Finally worked out how rcbrougton's solver solved
UTA original! Very interesting path taken too. Lots of innies and hidden cages that we never thought of - r7, r9, n9. Oh yeah - and did I mention combinations? Lots of combination conflicts from many different directions at once.
This walk-through is the sweetened-condensed version of rcbroughton's - lots of steps left out, changed the order of many - and lots of my own explanations included. The step numbers should correspond.
Oh yeah - and did I mention combination charts? Make sure you've got one that works
Good luck - a very rough road ahead
[Many thanks to Andrew for navigating through it!!]Just a quick reminder, UTA is a diagonals puzzle: 1-9 cannot repeat on diagonals. [edit: colours changed to get better match with burgundy]1. Must use 9 in cage 22(3) at r3c5
-> Removed 9 from r3c1234
19. 45 rule on r12 -> r2c1 = 7
19a. r34c1 = 13 = [49]/{58}
-> Removed candidate 369 from r3c1
-> Removed candidates 346 from r4c1
21. Only combinations {18/36/45} allowed in cage 9(2) at r2c8
-> Removed candidate 2 from r2c89
22. Only combinations {69/78} allowed in cage 15(2) at r7c1
-> Removed candidate 8 from r7c2
13. 45 rule on N3. Included cells r23c7 minus excluded cell r1c6 equals 11
-> Min of included cells is 12. Set min candidate in r2c7 (3)
-> min r3c7 = 7 - (can't have r23c7 = {66})
-> Max of excluded cells is 4. Set max candidate in r1c6
14. Only combinations {16/25/34} allowed in cage 7(2) at r2c6
-> Removed candidates 56 from r2c6
27. 3 locked in n3 in 9(2) or 5(2)
27a.if 9(2) = [1/4]->5(2) = {23};
27b.if 5(2) = {14} ->9(2) = {36})
-> Removed candidate 3 from r2c7, r1c789
28. Only combinations {16/25/34} allowed in cage 7(2) at r2c6
-> Removed candidate 4 from r2c6
7. 45 rule on row 9. Excluded cells r8c5 r8c8 equal 17
-> Only combination {89} allowed
8. Only combinations with 8 or 9 (not both) allowed in cage 17(3) at r8c8.
-> Removed candidates 89 from r9c89
9. Naked pair 89 found in row 8 at r8c58
9a.-> Removed value 8 from r5c5 (because both cells connected through D\)
9b.-> Removed value 8 from r5c5 r9c7 (because r9c7 is in the same cage as r8c5 and same n as r8c8)
9c.-> Removed value 8 from r5c5 r8c34 r8c679
9d.-> Removed value 9 from r5c5(as above) r9c7(as above) r8c6 r8c7 r8c9
9e.-> Removed combination {18} from cage 9(2) at r8c3
9f.-> Removed candidate 1 from r8c34
10. Value 4 locked in row 8 of combined cages 9(2) at r8c3 & 5(2) at r8c1
10a.ie 5(2) = {23} -> 9(2) = {45};
9(2) = [2/3] -> 5(2) = {14}
-> Removed 4 from r8c679
34. 45 rule on N7, N8. Excluded cells r89c7 equal 10
-> Only combinations {37/46} allowed
-> Removed candidates 125 from r8c7
-> Removed candidates 1256 from r9c7
-> Found a hidden cage 10(2) at r89c7
This is a tricky step - but absolutely critical. Get your combination tables ready!
35. 45 rule on row 9. Included cells r9c56789 equal 23 = h23(5) cage
35a.-> Cage h10(2) at r8c7 doesn't allow permutations in r9c89 with {36}
->17(3)n9 {368} combo eliminated
35b. h23(5)r9 must include {347} (r9c7)
-> {12569} excluded
35c. Possible combinations for h23(5)r9 must have a triple overlap of combination from r9c56:23(4) and then double overlap from r9c89:17(3), AND must agree with the {89} pair in r8 AND must agree with the h10(2)n9
ie. combinations h23(5)r9 = {r9c56}[r9c7]{r9c89}
-{12389} Blocked (must have 3 in r9c7 but no triple overlap of combination with r9c567:23(4))
-{12479} = {29}[4]{17} (7 in r9c7 blocked - no triple overlap of combination with r9c567:23(4))
-{12578} Blocked (only be {25}[7]{18} but no 8 in r9c89)
-{13469} Blocked (r9c7 = [3/4] but no triple overlap with r9c567:23(4))
-{13478} Blocked (only be {347} in r9c567: but no 8 left in r9c89)
-{13568} Blocked (only be {56}[3] in r9c567: but no 8 left in r9c89)
-{14567} = {56}[4]{17}/{16}[7]{45}
-{23459} = {29}[4]{35} (3 in r9c
7 blocked: no triple overlap with r9c567:23(4))
-{23468} Blocked ({28}[4]{36} blocked - see step 35a; 3 in r9c6: no triple overlap with r9c567:23(4))
-{23567} = {57}[3]{26}/{35}[7]{26}
-({56}[3]{27} blocked since 7 must be in r8c7 in h10(2) -> 2 7's n9)
-({25}[7]{36} blocked since no {36} possible r9c89 - step 35a)
-({26}[7]{35} blocked since 3 must be in r8c7 in h10(2) -> 2 3's n9)
35d.In summary h23(5)r9 =
{12479} = [{29}4{17}]
{14567} = [{56}4{17}/{16}7{45}]
{23459} = [{29}4{35}]
{23567} = [{57}3{26}/{35}7{26}]
-> Removed candidates 48 from r9c56
35e. from step 35d. r8c5 + r9c56 (part of 23(4) that's in n8) = {289/568/169/578/358}
35f. from step 35d. the only combinations in 23(4)n89 = {2489/4568/1679/3578} = {1679/2489/3578/4568}
35g. from step 35d. 17(3)n9 = {179/458/359/269} = {179/269/359/458} = [8/9..]
39. r8c8 = [8/9]. Only other place for [8/9] in nonet 9 is in row 7
39a. 15(2) at r7c1 = {69/78} = [8/9..]
39b.->Killer pair {89}: locked for r7
-> Removed candidate 8 from r7c3456
-> Removed candidate 9 from r7c56
39c. ->17(4)n89 = {1367/1457/2357/2456}
40. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 1 from r7c34
18. 45 rule on N7
-> Found a hidden cage 15(3) at r789c4
41. Only combinations {249/258/267/348/357/456} allowed in cage h15(3) at r7c4
-> Removed candidate 1 from r9c4
43. from step 41. h15(3)n8 = {249/258/267/348/357/456}
43a. {267} blocked
-from step 39c.the only combination in 17(4)n89 possible is {1457} with {145} in n8. All other combinations are blocked since they have 2 candidates in common with {267}.
-But{267-145} in h15(3) + 3 cells from 17(4) in n8 is blocked by r8c5 + r9c56 (see step 35e.)
43b.{357} blocked
-the only available combination in 17(4)n89 (step 39c.) without a 5 is {1367},
-but they have both {37} in common -> blocked
43c.
-> Only combinations allowed in h15(3) = {249/258/348/456} (no 7)
-> Removed candidate 7 from r78c4
-> Removed candidates 237 from r9c4 (since is the only cell with an 8 or 9)
44. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 2 from r7c3
45. Only combinations {27/36/45} allowed in cage 9(2) at r8c3
-> Removed candidate 2 from r8c3
47. 45 rule on N1, N2, N3, N4. Excluded cells r46c4 equal 7
-> Only combinations {16/25/34} allowed
-> Removed candidates 789 from r46c4
-> Found a hidden cage 7(2) at r46c4
48. h15(3) in n8
-> Cage h7(2) at r4c4 doesn't allow permutations with {456}
-> Only combinations {249/258/348} allowed (no 6)
-> Removed candidate 6 from r78c4
-> Removed candidates 456 from r9c4 (only cell with 8/9)
49. Only combinations {45/63/27} allowed in cage 9(2) at r7c3
-> Removed candidate 3 from r7c3
50. Only combinations {27/45/63} allowed in cage 9(2) at r8c3
-> Removed candidate 3 from r8c3
51. Naked pair 89 found in N8 at r8c5 r9c4
-> Removed value 9 from r9c56
-> Removed combination {2489} from cage 23(4) at r8c5
-> removed 2 from r9c56
51a.from step 35d. h23(5)r9
-> Only combinations =
{14567} = [{56}4{17}/{16}7{45}]
{23567} = [{57}3{26}/{35}7{26}]
=567{14/23}
51b.-> no 3 r9c89
51c.->17(3) = {179/269/458}
51d.->567 locked for r9
51e.->22(4)r9 = 89{14/23}
51f. combining the information from step 51a with the h10(2)n9
->r8c78 + r9c789 = [69417/38745/79326/39726] = 7{1469/2369/3458}
51g.-> 7 locked for n9 in h10(2) and 17(3)
51h. "45"n9 -> 4 innies = 18(4) = {1269/1458/2358} ({1359/1368/2349/3456} blocked by combined h10(2)-17(3) step 51f)
51i. 22(4)n78 = {1489/2389} = 89{14/23}
-since {14/23} must be in n7 -> Killer quad with 5(2)n7
-> no 4 r78c3
53. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 5 from r7c4
54. Only combinations {27/36/45} allowed in cage 9(2) at r8c3
-> Removed candidate 5 from r8c4
55. Value 5 locked in column 3 of N7
-> Removed 5 from r123456c3
56. Since 5 is locked in r78c3 -> Value 4 locked in r78c4 (since connected by 2 9(2)cages)
-> 4 locked for c4
-> Removed 4 from r123456c4
57. Only combinations {16/25} allowed in cage h7(2) at r4c4
-> Removed candidate 3 from r46c4
58. Value 4 locked for column 4 in N8
-> Removed 4 from r7c56
58a. 17(4)n89 = {1367/2357} = 37{16/25}
59. Must use 37 in cage 17(4) at r7c5
-> Removed candidate 3 from r8c4
60. Only combinations {27/45} allowed in cage 9(2) at r8c3
-> Removed candidate 6 from r8c3
61. Value 6 locked in row 7 of N7
-> Removed 6 from r7c56789
62. Value 2 locked in row 8 of combined cages 9(2) at r8c3 & 5(2) at r8c1
-> Removed 2 from r8c69
65. 45 rule on row 89. Included cells r8c679 = h14(3)
65a. = [176/671/761/356/365/536] ([563] not compatable with 17(4) combo's)
-> Cage h15(3) at r7c4 doesn't allow permutations in r8c56 with {39}
-> Only combinations {167/356}
-> Removed candidate 3 from r8c9
68a.From step 51h. we know that innies n9 = 18(4) = {1269/1458/2358}
68b. -> with r8c9 = {156} -> {r7c789} = {269/129/458/148/238}
68c. "45" r7 -> 5 innies = h21(5)
68d. combining steps 68b and c and keeping an eye on the combinations in the 17(4)n89: {1367/2357}
-> h21(5) = {r7c56}{r7c789}
= {13-269}/{36-129}/{13-458}/{35-148}/{17-238}
68e. -> h21(5) = {12369/12378/13458} = 13{269/278/458}
68f. -> no 3 r7c4
68g. and no 3 in r8c7 (since 3 locked in h21(5) r7 is in the same cage or nonet as r8c7)
68h. -> no 7 r9c7
68i.removed 6 from r7c3
69. from step 68h and 51a.
h23(5)r9 = {14567} = [{56}4{17}]
= {23567} = [{57}3{26}]
69a. -> Only combinations {3578/4568} allowed in cage 23(4) at r8c5
69b. = 58{37/46} (no 19)
69c. -> r8c5 = 8
69d. 5 is locked for n8 and r9 in r9c56 = 5[6/7](no 3)
75. Only combinations {249} with 9 locked in r9c4 allowed in cage h15(3)n8 at r7c4
-> Set candidate 9 in r9c4
75a. r78c4 = {24}:locked for n8, c4
75b. r8c8 = 9
placed for D\75c. r9c89 = {17/26}(no 3,4)
86. Only combination {16} allowed in cage h7(2) at r4c4
-> Removed candidate 5 from r4c4
-> Removed candidate 5 from r6c4
86a. {16}locked for c4,n5
87. "45"n5 -> r46c6 = 13 = [94]/{58}
88. Naked pair {57} found in r78c3
-> Removed value 7 from r456c3 r7c2
-> Removed combination {1457} from cage 17(4) at r4c3
-> Removed combination {23459} from cage 23(5) at r5c1
-> Removed combination {78} from cage 15(2) at r7c1 = {69} only
89. Naked pair {16} found in cage h7(2)c4
-> Removed combination {1469} from cage 20(4) at r1c3
-> cage 25(5) at r4c5 = 237{49/58}
91. Must use {1367} in cage 17(4) at r7c5
91a. 6 is locked for r8
-> Removed candidate 6 from r8c9
99 from step 68d. "45" r7 -> 5 innies = h21(5)
99a. -> {r7c56}{r7c789} = {13-458}/{17-238}
99b. -> 1 locked in r7c56 for r7, n8
99c. -> h21(5) = {12378/13458} = 138{27/45}
Now moving into n6 - and the key moves that finally unlock this puzzle
37. 45 rule on N7, N8, N9. Excluded cells r56c8 r6c9 = 20
37a.-> Only combinations {389/479/569/578} allowed
37b.-> Removed candidates 12 from r5c8
37c.-> Removed candidates 12 from r6c8
37d.-> Found a hidden cage 20(3) at r56c8 r6c9
80. 23(4)n69 = {2678/3578/4568}.
80a.From step 37, r56c8 + r6c9 = 20.
-r6c9 = 3..9 -> r56c8 = 11..17 -> r7c78 = 6..12
80b.from step 99a. ->{r7c789} = {458}/238}
from 80a. r7c78 = 6..12 -> = {45/48/28/38}
80c. from step 80 ->r56c8 = {68/56/67/57}
80d. But since r56c8 are part of a h20(3) (step 37a) -> {67/68} are blocked
80e. ->r56c8 = {56/57} = 5{6/7} = 11/12
80f.->5 locked for c8,n6 and no 5 in r7c78
80g. -> r6c9 = 8/9 only (from h20(3)n6)
80h. from 80e. the rest of 23(4)n69 = {38/48} = 8{3/4} (no 2)
80i. 15(3)n69 = [825/951] with r7c9 = {25}
80j. -> 15(3) = 5{28/19}: 5 locked for c9, n9
80k. -> r6c9 = {89}, r7c9 = {25}, r56c8 = {567} (no 8), r7c78 = {348} = 8{3/4}
80l. 23(4) n69 = {3578/4568} = 58{37/46} with 5 locked for c8 -> no 4 r2c9
80m. 9(2)n3 = {18/36}(no 4) = [1/3...]
80n. 5(2)n3 = {14/23} = [1/3..] -> Killer pair {13} locked for n3
109/114. 45 rule on N3. Included cells r23c7 minus excluded cell r1c6 equals 11
-1 in r1c6 -> r23c7 = 12 = {57}
({48} blocked since requires 9(2)n3 = {36} and 5(2) = {23} = 2 3's)
-2 in r1c6 -> r23c7 = 13 = {49}
({58} blocked since requires 9(2)n3 = {36} and 5(2) = {14} = but this leaves {279} for r1 -> 2 2's in r1; {67} blocked by r8c7)
-3 in r1c6 -> r23c7 = 14 = {59}
({68} blocked by 9(2)n3)
-4 in r1c6 -> r23c7 = 15 = blocked
({69} requires 9(2) = {18}, 5(2) = {23} -> 2 4's in r1)
109a. r23c7 = [57/49/59]
-> Removed candidate 6 from r2c7
-> Removed candidate 8 from r3c7
-> removed 4 from r1c6
110. Only combinations {25/34} allowed in cage 7(2) at r2c6
-> Removed candidate 1 from r2c6
111. Only combinations {589/679} allowed in cage 22(3) at r3c5
-> Removed candidate 5 from r3c6
112. Value 1 locked for column 7 in N6
-> Removed 1 from r4c89 r5c9
115. Value 4 locked in column 5 of N2
-> Removed 4 from r456c5
90. Must use 9 in cage 27(5) at r1c1
-9 for 27(5)n12 only in n1
-> no 9 in r1c3
117. Value 4 locked for column 5 in cage 20(4) at r1c3
-> Removed 4 from r1c3
117a. 20(4) must have 4 = {1478/2459/2468/3458/3467} ({1469} blocked since no candidates in r1c4)
117b. 45 on r123 -> r4c12 = 12 = [57/84/93]: r4c2 = {347}
117c. 45 on n1 -> r23c4 - 7 = r1c3.
-r1c3 = 1 -> r23c4 = 8 = {35} -> rest of 20(4) = {478}
-r1c3 = 2 -> r23c4 = 9 = no options -> no 2 in r1c3
-r1c3 = 3 -> r23c4 = 10 = [37] -> rest of 20(4) = {458} ({467} blocked by r23c4)
-r1c3 = 6 -> r23c4 = 13 = {58} -> rest of 20(3) = {347} ({248} blocked by r23c4)
-r1c3 = 8 -> r23c4 = 15 = [87] -> rest of 20(3) = {345} ({147} blocked by r23c4;{246} blocked since no candidates in r1c4)
117d. In summary, r23c4 = {35/58}[37/87]
117e. In summary, rest of 20(3)
in n2 = 4{78/58/37/35} (no 1269)
117f. -> r2c3 = 9 (hsingle r2)
117g. -> 9 for n2 locked in 22(3) -> no 9 in r3c7
117h. r3c7 = 7
126a. naked quint on {34578} in n2 -> r12c6 = [12], r3c56 = {69} ->
r2c7 = 5, r89c7 = [64],
r7c78 = {38} locked for r7, r7c34 = [54], r78c6 = [73], r56c8 = {57}, 15(3)n56 = 9{28/46}
(note: may still have {258} combo so this next bit may be wrong) with 9 locked for n6, c7 -> r1c9 = 9, r1c78 = {28}, 9(2)n3 = {36}, r3c89 = [41] etc