“The usual uncaged killer rules, the number is in the top left corner with top dominating left as usual. There are no single cages. Note each question mark represents a digit so, for example, a double cage with a single ? does not contain a 9.”

I assume that there are no diagonally-connected cages.

Prelims, based just on positions of totals.

a) R1C12 must be 5(2) cage = {14/23}

b) R1C78 must be 8(2) cage = {17/26/35}, no 4,8,9

c) R2C45 must be 3(2) cage = {12}

d) R1C345 must be 21(3) cage = {489/579/678}, no 1,2,3

e) R12C6 must be 12(2) cage = {39/48/57}, no 1,2,6

f) R23C3 must be 5(2) cage = {14/23}

g) There must be a cage starting at R7C3 including R7C4

h) R6C34 must be 11(2) cage = {29/38/47/56}, no 1

i) R5C45 must be ?(2) cage, no 9

j) R3C45 must be 8(2) cage = {17/26/35}, no 4,8,9

k) R34C6 must be 8(2) cage = {17/26/35}, no 4,8,9

l) R4C45 must be 14(2) cage = {59/68}

m) R45C3 must be ??(2) cage

n) R56C6 must be ??(2) cage

o) There must be 18(?) cage in R9C123, possibly also including R9C4

p) R8C12 must be 14(2) cage = {59/68}

q) R7C12 must be 8(2) cage = {17/26/35}, no 4,8,9

r) R56C1 must be ??(2) cage

s) R56C2 must be ?(2) cage, no 9

t) R23C7 must be 12(2) cage = {39/48/57}, no 1,2,6

u) R4C78 must be ?(2) cage, no 9

v) R5C78 must be 14(2) cage = {59/68}

w) R45C9 must be 9(2) cage = {18/27/36/45}, no 9

x) R7C67 must be 12(2) cage = {39/48/57}, no 1,2,6

y) R6C78 must be ?(2) cage, no 9

z) R67C9 must be ??(2) cage

aa) There must be a 17(?) cage in R8C34, possibly also including R9C4

ab) R678C5 must be 21(3) cage {489/579/678}, no 1,2,3

ac) R7C34 must be ?(2) cage, no 9

ad) R9C56 must be ?(2) cage, no 9

ae) R8C67 must be ?(2) cage, no 9

af) R78C8 must be ?(2) cage, no 9

ag) R89C9 must be 10(2) cage = {19/28/37/46}, no 5

ah) R9C78 must be 14(2) cage = {59/68}

and two more based on a clash

ai) 17(3) cage in R8C34 + R9C4 (cannot be 17(2) cage in R8C34 which would clash with R8C12)

aj) 18(3) cage at R9C1

This just leaves a pair of cages in N14 which can be 13(2) and 21(4) or 13(3) and 21(3) and a pair of cages in N3 which can be 9(2) and 16(3) or 9(3) and 16(2)

Steps resulting from Prelims

1a. Naked quad {1234} in R1C12 + R23C3, locked for N1

1b. Naked pair {12} in R2C45, locked for R2 and N2, clean-up: no 3,4 in R3C3, no 6,7 in R3C45, no 6,7 in R4C6

1c. Naked pair {35} in R3C45, locked for R3 and N2, clean-up: no 7,9 in R12C6, no 7,9 in R2C7, no 3,5 in R4C6

1d. Naked pair {48} in R12C6, locked for C6 and N2, clean-up: no 4,8 in R7C7

2. 9 in N2 only in R1C45 -> 21(3) cage at R1C3 = {579} (only remaining combination) -> R1C3 = 5, R1C45 = {79}, locked for R1 and N2, clean-up: no 1,3 in R1C78, no 6 in R6C4

2a. R3C6 = 6 -> R4C6 = 2, clean-up: no 7 in R5C9, no 9 in R7C3

2b. Naked pair {26} in R1C78, locked for R1 and N3, clean-up: no 3 in R1C12

2c. Naked pair {14} in R1C12, locked for R1 and N1 -> R12C6 = [84], R23C3 = [32], clean-up: no 8,9 in R3C7, no 8,9 in R7C4

3. R1C9 = 3, no 6 in R2C9 -> 9(3) cage in R123C9 = {135} (only remaining combination) -> R23C9 = [51], clean-up: no 4,6,8 in R45C9, no 7,9 in R89C9

3a. R2C7 = 8 -> R3C7 = 4, clean-up: no 6 in R5C8, no 8 in R9C8

3b. R23C8 must be 16(2) cage = {79}, locked for C8, clean-up: no 5 in R5C7, no 5 in R9C7

3c. R45C9 = [72], clean-up: no 8 in R89C9

3d. Naked pair {46} in R89C9, locked for C9 and N9

3e. R9C7 = 9 -> R67C9 = [98], R9C8 = 5, clean-up: no 3,7 in R7C6

3f. R5C78 = [68] -> R1C7 = 2

4. 13(?) cage at R2C1, min R23C1 = 13 -> 13(2) cage in R23C1 = [67], clean-up: no 1,2 in R7C2, no 8 in R8C2

4a. That leaves R23C2 + R4C12 as the remaining 21(4) cage, R23C2 = [98] = 17 -> R4C12 = 4 = {13}, locked for R4 and N4, clean-up: no 5 in R8C1

4b. R4C78 = [54], clean-up: no 9 in R4C45

4c. Naked pair {68} in R4C45, locked for R4 and N5 -> R4C3 = 9

4d. Naked pair {13} in R6C78, locked for R6, clean-up: no 8 in R7C3

4e. Naked triple {457} in R6C456, locked for R6 and N5

4f. R6C3 = 6 -> R6C4 = 5, R6C6 = 7

4g. R6C5 = 4 -> R78C5 = 17 = [98]

4h. R7C6 = 5 -> R7C7 = 7, clean-up: no 1,3 in R7C1, no 3 in R7C2

5. 45 rule on N7 2 innies R78C7 = 5 = {14}, locked for C7 and N7

and the rest is naked singles.