total will exceed 21)

--> R4C2+R6C8<>[21] --> R6C2+R4C8=[21]

--> (step 3) R1C5..R5C1=[13331], R1C5..R5C9=[12112], R5C1..R9C5=[12112]

Hidden singles: D\2469=[1211]

Step 5:

The 12(?) cage must be of length 4/5 (length 3 must have singleton)

If the length is 4, with [2] already in R5C9, 12(4)=[2424]

If the length is 5, with [2] already in R5C9+R9C5, 12(5)=[23232]

--> D\7=[2], R6C8+R8C6=[33/44] (<>{56789})

Step 6:

Now the 21(?) cage already has the cell values {123} within D\123456, each must repeat at least once

--> 6 cells of 21(?) must be {112233} totalling 12, 9 short of 21

--> At least one value from {456789} must appear in 21(?) and repeat at least once

(otherwise max total = 1+1+1+2+2+2+3+3+3 = 18 < 21)

--> To make up the remaining total of 9, there is only one possible way: {144}

--> 21(?) must be of length 9 --> 21(9)=D\={111223344}

--> D\1=[4], D\58={34} (<>{56789})

Step 7:

Consider the bottom 30(?) cage: it must start from R8C1 and extend towards R2C9 with a min length of 4

Now [2] of N1 is locked in R1C3+R3C1

--> R3C8+R8C3, collectively seeing R1C3+R3C1, cannot be both [2]

--> 30(?) R9C2 cannot contain [2] as a repeating cell value

--> R8C3<>[2]

Hidden singles: R1C3=R8C1=[2] --> R3C8=[2]

Step 8:

Consider the 46(?) cage: it must start from R4C9 and extend towards R9C4 with a length of 6

--> The only possiblility is 46(6)=[{689}{689}] (<>{34578})

--> R4C9+R5C8+R6C7={689} (N6), R7C6+R8C5=R9C4={689} (N8)

Step 9:

Consider the 38(?) cage: it must start from R7C9 and extend towards R1C3 with a min length of 5

It cannot be of length 7 (no second [2] to match R1C3=[2])

It cannot be of length 5 (max R6C8=4 and it must repeat, so max total would be 4+4+9+9+9=35<38)

--> 38(?) must be of length 6, from R7C9 to R2C4 --> R2C4=[3] must repeat within the cage

--> R6C8<>[4] (otherwise max total would be 3+3+4+4+9+9=32<38)

--> (step 5) R6C8+R8C6=[33] --> (step 6) D\58=[34]

--> with R2C4+R6C8=[33], remaining total=38-3-3=32

--> with max R5C7=7, the only way to make this up is {7799}

--> R5C7=[7], R3C5+R4C6+R7C9 must be from {79} (<>{34568})

Hidden singles: R1C9=R7C1=R9C7=[3]

Step 10:

Consider the 27(?) cage: it must start from R9C7 and extend towards R3C1 with R4C2+R8C6+R9C7=[333]

It cannot be of length 4/5/6 (to avoid singletons max total would be 3+3+3+5+5+5=24<27)

--> 27(?) must be of length 7, from R9C7 to R3C1 --> R3C1+R5C3+R6C4+R7C5=27-3-3-3=18

--> To make up this total of 18 without singletons, there is only one possible way: {4455}

--> R3C1=[5], R5C3+R6C4+R7C5 must be from {45} (<>{6789})

Step 11:

Consider the 25(?) cage: it must start from R3C9 and extend towards R9C3 with R4C8+R5C7+R6C6+R8C4=[1711]

It cannot be of length 3/5/6 (singletons would appear)

It cannot be of length 4 (max length would be 1+1+7+7=16<25)

--> 25(?) must be of length 7, from R3C9 to R9C3 --> 25(7) = {1114477} (only possible way without singleton)

--> R7C5=[4], R3C9+R9C3={47} (<>{5689})

--> [4] of N2 locked in R3C46 (R3) --> R3C9+R9C3=[74]

--> (step 10) R5C3+R6C4=[54] --> R4C7+R6C9=[45]

--> (step 9) R3C5+R4C6+R7C9=[979] --> R7C4+R9C6=[75]

Hidden singles: R2C9=R3C6=R5C2=[4], R1C4=R4C5=[5], R1C2=R2C5=R9C8=[7], R5C6=R9C4=[9]

Step 12:

Consider the top 30(?) cage: it must start from R2C1 and extend towards R9C8 with a min length of 4

It cannot be of length 5/6/7/8 (min total would be 6+8+6+6+8=34>30)

--> 30(?) R2C1 must be of length 4, from R2C1 to R5C4 --> 30(4) R2C1 = [9696]

(only possible way without singleton)

--> 30(?) R9C2 must be of length 4, from R9C2 to R6C5 --> 30(4) R9C2 = [8778]

Naked singles to finish.

- Code: Select all
`472516893`

918372654

563894127

839257416

145639782

726481935

351748269

297163548

684925371

21(9) = [413231241]

7(5) = [12112]

30(4) = [9696]

25(7) = [7171414]

46(6) = [689869]

11(5) = [13331]

12(5) = [23232]

38(6) = [397739]

30(4) = [8778]

7(5) = [12112]

27(7) = [5354433]

Total = 63 repeats