SudokuSolver Forum

3x3:d:k:2304:1:3330:3330:4:3333:3333:7:2312:2304:2304:11:12:3085:14:2312:2312:17:3346:19:2580:21:3085:23:2072:25:3354:3346:28:29:2580:31:2072:33:34:3354:36:37:38:39:40:41:42:43:44:3373:46:47:3888:49:3634:51:52:3381:3373:55:3888:57:58:59:3634:61:3381:63:2368:2368:66:67:68:69:2374:2374:2368:73:3402:3402:76:3405:3405:79:2374:

184936752
265871349
937254168
492317685
673582491
518649237
849123576
751468923
326795814

HS3X Walkthough

0. Preliminaries
0a. 9(3) @ r1c1, r1c9, r9c1, r9c9 = {126|135|234} -> no 7,8,9
0b. 13(2) @ r1c3, r1c6, r3c1, r3c9, r6c1, r6c9, r9c3, r9c6 = {49|58|67} -> no 1,2,3
0c. 12(2) @ r2c5 = {39|48|57} -> no 1,2,6
0d. 10(2) @ r3c3 = {19|28|37|46} -> no 5
0e. 8(2) @r3c7 = {17|26|35} -> no 4,8,9
0f. 15(2) @r6c4 = {69|78} -> no 1,2,3,4,5
0g. 14(2) @r6c6 = {59|68} -> no 1,2,3,4,7

1. hidden killer pair {89} in 15(2) and r5c5 for D/ -> r5c5 = {89}

2. 7 locked in 10(2) for D\ -> 10(2) = {37} -> 3,7 locked for D\
2a. no 3,7 in r4c3 and r3c4
2b. no 4,5 in r2c1 and r8c9

3. 4 locked in r1c1 r2c2 r8c8 r9c9 for D\ -> either 9(3) @ r1c1 or 9(3) @r9c9 is {234}
3a. 45 on D\ (1 innie r5c5, 2 outies r2c1 r8c9) : r5c5 = r2c1 + r8c9 + 3 -> r2c1 + r8c9 = 5 or 6 -> r2c1 r8c9 = {23|33} (step 3)
3b. -> 3 locked in r2c1 r8c9 -> no 3 in r2c9 and r8c1

4. 45 on D/ (1 innie r5c5, 2 outies r2c7 r8c3) : r5c5 = r2c7 + r8c3 + 4 -> r2c7 + r8c3 = 4 or 5 -> no 5,6 in r2c7 and r8c3
4a. no 4 in r1c9 (blocked by r2c1)
4b. no 4 in r9c1 (blocked by r8c9)
4c. 4 locked in r8c2 r2c8 for D/ -> no 4 in r2c2 and r8c8
4d. -> either 9(3) @ r9c1 or 9(3) @r1c9 is {234} -> no 4 in r8c3 r2c7 (step 4)

5. 8,9 locked r5c5 r6c46 for n5
5a. 7 locked in r4c46 r6c4 for n5

6. 9(3) @r1c1 : no 2 in r1c1
6a. 9(3) @r9c9 : no 2 in r9c9
6b. 2 locked in r2c2 r8c8 for D\ -> no 2 in r8c2 r2c8

7. 45 on D/ (5 innies) : r9c1 r8c2 r5c5 r2c8 r1c9 = 22(5) = 4{1269|1359|1368|2358}
7a. 3 in r8c2 (only possible for {23458}) places 2 in r1c9 -> blocked by r8c9 -> no 3 in r8c2
7b. 3 in r2c8 (only possible for {23458}) places 2 in r9c1 -> blocked by r2c1 -> no 3 in r2c8

8. no 4 in r1c1
8a. r1c1 = 4 -> r2c23 = [32], r3c3 = 7, r4c4 = 3, r2c7 = 1 -> r8c2 = 4, r9c1 = 2, r8c3 = 3, r1c9 = 3, r2c8 = 5 -> can't place anything in 8(2) -> no 4 in r1c1
8b. -> r9c9 = 4, r8c89 = [23], r8c3 = 1 -> no 4 in r8c2, no 5,6 in r9c1
8c. -> r2c8 = 4, r2c7 = 3, r1c9 = 2, r9c1 = 3, r8c2 = 5, r2c1 = 2 -> no 5 in r1c1

9. 1,6 locked in r1c1 r2c2 for D\ and n1
9a. no 7 in r4c1 and r1c4, 14(2) in D\ = {59} -> r5c5 = 8
9b. 15(2) in D/ = {69}, 8(2) in D/ = {17}
9c. r34 : 7 locked in 10(2) and 8(2) for r34
9d. r67 : 9 locked in 15(2) and 14(2) for r67

10. cleanup: no 6 in r4c1, no 9 in r1c6, no 8 in r6c1, no 4 in r67c1, no 8 in r9c4, no 9 in r9c3467
10a. 13(2) @ r3c9 = {58} -> 5,8 locked for c9
10b. 13(2) @ r6c9 = {67} -> 6,7 locked for c9
10c. r67 : 6 locked in 15(2) and 13(2) @ r6c9 for r67
10d. -> r67c1 = [58], r6c6 = 9, r7c7 = 5, r6c49 = [67], r7c39 = [96], r9c34 = [67], r9c67 = [58], r9c2 = 2, r4c4 = 3, r3c3 = 7, r3c7 = 1, r4c6 = 7, r25c9 = [91]
10e. 13(2) @r3c1 = {49} -> 4,9 locked for c1
10f. r58c1 = [67], r7c2 = 4, r1c1 = 1, r2c2 = 6, r8c7 = 9, r79c8 = [71], r9c5 = 9
10g. r1c67 = [67], r8c5 = 6, r23c5 = [75], r34c9 = [85], r13c8 = [56], r1c34 = [49], r34c1 = [94], ...

the rest is singles

0. Prelim
All 9/3={126|135|234}
All 13/2={49|58|67}
12/2 @ r2c5={39|48|57}
10/2 @ r3c3={19|28|37|46}
8/2 @ r3c7={17|26|35}
15/2 @ r6c4={69|78}
14/2 @ r6c6={59|68}

1.
/567 form KHP {89} @ d/
=> r5c5 from {89}
=> 7 @ d\ locked @ 10/2 @ r3c3={37} (NP @ d\)
=> \1289 has no 3
=> 9/3 @ r1c1,r8c8 can't be [243|342|153|351]
=> r2c1,r8c9 can't have {45}

2.
Innie-outies @ d\: r5c5=r2c1+r8c9+3
=> r2c1+r8c9=5|6 from {1236}={23|33}
But 4 @ d/ locked @ /1289
=> one of 9/3 @ r1c9,r8c2 must be {234}
=> one of the 6 cells r1c9+r2c78+r8c23+r9c1 must be 3
=> r2c1+r8c9, seeing all these 6 cells, can't be {33}

3.
Now r2c1+r8c9={23}
=> r5c5=2+3+3=8 (NE @ d\/)
=> 15/2 @ r6c4 from {679}={69} (NP @ d/)
=> 14/2 @ r6c6 from {569}={59} (NP @ d\)
=> 7 @ d/ locked @ 8/2 @ r3c7={17} (NP @ d/)
Generalised x-wing: 7 @ r34,n5 locked @ \34+/34
Generalised x-wing: 9 @ r67,n5 locked @ \67+/67

4.
Outies @ d/: r2c7+r8c3=4={13|22}
But r2c1+r8c9={23} => r2c7+r8c3 can't be {22}, must be {13}
Now r3c37=[37|71]
=> r2c7+r8c3 can't be [13], must be [31]
=> r2c1+r8c9=[23]
=> 9/3 @ r1c9=[234], 9/3 @ r8c2=[513], 9/3 @ r8c8=[234]

5.
13/2 @ r3c9 from {5689}={58} (NP @ c9)
=> 13/2 @ r6c9={67} (NP @ c9)
Generalised x-wing: 6 @ r67 locked @ /67+r67c9
=> 13/2 @ r6c1 from {4578}=[58]
=> 14/2 @ \67=[95]
=> 15/2 @ /67=[69]
=> 13/2 @ r6c9=[76]
=> 13/2 @ r9c3=[67]
=> 13/2 @ r9c6 from {589}=[58]
=> 10/2 @ r3c3=[73]
=> 8/2 @ r3c7=[17]
=> r25c9=[91]
=> 13/2 @ r1c6 from {4678}=[67]
=> 9/3 @ r1c1=[126]
=> 12/2 @ r2c5=[75]
=> 13/2 @ r3c9=[85]
=> r13c8=[56]
=> 13/2 @ r1c3 from {489}=[49]
=> 13/2 @ r3c1=[94]

All naked singles from here.

The 12/2 @ r2c5 could be changed to 9/2, 10/2 & 11/2, each with a different unique solution. The key thing is without that cage the puzzle has 4 solutions, each with a different value @ r3c5, a pretty interesting property.