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Assassin 2

 
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mhparker
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Joined: 20 Jan 2007
Posts: 345
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PostPosted: Tue Dec 11, 2007 5:16 pm    Post subject: Assassin 2 Reply with quote

Hi folks,

Hey, I'm the first to post a WT for this one! Dancing Come on guys, where are you all? Laughing

The question is: Was this the puzzle where Ruud developed his love of 3-cell cages?

Estimated rating: 0.75. Unfortunately, I messed it up a bit and strayed from the optimum route near the start. Clearly, I can't do 0.75-rated puzzles properly any more... Rolling Eyes

Anyhow, here's my (non-optimal) WT:


Assassin 2 Walkthrough

Prelims:

a) 21(3) at R1C6 = {489/579/678} (no 1..3)
b) 11(3) at R2C7, R5C1 and R7C5 = {128/137/146/236/245} (no 9)
c) 9(3) at R3C1, R3C6 and R7C5 = {126/135/234} (no 7..9)
d) 20(3) at R3C3, R6C1, R8C4 and R8C6 = {389/479/569/578} (no 1,2)

1. Innie/Outie (I/O) diff. C789: R19C7 = R37C6 + 14
1a. -> R37C6 = {12}, locked for C6; R19C7 = {89}, locked for C7

2. Outies C89: R258C7 = 7(3) = {124}, locked for C7

3. 9(3) at R3C6 = {135} (last combo)
3a. -> R3C6 = 1; R34C7 = {35}, locked for C7

4. Naked single (NS) at R7C6 = 2

5. Outies R12: R3C258 = 22(3) = {589/679} (no 1..4)
5a. 9 locked for R3

6. Innies N3: R13C7+R3C9 = 17(3) = [836/854/935/953]
6a. -> R3C9 = {3..6} (no 2,7,8)

7. Hidden single (HS) in R3 at R3C1 = 2
7a. -> split 7(2) at R4C12 = {16/34} (no 5)

8. Outies N12: R1C7+R4C3 = 17(2) = {89} (no 3..7)
8a. no 8,9 in R1C3 (CPE)

9. Innies N1: R13C3 = 12(2) = [48]/{57} (no 1,3,6; no 4 in R3C3)

10. 11(3) at R5C1 = {128/245} (no 3,6,7)
(Note: {137/146/236} all blocked by R4C12 (step 7a))
10a. 2 locked for R5 and N4

11. 12(3) at R5C7 = {138/345} (no 6,7,9)
(Note: {147/156} both blocked by 11(3) at R5C1 (step 10))
11a. 3 locked for R5 and N6
11b. can only have 1 of {14}, which must go in R5C7
11c. -> no 1,4 in R5C89

12. R34C7 = [35]
12a. -> no 4 in R3C9 (step 6)

13. HS in R3 at R3C4 = 4
13a. -> R34C3 = [79] (last permutation)
13b. -> R1C3 = 5 (step 9)

14. 12(3) at R5C7 = {138} (last combo)
14a. -> R5C7 = 1; R5C89 = {38}, 8 locked for R5 and N6

15. 11(3) at R5C1 (step 10) = {245} (last combo), locked for R5 and N4
15a. cleanup: no 3 in R4C12 (step 7a)

16. Naked pair (NP) at R4C12 = {16}, locked for R4 and N4

17. 15(3) at R3C9 = {267} (no 4,5) (last combo)
17a. -> R3C9 = 6; R4C89 = {27}, locked for R4 and N6
17b. -> R1C7 = 8 (step 6)

18. R679 = [679]

19. NS at R3C8 = 5
19a. -> split 6(2) at R2C78 = {24}, locked for R2 and N3

20. NP at R6C89 = {49}, locked for R6
20a. -> R7C9 = 1 (cage sum)

21. HS in C4 at R6C4 = 1
21a. -> split 11(2) at R4C6+R5C5 = [47] (last combo/permutation)

22. HS in C4 at R1C4 = 2
22a. -> R2C4 = 8 (cage sum)

23. R3C25 = [89]
23a. split 7(2) at R2C23 = {16} (last combo), locked for R2 and N1
23b. split 8(2) at R12C5 = [35]

24. R4C45 = [38]
24a. -> R5C4 = 6 (cage sum)

25. 16(3) at R5C6 = [925]

26. R12C6 = [67]

27. 17(3) at R1C8 = [179] (last combo/permutation)

28. NS at R2C1 = 3

29. 20(3) = {389/578} (no 4,6)
29a. 7 of {578} must go in R6C2
29b. -> R6C1 = 8

Now all singles and simple cage sums to end.
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Cheers,
Mike


Last edited by mhparker on Tue Jan 15, 2008 8:10 pm; edited 1 time in total
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mhparker
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Joined: 20 Jan 2007
Posts: 345
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PostPosted: Tue Dec 11, 2007 5:36 pm    Post subject: Reply with quote

Hi folks, it's me again...

I wrote:
Clearly, I can't do 0.75-rated puzzles properly any more...

Fortunately, this one should be a tad more difficult...

Assassin 2X (A2X) (Est. rating: 1.75)



3x3:d:k:4352:4352:3842:3842:4612:3589:3589:3335:3335:4352:4618:4618:3842:4612:3589:4367:4367:3335:4626:4618:3860:3860:4612:3095:3095:4367:3354:4626:4626:3860:3614:3614:2592:3095:3354:3354:2596:2596:2596:3614:2592:5417:3626:3626:3626:3373:3373:4143:2592:5417:5417:4403:3380:3380:3373:4151:4143:4143:3130:4403:4403:6205:3380:4415:4151:4151:3906:3130:2628:6205:6205:3399:4415:4415:3906:3906:3130:2628:2628:3399:3399:

Note: It's a Killer-X, where no repeated digits are allowed on either diagonal.

Yeah, looks pretty straightforward, I know... Wink
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Cheers,
Mike


Last edited by mhparker on Tue Jan 08, 2008 6:09 pm; edited 4 times in total
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Caida
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PostPosted: Tue Dec 11, 2007 10:45 pm    Post subject: Reply with quote

I managed to start A2X pretty well - even got a couple numbers placed. I have most of n5 complete with 3 numbers placed and the others are reduced to pairs

But I have now ground to a halt.

Does the fact that it is a Killer-X help at all with solving it? I'm not seeing it lead to anything so far.

Would it help to do the original A2?

Cheers,

Caida
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mhparker
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PostPosted: Wed Dec 12, 2007 9:36 am    Post subject: A2X-Lite Reply with quote

Hi Caida,

Thanks for giving it a go. Here are some answers to your questions.

Caida wrote:
I managed to start A2X pretty well...But I have now ground to a halt.

Yes, it's a toughie. Unlike several recent Assassin V1's, which consist of finding one (or at most, two) key move(s), apart from which the rest is plain sailing, this is more like a traditional V2, which (whilst having easier sections) generally involves more of a fight to dig away at the candidates. This puzzle does contain at least one key move, however, that will make your life easier if you find it/them.

Caida wrote:
Does the fact that it is a Killer-X help at all with solving it? I'm not seeing it lead to anything so far.

Yes, but they are further down the road.

Caida wrote:
Would it help to do the original A2?

No, because it doesn't require a move similar to that/those needed to unlock this puzzle.

BTW, I have increased the estimated rating for the A2X to 1.75, which I now suspect may be nearer to the mark.


In the meantime, maybe this is more like the puzzle you're looking for?:


Assassin 2X Lite (A2X-Lite) (Est. rating: 1.25)



3x3:d:k:4096:4096:2050:2050:3588:4613:4613:3591:3591:4096:5130:5130:2050:3588:4613:4879:4879:3591:2066:5130:4884:4884:3588:4631:4631:4879:4634:2066:2066:4884:2590:2590:3616:4631:4634:4634:3620:3620:3620:2590:3616:5417:3114:3114:3114:4397:4397:4655:3616:5417:5417:1587:3892:3892:4397:3127:4655:4655:3898:1587:1587:4925:3892:4927:3127:3127:4674:3898:3908:4925:4925:2119:4927:4927:4674:4674:3898:3908:3908:2119:2119:

Note: It's a Killer-X, where no repeated digits are allowed on either diagonal.

It's still no walkover, but should be more fun to solve than the original A2X.

Enjoy!
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Mike
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Caida
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PostPosted: Wed Dec 12, 2007 4:02 pm    Post subject: Re: A2X-Lite Reply with quote

mhparker wrote:

In the meantime, maybe this is more like the puzzle you're looking for?:


This was much easier - nicer for my first attempt at a "Killer-X"

Thanks!!

Here's my walkthrough for A2XLite (kept in tiny text as it is still brand new). Any comments/suggestions/corrections are always appreciated!

I would consider rating A2XLite a 1.0 (but would like to hear what others have to say).

edited for typos and notes added Thanks Andrew!!
Also had some faulty logic at my step 11 - I made an elimination too early - have reworked my walkthrough to fix this.
A couple more edits completed

Assassin 2XLite walkthrough

Preliminaries:

a. 8(3)n12 and n14 and n9 = {125/134} (no 6..9)
a1. -> 1 locked in n9 -> no 1 elsewhere in n9
a2. -> r4c4 no 1 (CPE with 8(3)n12 and 8(3)n24)
Andrew pointed out that I missed the obvious CPEs of r1c56 and r56c1 oops!!
b. 20(3)n1 = {389/479/569/578} (no 1,2)
c. 19(3)n124 and n3 and n7 and n9 = {289/379/469/478/568} (no 1)
d. 10(3)n5 = {127/136/145/235} (no 8,9)
e. 21(3)n5 = {489/579/678} (no 1,2,3)
f. 6(3)n698 = {123} (no 4..9)
f1. -> killer pair {23} locked in n9 in r7c7 and 8(3)n9 -> no 2,3 elsewhere in n9

1. Innies n1: r1c3+r3c13 = 9(3) = {126/135/234} (no 7..9)
1a. -> 19(3)n124: r3c4 and r4c3 no 2,3 (requires 89 or 79)

2. Innies n9: r7c79+r9c7 = 18(3) = [2]{79}/[3]{69/78}
2a. -> r7c9 and r9c7 no 4,5

3. Innies r1234: r4c456 = 8(3) = {125/134} (no 6..9) -> 1 locked in r4c56 for r4 and n5
3a. cage overlap: r5c4 – r4c6 = 2
3b. -> r5c4 no 2

4. Innies r5: r5c456 = 19(3) ={379/469/478/568} (no 2)
Note: {289} blocked by r5c4

5. Innies r6789: r6c456 = 18(3)
5a. cage overlap: r5c6 – r6c4 = 3
5b. -> r5c6 no 4; r6c4 no 7,8,9

6. Innies c1234: r456c4 = 11(3) = {236/245} (no 7) -> 2 locked for c4 and n5
6a. -> r5c4 no 4 (step 3a)
Andrew noticed that I missed eliminating 5 from r4c6 also b/c of step 3a
6b. cage overlap: r6c4 – r4c5 = 1
6c. -> r6c4 no 3
6d. -> r5c6 no 6 (step 5a)
6e. r4c4 no 5 (from h8(3) step 3)
6f. h11(3)c4 = [236/362]
Note: [254] blocked from h11(3) by h8(3) and 10(3) and [452] blocked by step 5
You can't have 254 b/c this would put a 2 in the h8(3)r4 - but if 8(3) has a 2 then it must also have a 5 - but it can't have a 5 b/c it is in the h11(3)c4.
452 won’t work because this puts a 2 in r6c4 so then you need a 5 in r5c6 (step 5a) but the 5 is already placed in r5c4.

6g. 2,3,6 locked for n5 in c4
6h. -> r4c6 no 5 (step 3a)
6i. -> r5c6 no 7, 8 (step 5a)
6j. 21(3)n5 = [9]{48}/[5]{79}/[9]{57} -> 9 locked in 21(3) for n5

7. 14(3)n5 = [176/482]
7a. -> r5c5 no 4,5

8. Innies c6789: r456c6 = 18(3) = [198/459/495]
8a -> r6c6 no 4,7
8b. -> r6c5 no 5,8,9
8c. -> 7 locked in n5 in c5 -> no 7 elsewhere in c5
8d. -> 9 locked in n5 in c6 -> no 9 elsewhere in c6
8e. -> 8 locked in n5 in diagonal top left to bottom right - > no 8 elsewhere on that diagonal

9. 8(3)n14: r3c1 = 1
9a. -> killer quadruple {2345} locked for r4 8(3)n14 and h8(3)n5 -> no 2,3,4,5 elsewhere in r4

10. 8(3)n12 = [2]{15}/[3]{14}
10a. -> r1c3 no 4,5
10b. -> 1 locked in r12c4 for c4 and n2 -> no 1 elsewhere in c4 and n2

Changing walkthrough b/c of faulty early elimination to original step 11 (I had said that Innies c5: r456c5 = 16(3) = [574] Andrew pointed out that I missed [187] which is still valid. This was a BIG mistake on my part).

new11. hidden single: r9c9 = 1 (only 1 on \d)
new11a. hidden single: r2c4 = 1
new11b. innies n1 (step 1): r3c3 no 2,3,4

new12. 19(3)n124 = {469/568} (no 7) -> 6 locked in r34c3 -> no 6 elsewhere in c6
new12a. -> r3c4 no 9
new12b. -> 9 locked in n2 in c5 -> no 9 elsewhere in c5
new12c. -> 7 locked in n2 in c6 -> no 7 elsewhere in c6
new12d. 14(3)n2 = {239} -> locked for n2 and c5
new12e. -> {23} locked in n8 in r789c6
new12f. -> r89c6 <> {23} -> r7c6 = {23} (no 1) -> pair {23} locked in r7 in c67
new12g. -> r6c7 = 1 (only 1 in 6(3))
new12h. single: r1c8 = 1
new12i. -> r8c6 no 1 (not possible to have {12/13} in 15(3))
new12j. -> hidden single r4c6 = 1
new12k. -> r5c4 = 3 (step 3a)
new12l. -> r4c45 = [25]

Now the rest of my steps are valid

11. Innies c5: r456c5 = 16(3) = [574]
11a. -> r4c456 = [251]
11b. -> r5c456 = [379]
11c. -> r6c456 = [648]
11d. -> 278 locked for diagonal l->r
11e. -> 167 locked for diagonal r->l

12. 6(3)n689 = [123]
12a. -> 3 locked for diagonal l->r
12b. -> 8(3)n9 = {125} -> locked for n9

13. pair {34} locked in r4c12 for n4
13a. 14(3)n4 = {158} -> locked for n4 and r5
13b. hidden single: r4c3 = 6
13c. triple {279} locked in r6c123 for r6

14. 15(3)n69 = {35}[7]
14a. -> r7c9 = 7
14b. -> r9c7 = 8 (step 2)

15. 15(3)n89 = {34}[8]
15a. -> {34} locked for n8 and c6 -> no 3,4 elsewhere in n8 and c6
Made some changes here – based on my new steps #11 and 12
new15b. -> triple {567} locked in r123c6 -> no 5,6,7 elsewhere in n2
new 15c. -> 8(3)n12 = [341]


16. this is now step new15b
16a. this is now step new12d
16b. single: r3c4 = 8
16c. 19(3)n124 = [586] -> 5 locked for diagonal l->r
16d. this is now step new11
16e. r9c5 = 6
16f. this is now step new12h
16g. this is now step new15c

17. 18(3)n23 = {567} no 2,9
17a. 18(3)n478 = [945]
17b. -> 4 locked for diagonal r->l
17c. 18(3)n234 = [729]
17d. -> 2 locked for diagonal r->l

Everything left is cage sums and singles

I'll have to give the original A2X another go in a bit.
But would really love if someone else did a walkthrough so I could take a peek Wink

Cheers,

Caida


Last edited by Caida on Sun Dec 23, 2007 12:02 am; edited 10 times in total
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Nasenbaer
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PostPosted: Wed Dec 12, 2007 8:27 pm    Post subject: Reply with quote

Beaten again. Oh well. Wink

Yes, A2X-Lite is much easier. And here is my walkthrough. It's the way I solved it so no nice shortcuts and a lot of jumping around. Don't know about the rating, haven't looked into that.


A2X Lite

1. n12: 8(3) = 1{25|34} -> no 6,7,8,9 -> 1 locked for 8(3) -> no 1 in r1c56

2. n1: 20(3) = {389|479|569|578} -> no 1,2

3. n3: 19(3) = {289|279|469|478|568} -> no 1

4. n14: 8(3) = 1{25|34} -> no 6,7,8,9 -> 1 locked for 8(3) -> no 1 in r56c1

5. n124: 19(3) = {289|279|469|478|568} -> no 1

6. n5: 10(3) = {127|136|145|235} -> no 8,9

7. n5: 21(3) = {489|579|678} -> no 1,2,3

8. n689: 6(3) = {123} -> no 4,5,6,7,8,9 -> 1,2,3 locked for 6(3)

9. n9: 19(3) = {289|279|469|478|568} -> no 1

10. n7: 19(3) = {289|279|469|478|568} -> no 1

11. n9: 8(3) = 1{25|34} -> no 6,7,8,9 -> 1 locked for 8(3) and n9

12. n9: {23} locked in 8(3) and r7c7 for n9

13. n9: 19(3) = {469|478|568}
13a. -> {45} locked in 19(3) and 8(3) for n9

14. r7c79 + r9c7 = h18(3) = {279|369|378} -> no 1,2,3,4,5 in r7c9 and r9c7

15. n1: r1c3 + r3c13 = h9(3) = {126|135|234} -> no 7,8,9 in r3c3

16. n124: 19(3): cleanup -> no 2,3 in r3c4 and r4c3

17. 45 on r12: r3c258 = h19(3) = {289|279|469|478|568} -> no 1

18. 45 on r1234: r4c456 = 8(3) = 1{25|34} -> no 6,7,8,9 -> 1 locked for 8(3), r4 and n5

19. (step 4) -> r3c1 = 1

20. (step 1) -> 1 locked in r12c4 for c4 and n2

21. r9c9 = 1 (single for D\)

22. r2c4 = 1 (single for r2)

23. {12345} locked in r4c12456 for r4

24. n36: 18(3) = {279|369|378|468|567} -> r3c9 = {2345}

25. n5: 10(3): cleanup -> no 4 in r4c5

26. 45 on r6789: r6c456 = 18(3) = {279|369|378|468|567} ({459} blocked by h8(3))
26a. -> cleanup -> no 7,8,9 in r6c4 (r6c56 must be at least 12 from the 21(3) cage)

27. n5: 14(3): cleanup -> no 2,4 in r5c5

28. n1: h9(3): cleanup -> no 4 in r1c3, no 2,4 in r3c3

29. n12: 8(3): cleanup -> no 3 in r1c4

30. n124: 19(3) = {379|469|568}

31. 45 on c1234: r456c4 = h11(3) = 2{36|45} -> 2 locked for h11(3), c4 and n5

32. n12: 8(3): cleanup -> no 5 in r1c3

33. n1: h9(3): cleanup -> no 3 in r3c3

34. n124: 19(3) = 6{49|58} -> no 7 in r4c3, no 7,9 in r3c4

35. r4: 7 locked in r4c789 for r4 and n6

36. n1: 20(3): {569} blocked by r3c3 -> no 6 in 20(3)

37. n1: 16(3) = {268|349|358|457} ({259|367} blocked by h9(3))

38. n5: 10(3): {145} blocked by r1c4 -> 10(3) = 3{16|25} -> no 4 in r45c4 -> 3 locked for 10(3) and n5

39. n5: 14(3): cleanup -> no 5,6 in r5c5

40. n5: h8(3): cleanup -> no 5 in r4c4, no 3 in r4c5

41. n5: 10(3): cleanup -> no 5 in r5c4 -> 3 locked in r45c4 for c4 and n5

42. {23} locked in r4c4 and r7c7 for D\ -> no 2,3 in r1c1 and r2c2

43. 45 on c6789: r456c6 = h18(3) = {189|459|468|567}

44. n5: h8(3) and 10(3): no 5 in r4c6 (would mean r4c45 = [21], r5c4 = 7 -> not possible)
44a. 14(3): {257} not possible
44b. h18(3) at r4c6: {567} not possible -> no 7 in r56c6
44c. 21(3): no 5,6 at r6c5
44d. h18(3) at r4c6: {468} not possible (would force 6 in r5c4) -> no 4,6 in r56c6
44e. -> 9 locked in r56c6 for 21(3), c6 and n5
44f. no 8 in r6c5
44g. 14(3): {158} not possible, blocked by h18(3) at r4c6 -> no 4,5 in r6c4
44h. combination check: no 5 in r6c6, no 8 in r5c6, no 2 in r5c4 (h18(3) at r6c4 = [675] clashes with 14(3) = [176])

45. n5: h11(3) = {236} -> 2,3,6 locked for c4 and n5

46. n5: 7 locked in r56c5 for n5 and c5

47. n8: 7,9 locked in r789c4 for n8 and c4

48. n2: 9 locked in 14(3) -> 14(3) = {239} -> 2,3,9 locked for c5 and n2

49. n8: 15(3) = 6{18|45} -> 6 locked for c5 and n8

50. n23: 18(3) = {378|468|567} (no 1,2,9}

51. -> r1c8 = 1

52. n3: 14(3) = 1{49|58|67} -> no 2,3

53. 45 on r5: r5c456 = h19(3) = {379|568}

54. n4: 14(3) = {149|158|347} (other combinations blocked by h19(3) and r4c12)
54a. -> {45} locked in 14(3) and r4c12 for n4

55. n6: 2 locked in 12(3) for r5 and c6
55a. n6: 12(3) = 2{19|46} (other combinations blocked by h19(3)) -> no 9 at r5c7

56. r7: 2 locked in r7c67 for 6(3) and r7

57. r5: 14(3): {149} blocked by h19(3) and 12(3) -> no 9

58. 8 locked in r5c5 and r6c6 for n5 and D\

59. n6: 5 locked in r6c89 for n6, r6 and 15(3)
59a. 15(3) = 5{37|46} -> no 6 in r6c89

60. n9: h18(3): no 6,7 in r9c7

61. n89: 15(3) = {159|249|258|348} -> no 8 in r89c6

62. n6: 8 locked in r4c789 for n6 and r4

63. n124: 19(3): 6 locked in r34c3 for 19(3) and c3
63a. no 5 in r3c4

64. n3: 14(3): {167} blocked by r7c9 -> no 6,7

65. n3: 19(3): {289|478} blocked by 14(3) -> no 2
65a. {89} locked in 14(3) and 19(3) -> no 8,9 in r13c7

66. n7: 19(3): no 2 in r8c1 (blocked by r9c7)


67. 45 on n3: r1c7 + r3c79 = h12(3) = 2{37|46} -> 2 locked in r3c79 for h12(3), r3 and n3

68. n6: 3 locked in r6c789 for n6 and r6

69. n47: 17(3): {458|359} not possible -> no 5

70. 45 on c89: r258c7 = h15(3) = 5{19|28|46} -> no 3,7

71. from step 17: r3: h19(3) = 9{37|46} -> no 5,8 -> no 4 in r3c8

72. 45 on r89: r7c258 = h16(3) = {169|178|349|358|367|457}

73. 45 on c12: r258c3 = h16(3) = {178|349|358|457}

74. n78: 18(3) = {279|378|459} -> no 7,8,9 in r9c3

75. n8: 4 or 5 in r89c6 (15(3)) -> 15(3) = {168}

76. r456c5 = [574], r456c6 = [198], r456c4 = [236], r7c67 = [23], r6c7 = 1, r7c9 = 7

77. n6: 12(3) = {246}

78. r4c3 = 6, r3c34 = [58]

79. r1c34 = [34]

80. n1: 20(3) = {479}

81. r1c167 = [657], r23c6 = [67], r34c7 = [29], r4c89 = [78], r123c9 = [943], r123c5 = [239]



Edit: Thanks for your comments, Andrew. They are included in blue.
Cheers,
Nasenbaer


Last edited by Nasenbaer on Fri Dec 21, 2007 10:25 pm; edited 2 times in total
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Caida
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PostPosted: Thu Dec 13, 2007 4:29 pm    Post subject: Walkthrough A2X Reply with quote

After several false starts I have finally managed to finish A2X.
Most of my earlier problems came from faulty math and eliminating combos prematurely.

I don't know that I would rate this as a 1.75 (I managed to finish it through perseverance) but I could see rating it as 1.5.

Interested to see other approaches.

Here's my walkthough - after several restarts I have put what I found were the key moves (my steps 6,7,8) as close to the beginning as I could.

Any comments/suggestions/corrections appreciated!
Edited for typos - thanks Andrew!!

Assassin 2X walkthrough

Prelims:

a. 10(3)n4 and n5 and n89 = {127/136/145/235} (no 8,9)
b. 21(3)n5 = {489/579/678} (no 1,2,3)
c. 24(3)n9 = {789} (no 1..6) -> 7,8,9 locked for n9

1. Innies r1234: r4c456 = 10(3) = {127/136/145/235} (no 8,9)
1a. overlapping cages: r5c4 – r4c6 = 4
1c. -> r5c4 no 1,2,3,4
1d. -> r4c6 no 6,7

2. Innies r6789: r6c456 = 14(3)
2a. overlapping cages: r5c6 – r6c4 = 7
2b.-> r5c6 = 8,9 (no 4,5,6,7)
2c. -> r6c4 = 1,2 (no 3,4,5,6,7)

3. Innies c1234: r456c4 = 10(3) = {127/136/145/235} (no 8,9)
3a. overlapping cages: r4c5 – r6c4 = 4
3b. -> r4c5 = {56} (no 1,2,3,4,7)
3c. -> r4c6 no 4,5 (step 1a)

4. Innies c6789: r456c6 = 20(3) = [3]{89} -> 3 locked for diagonal -> no 3 elsewhere in diagonal top right (tr) -> bottom left (bl)
4a. -> {89} locked for c6 and n5
4b. r5c4 = 7 (step 1a)
4c. -> r6c5 = 4 (cage 21(3))
4d. h10(3)c4 (step 3) = {1[7]2}
4e. -> {12} locked for c4 in n5
4f. -> {56} locked for c5 in n5

5. 10(3)n4 = {136/145/235} {1 | 2} {5 | 6}
5a. killer pair {56} in r5 in 10(3)n4 and r5c5 -> no 5,6 elsewhere in r5

6. Innies n478: r4c123+r789c6 = 36
6a. -> max r4c123 = 24 -> min r789c6 = 12
6b. -> max r89c6 = 9 + max r7c6 = 7
6c. -> max r789c6 = 16 -> min r4c123 = 20 (no 1,2)

7. since 10(3)n4 must contain either 1 or 2 (but not both) and r4c123 doesn’t have either than
7a. -> r6c123 must have the other 1 or 2 for n4
7b. -> killer pair {12} in r6c123 and r6c4 -> no 1,2 elsewhere in r6
7c. -> killer pairs of {12} are also in r5c123 + r6c123 and in r5c123 and r5c789

I don’t think this next bit is t&e – but don’t know how to write it without making it look t&eish

8. if r6c4 = 1 -> r5c6 = 8 and 10(3)n4 = {1..} -> 14(3)n6 must have a 2 and 9 but no 1 or 8 (239)
8b. if r6c4 = 2 -> r5c6 = 9 and 10(3)n4 = {1..} -> 14(3)n6 must have a 1 and 8 but no 9 (158) -> no 5 available
8c. -> 14(3)n6 = 239 -> locked for r5 and n6
8d. -> h10(3)r4 = [253]
Andrew pointed out that I forgot to note the hidden single in r4c4
8e. -> 10(3)n5 = [361]
8f. -> 21(3)n5 = [849]
8g. -> 10(3)n4 = {145} -> locked for n4
8h. -> 2,6,9 locked for diagonal from top left (tl) -> bottom right (br)
8i. -> 1,6,3 locked for diagonal from top right (tr) -> bottom left (bl)

9. Outies r12: r3c258 = 22(3) = {589/679} (no 1..4) -> 9 locked for r3

10. Outies c12: r258c3 = 19(3) = [649/946/748/847/658/856] (no 1,2,3)
10a. -> r28c3 no 4,5

11. 17(3)n3 = {269/278/359/368/458/467} (no 1)
Note: combo {179} blocked by 24(3)n9
11a. Outies c89: r258c7 = 17(3) = [827/728/629/638/539]
11b. -> r2c7 no 2,3,4,9
11c. -> r5c7 no 9
11d. -> 17(3)n3: r2c8 no 5,7,8,9

12. Innies n1: r1c3+r3c13 = 10(3) = {127/136/145/235} (no 8,9)

13. Innies n9: r7c89+r9c7 = 8(3) = {125/134} (no 6) -> 1 locked for n9

14. 13(3)n69 = [571/751/562/652]
14a. -> 5 locked in n6 in r6c89 -> no 5 elsewhere in n6
14b. -> r6c89 no 8
14c. -> r7c9 no 3,4,5

15. {14} locked in n6 in r4c789
15a. -> r4c89 must contain at least 1 of {14} (could contain both)
15b. -> 13(3)n36 = [148/184/418/481/814/841/517/571/247/274/346/364]
15c. -> r3c9 no 6,7

16. {23} locked in n4 in r6c123
16a. -> r6c12 must contain at least 1 of {23} (could contain both)
15b. -> 13(3)n47 = [238/283/328/382/823/832/274/724/265/625/364/634]
15c. -> r7c1 no 1679

16. 17(3)n689 = [674/764/863/845/854] (no 1,2)
16a. -> 1 in diagonal tl-> br locked in n1 -> no 1 elsewhere in n1

17. h8(3)n9 (step 13) = [314/413/512/521]
17a. -> r9c7 no 5

18. 18(3)n14 = [567/576/468/486/369/396/378/387/279/297]
18a. -> r3c1 no 6,7
18b. 18(3)n1: min r2c3+r3c2 = 11 -> r2c2 no 8


19. Innies n7: r7c13+r9c3 = 12(3) = {129/138/147/237/246/345}
19a. -> r7c1 no 8
19b. -> r9c3 no 8, 9

20. 12(3)n236 = {147/246} (no 5,8)
Note combo [651] blocked by h22(3)r3; combo [156] blocked by h17(3)c7 and r6c7

21. Innies n3: r1c7+r3c79 = 15(3) = {258/348/357/456}
21a. -> r1c7 and r3c9 no 1,2,4,7,9
21b. -> single: r8c7 = 9
21c. -> pair {78} locked in n9 for c8
21d. -> h17(3)c7 = [539/629]
21e. -> r2c7 no 7,8
21f. -> 17(3)n3 = [629] -> 2 locked for diagonal
21g. -> r5c789 = [239]

22. h22(3)r3 = [679/589]
22a. -> r3c2 no 7,8

23. 9 locked for diagonal in n7 -> no 9 elsewhere in n7

24. 18(3)n1 = [495/396/486/576]
24a. -> r2c2 no 1,7
24b. triple {345} locked for diagonal in r2c2, r7c7, r9c9 -> no 3,4,5 elsewhere in diagonal (not in r1c1 or r3c3)

25. 15(3)n124 = [159/168/186]
25a. -> r3c3 = 1
25b. -> r3c4 no 3,4
25c. -> r4c3 no 7

26. Innies n7: r7c3 no 8, 9
26a. single: r9c1 = 9
26b. 17(3)n7: r8c1 and r9c2 = {17/26/35} (no 4,8)
26c. 8 in r9 locked in n8 -> no 8 elsewhere in n8

27. 12(3)n236 = {47}[1]
27a. -> {47} locked in r3
27b. -> h22(3)r3 = [589]
27c. -> 15(3)n124 = [168]
27d. -> 13(3)n36 = [3]{46} -> {46} locked for n6 and r4

singles and cages sums left


Solution:

867392514
349517628
251684793
798253146
514768239
632149857
425976381
186435972
973821465

Cheers,

Caida


Last edited by Caida on Thu Jan 03, 2008 6:36 pm; edited 2 times in total
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Afmob
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PostPosted: Thu Dec 13, 2007 10:36 pm    Post subject: Reply with quote

Seems my key moves are not much different from Caida's ones. I wonder if my step 8c can be considered a Killer XY-Wing?

A2X Walkthrough:

1. N5
a) Innies+Outies R6789: 7 = R5C6 - R6C4
-> R5C6 = (89), R6C4 = (12)
b) Innies R5 = 21(3) <> 1,2,3
c) Innies C6789 = 20(3) <> 1,2
d) 10(3) <> 7 because 1,2 only possible @ R6C4
e) Innies R5 = 21(3) must have 4,5 xor 6 and R5C5 = (456) -> R5C4 <> 4,5,6
f) Innies C1234 = 10(3) = {127} -> R5C4 = 7, (12) locked for C4+N5

2. N5+R5
a) 21(3) = {489} locked for N5
b) Innies C5 = 15(3) = {456} -> R6C5 = 4, (56) locked for C5+N5
c) Innies C6789 = 20(3) = {389} -> R4C6 = 3, (89) locked for C6
d) Killer pair (56) locked in 10(3) + R5C5 for R5

3. N9
a) Innies = 8(3) = 1{25/34} -> 1 locked
b) 6 locked in 13(3) = 6{25/34}

4. C123
a) Outies C12 = 19(3) <> 1
b) Outies C12 = 19(3): R28C3 <> 2,3 because R5C3 <= 6
c) Innies N1 = 10(3) <> 8,9
d) 15(3) @ R8C4: R9C3 <> 9 because R89C4 >= 7
e) 13(3): R7C1 <> 9 because {139} blocked by Killer pair (13) of 10(3)

5. R123
a) Outies R12 = 22(3) = 9{58/67} -> 9 locked for R3
b) 12(3) <> 8

6. C789
a) 17(3) @ N9: R6C7+R7C6 <> 1,2 because R7C7 <= 5
b) 17(3) @ N9: R6C7 <> 3,5 because R7C67 <> 8,9

7. N12+C6 !
a) ! R789C6 <= 16 because R89C6 <= 9 -> R123C6 >= 9 (step 1c)
b) ! Innies+Outies N12: 11 = R4C123 - R123C6
-> R4C123 >= 20 <> 1,2

8. R456 !
a) R6C123 must have 1 xor 2 @ N4 because 10(3) <> {127}
b) Killer pair (12) locked in R6C123+R6C4 for R6
c) ! Consider placement of 6 in R5 -> R6C123 <> 1
- i) 6 in 10(3) @ N4 = {136} -> R6C123 <> 1
- ii) 6 in 10(3) @ N5 = {136} -> R6C4 = 1 -> R6C123 <> 1
d) Hidden Single: R6C4 = 1 @ R6 -> R5C5 = 6
e) R4C4 = 2, R4C5 = 5
f) Innies R6789 = 14(3) = {149} -> R6C6 = 9 -> R5C6 = 8
g) 1 locked in 10(3) @ N4 = {145} locked for R5+N4
h) 14(3) @ N6 = {239} locked for N6
i) 5 locked in 13(3) @ R6 (N6) -> 13(3) = 5{17/26}, R7C9 <> 5

9. N1
a) 17(3) @ N9 <> 1
b) 1 locked in D\ for N1
c) Innies N1 = 10(3): R3C3 <> 4,7 because R1C3+R3C1 <> 1
d) 18(3) @ R3C1: R3C1 <> 6,7 because R4C12 >= 13

10. C123
a) Outies C12 = 19(3) must have 4 xor 5 because R5C3 = (45) -> R28C3 <> 4,5
b) 13(3): R7C1 <> 1,6,7 because R6C12 <> 4,5

11. N9+D\
a) 10(3): R9C7 <> 2 because R89C6 <> 3 and Killer pair (17) of 12(3) @ N8 blocks {127}
b) Innies = 8(3): R9C7 <> 5 because R7C79 >= 4
c) 13(3) @ R8C9: R8C9+R9C8 <> 5 because R9C9 <> 2,6
d) 5 locked in R7C7+R9C9 for D\
e) 15(3) @ D\ must have 1 xor 3 because R3C3 = (13) -> R3C4 <> 3

12. C89 !
a) ! Innies+Outies C9: 7 = R1469C8 - R5C9
-> R5C9 <> 2,3 because R6C8 >= 5
b) R5C9 = 9

13. N12 !
a) ! Innies+Outies N1: 23 = R4C123+R3C4 - R1C3
-> R4C123 >= 22 because 9 locked there @ N4
-> R1C3 <> 2
b) 15(3) @ R1C3 <> 9
c) 9 locked in 18(3) @ N2 = 9{18/27} -> 9 locked for C5
d) 3 locked in 15(3) @ R1C3 = 3{48/57} for C4; R1C3 <> 3
e) 15(3) @ R1C3: R1C3 <> 5 because R12C4 <> 7
f) Innies N1 = 10(3) = 1{27/45} because R1C3 = (47) -> R3C3 = 1, R3C1 = (25)

14. C789
a) Hidden Single: R3C9 = 3 @ R3
b) 13(3) @ R3C9 = {346} -> (46) locked for R4+N6
c) Innies N3 = 15(3) = 3{48/57}, R1C7 <> 4
d) Hidden Single: R2C7 = 6 @ C7, R8C7 = 9 @ C7
e) 24(3) = {789} -> (78) locked for C8
f) 17(3) @ N3 = {269} -> R2C8 = 2, R3C8 = 9
g) 13(3) @ R6C8 = {157} -> R7C9 = 1, R6C8 = 5, R6C9 = 7
h) Innies N9 = 8(3) = {134} -> (34) locked for C7+N9
i) Innies N3 = {357} locked for N3
j) 17(3) @ R6C7 = 8[54/63] -> R6C7 = 8
k) 10(3) <> 7 because R9C7 = (34)

15. N12
a) 15(3) @ R3C3 = 1[59/68]
b) 18(3) @ R3C1 = {279} -> R3C1 = 2, (79) locked for N4
c) 15(3) @ R3C3 = {168} -> R4C3 = 8, R3C4 = 6

16. R123
a) 14(3) = {257} -> R1C6 = 2
b) 18(3) @ N2 = {189} locked for C5+N5
c) 15(3) = {357} -> R1C3 = 7, (35) locked for C4+N2
d) 18(3) @ N1 = {459} -> R2C2 = 4, R2C3 = 9, R3C2 = 5

17. N7
a) 16(3) = 6[28/37]
b) Hidden Single: R9C1 = 9 @ D/, R6C1 = 6 @ C1
c) 15(3) = {348} -> R9C3 = 3
d) 16(3) @ R6C3 = {259} -> R6C3 = 2, R7C3 = 5, R7C4 = 9

18. Rest is singles (without considering the diagonals).

Rating: 1.5, had some difficult moves but not too many of them.


Last edited by Afmob on Mon Dec 31, 2007 7:30 am; edited 3 times in total
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Andrew
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PostPosted: Sat Dec 15, 2007 7:16 am    Post subject: Reply with quote

I finished A2X-Lite (I prefer Light) yesterday evening and went through Caida's and Nasenbaer's walkthroughs today. Since neither of them had my breakthrough move, I'm posting my walkthrough now. Must admit I didn't initially spot step 21 but when I did I reworked it to put it in the right place.

I'd originally been thinking of rating it as an easier 1.25 but after I found step 21 I'll make it a high 1.0.

Here is my walkthrough for A2X-Lite.

Prelims

a) 8(3) cage at R1C3 = 1{25/34}, CPE no 1 in R1C56
b) 20(3) cage in N1 = {389/479/569/578}, no 1,2
c) 19(3) cage in N3 = {289/379/469/478/568}, no 1
d) 8(3) cage at R3C1 = 1{25/34}, CPE no 1 in R56C1
e) 19(3) cage at R3C3 = {289/379/469/478/568}, no 1
f) 10(3) cage in N5 = {127/136/145/235}, no 8,9
g) 21(3) cage in N5 = {489/579/678}, no 1,2,3
h) 6(3) cage at R6C7 = {123}
i) 19(3) cage in N9 = {289/379/469/478/568}, no 1
j) 19(3) cage in N7 = {289/379/469/478/568}, no 1
k) 8(3) cage in N9 = 1{25/34}, 1 locked for N9

1. Killer pair 2,3 in R7C7 and 8(3) cage, locked for N9

2. 19(3) cage in N9 = {469/478/568}
2a. Killer pair 4,5 in 8(3) cage and 19(3) cage, locked for N9
2b. Min R7C9 = 6 -> max R6C89 = 9, no 9
2c. Min R9C7 = 6 -> max R89C6 = 9, no 9

3. 45 rule on N1 3 innies R1C3 + R3C13 = 9 = {126/135/234}, no 7,8,9

4. 45 rule on R12 3 outies R3C258 = 19 = {289/379/469/478/568}, no 1

5. 45 rule on R5 3 innies R5C456 = 19 = {289/379/469/478/568}, no 1
5a. 2 of {289} must be in R5C4 -> no 2 in R5C5

6. 45 rule on R1234 3 innies R4C456 = 8 = 1{25/34}, 1 locked for R4 and N5
6a. R3C1 = 1 (only remaining position for 1 in 8(3) cage), R4C12 = {25/34}
6b. Naked quint {12345} in R4C12456, locked for R4
6c. R13C3 = 8 (step 3) = [26/35], no 4, no 2 in R3C3

7. 8(3) cage at R1C3 = 1{25/34}, 1 locked in R12C4, locked for C4 and N2
7a. 3 of {134} must be in R1C3 -> no 3 in R12C4

8. R9C9 = 1 (hidden single in D\), R8C9 + R9C8 = {25/34}
8a. 1 in N3 locked in R1C78, locked for R1
8b. R2C4 = 1 (hidden single in C4), R1C34 = {25}/[34]
[Hidden single in R2 would have been slightly quicker but that’s not what I spotted.]

9. 45 rule on R1234 1 outie R5C4 = 1 innie R4C6 + 2, no 2 in R5C4

10. 45 rule on R6789 1 outie R5C6 = 1 innie R6C4 + 3, no 4 in R5C6, no 7,8,9 in R6C4

11. 10(3) cage in N5 = {127/136/145/235}
11a. 1 of {145} must be in R4C5 -> no 4 in R4C5

12. 45 rule on C1234 3 innies R456C4 = 11 = {236} (cannot be {245} which clashes with R1C4), locked for C4 and N5, clean-up: no 5 in R1C3 (step 8b), no 3 in R3C3 (step 6c), no 5 in R4C6 (step 9), no 7,8 in R5C6, no 3 in R6C4 (both step 10)

13. Naked pair {23} in R4C4 + R7C7, locked for D\

14. 21(3) cage in N5 = {489/579}, 9 locked for N5

15. 14(3) cage in N5 = {167/248} (cannot be {158} because 5,8 only in R5C5, cannot be {257} because R4C6 only contains 1,4), no 5
15a. 4 of {248} must be in R4C6 -> no 4 in R5C5

16. 45 rule on C5 3 innies R456C5 = 16 = {178/457}, no 9, 7 locked for C5 and N5
16a. 5 of {457} must be in R4C5 -> no 5 in R6C5

17. 9 in N5 locked in R56C6, locked for C6
17a. 45 rule on N5 3 remaining innies R456C6 = 18 = {189/459}
17b. 4 of {459} must be in R4C6 -> no 4 in R6C6

18. 21(3) cage in N5 (step 14) = {489/579}
18a. 4 of {489} must be in R6C5 -> no 8 in R6C5
[At this stage I missed 8 locked in R5C5 + R6C6 for D\, fortunately it didn’t matter.]

19. 19(3) cage at R3C3 = {469/568} (cannot be {478} because R3C3 only contains 5,6), no 7, 6 locked in R34C3, locked for C3
19a. 4 of {469} must be in R3C4 -> no 9 in R3C4
19b. 7 in R4 locked in R4C789, locked for N6
19c. 7,9 in C4 locked in R789C4, locked for N8

20. 9 in N2 locked in R123C5 = {239} (only remaining combination), locked for C5 and N2
20a. 6 in C5 locked in R789C5, locked for N8

21. 2,3 in N8 locked in R789C6, R89C6 cannot be {23} -> R7C6 = {23}
21a. Naked pair {23} in R7C67, locked for R7
21b. R6C7 = 1 (only remaining position for 1 in 6(3) cage)

22. R1C8 = 1 (hidden single in R1), R12C9 = 13 = {49/58/67}, no 2,3

23. R5C456 (step 5) = {379/568}
23a. R5C789 = {246} (only remaining combination, cannot be {345} which clashes with R5C456), locked for R5 and N6 -> R5C4 = 3, R4C4 = 2, R4C5 = 5, R5C6 = 9, R6C4 = 6, R6C6 = 8, R5C5 = 7, R6C5 = 4, R4C6 = 1, 1,6,7 locked for D/, 2,7,8 locked for D\ -> R7C7 = 3, R7C6 = 2, clean-up: no 6,7 in R2C9 (step 22), no 4 in R8C9 + R9C8 (step 8)
23b. Naked pair {25} in R8C9 + R9C8, locked for N9

24. R4C3 = 6 (hidden single in R4), R3C3 = 5, R3C4 = 8 (step 19)

25. Naked pair {35} in R6C89, locked for R6, R7C9 = 7
25a. 45 rule on N9 1 remaining innie R9C7 = 8, R9C5 = 6
25b. R9C7 = 8 -> R89C6 = 7 = {34}, locked for C6 and N8

26. R1C4 = 4 (hidden single in C4), R1C3 = 3 (step 8b), clean-up: no 9 in R2C9 (step 22)

27. 18(3) cage at R1C3 = {567} (only remaining combination), no 2,9 in R1C7
27a. 5 in C6 locked in R12C6 -> no 5 in R1C7

28. R2C7 = 5 (hidden single in C7), clean-up: no 8 in R12C9 (step 22)

and the rest is naked singles, including eliminations along the diagonals, and cage sums

6 8 3 4 2 5 7 1 9
2 9 7 1 3 6 5 8 4
1 4 5 8 9 7 2 6 3
4 3 6 2 5 1 9 7 8
5 1 8 3 7 9 4 2 6
7 2 9 6 4 8 1 3 5
8 6 4 5 1 2 3 9 7
9 5 1 7 8 3 6 4 2
3 7 2 9 6 4 8 5 1

Maybe some of the steps I used here may be helpful for A2X, when I've got time to look at it.


Last edited by Andrew on Sat Dec 22, 2007 8:30 pm; edited 1 time in total
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mhparker
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PostPosted: Sun Dec 16, 2007 4:26 pm    Post subject: Reply with quote

Afmob wrote:
I wonder if my step 8c can be considered a Killer XY-Wing?

Interesting point (and great move!), Afmob. As you may have seen, I've gone into more detail on this question here.
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Mike
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Location: Lethbridge, Alberta

PostPosted: Fri Dec 28, 2007 8:59 pm    Post subject: Reply with quote

I finished A2X on Christmas Eve but only managed to go through Caida's and Afmob's walkthroughs yesterday evening.

Afmob wrote:
Seems my key moves are not much different from Caida's ones.

There seems to be a very narrow solving path around Caida's step 8, Afmob's step 8 and my step 21.

Mike wrote:
Yes, it's a toughie. Unlike several recent Assassin V1's, which consist of finding one (or at most, two) key move(s), apart from which the rest is plain sailing, this is more like a traditional V2, which (whilst having easier sections) generally involves more of a fight to dig away at the candidates. This puzzle does contain at least one key move, however, that will make your life easier if you find it.

I assume Mike was referring to the narrow solving path mentioned above. It certainly took a lot more work after that. I've a feeling that my later stages were longer than those of Caida and Afmob although I haven't tried to compare them to see where they were quicker.

I'll agree with both of them and rate A2X as 1.5.

Here is my walkthrough.

Prelims

a) R5C123 = {127/136/145/235}, no 8,9
b) 10(3) cage in N5 = {127/136/145/235}, no 8,9
c) 21(3) cage in N5 = {489/579/678}, no 1,2,3
d) 10(3) cage at R8C6 = {127/136/145/235}, no 8,9
e) 24(3) cage in N9 = {789}, locked for N9

1. 45 rule on N1 3 innies R1C3 + R3C13 = 10 = {127/136/145/235}, no 8,9
1a. Max R3C1 = 7 -> min R4C12 = 11, no 1

2. 45 rule on N9 3 innies R7C79 + R9C7 = 8 = 1{25/34}, no 6, 1 locked for N9

3. 45 rule on R5 3 innies R5C456 = 21 = {489/579/678}, no 1,2,3

4. 45 rule on R1234 3 innies R4C456 = 10 = {127/136/145/235}, no 8,9

5. 45 rule on C1234 3 innies R456C4 = 10 = {127/136/145/235}, no 8,9

6. 45 rule on C6789 3 innies R456C6 = 20 = {389/479/569/578}, no 1,2

7. 45 rule on C1234 1 outie R4C5 = 1 innie R6C4 + 4, R4C5 = {567}, R6C4 = {123}

8. 45 rule on C6789 1 outie R6C5 = 1 innie R4C6 + 1, no 9 in R6C5

9. 45 rule on R1234 1 outie R5C4 = 1 innie R4C6 + 4 -> R5C4 = 7, R4C6 = 3, locked for D/, R6C5 = 4 (step 8)

10. Naked pair {56} in R45C5, locked for C5 and N5
10a. Naked pair {12} in R46C4, locked for C4
10b. Naked pair {89} in R56C6, locked for C6

11. Killer pair 5,6 in R5C123 and R5C5, locked for R5

12. 45 rule on R12 3 outies R3C258 = 22 = 9{58/67}, 9 locked for R3

13. 45 rule on C12 3 outies R258C3 = 19 = {289/379/469/478/568}, no 1
13a. 2,3 of {289/379} must be in R5C3 -> no 2,3 in R28C3

14. 45 rule on C9 4 outies R1469C8 = 1 innie R5C9 + 7
14a. Min R1469C8 = 10 -> min R5C9 = 3

15. 17(3) cage at R6C7 = {179/269/278/359/368/458/467}
15a. 8,9 of {179/269/278/359/368/458} must be in R6C7-> no 1,2,3,5 in R6C7
15b. 1,2 of {179/269/278} must be in R7C7 -> no 1,2 in R7C6

16. 45 rule on C89 3 outies R258C7 = 17 = {179/269/278/359/368/458/467}
16a. 3,4 of {359/368/458/467} must be in R5C7 -> no 3,4 in R2C7

17. 17(3) cage in N3 = {179/269/278/458/467}
17a. 4 of {458} must be in R2C8 -> no 5 in R2C8

18. 45 rule on R89 3 outies R7C258 = 17 = {179/269/278/359/368} (cannot be {458/467} because 4,5,6 only in R7C2), no 4
18a. 3 of {359/368} must be in R7C5 -> no 3 in R7C2

19. 45 rule on N478 6 innies R4C123 + R789C6 = 36
19a. Max R89C6 = 9 (because of 10(3) cage at R8C6), max R7C6 = 7 -> max R789C6 = 16 -> min R4C123 = 20, no 1,2

20. R5C123 = {136/145/235} [1/2]
20a. Hidden killer pair 1,2 in R5C123 and R6C123 for N4 -> R6C123 must contain 1/2
20b. Killer pair 1,2 in R6C123 and R6C4, locked for R6

21. Chaining steps 20a and 20b, R5C123 and R6C4 must contain the same value of 1/2
21a. 45 rule on R6789, 1 outie R5C6 = 1 innie R6C4 + 7 -> R5C6 + R6C4 = [81/92]
21b. -> R5C123 + R5C6 must contain 1,8 or 2,9
21c. R5C789 = {239} (only remaining combination, cannot be {149/248} which clash with R5C123 + R5C6), locked for R5 and N6 -> R56C6 = [89], 9 locked for D\

22. R1469C8 = R5C9 + 7 (step 14)
22a. Min R1469C8 = 11 -> no 3 in R5C9 -> R5C9 = 9

23. R5C123 = {145} (only remaining combination), locked for R5 and N4 -> R5C5 = 6, locked for both diagonals, R4C5 = 5, R4C4 = 2 (step 4), locked for D\, R6C4 = 1, locked for D/
23a. R12C9 cannot be {13} -> no 9 in R1C8

24. 17(3) cage at R6C7 (step 15) = {368/458/467}, no 1
24a. 1 on D\ locked locked in R1C1 + R2C2 + R3C3, locked for N1

25. R7C79 + R9C7 (step 2) = 1{25/34}
25a. 5 of {125} must be in R7C7 -> no 5 in R7C9 + R9C7

26. 5 in R6 locked in R6C89 for 13(3) cage at R6C8 = 5{17/26}, no 3,4,8

27. Min R4C12 = 13 -> max R3C1 = 5

28. R1C3 + R3C13 (step 1) = {127/136/145/235}
28a. 1 of {127/145} must be in R3C3 -> no 4,7 in R3C3

29. 2,3 in R6 locked in R6C123 -> at least one of 2,3 must be in R6C12
29a. 13(3) cage at R6C1 = {238/247/256/346} (cannot be {139} because 1,9 only in R7C1), no 1,9
29b. 4 of {247} must be in R7C1 -> no 7 in R7C1
29c. 6 of {256/346} must be in R6C12 -> no 6 in R7C1

30. 1,4 in R4 locked in R4C789 -> at least one of 1,4 must be in R4C89
30a. 13(3) cage at R3C9 = {148/157/247/346}
30b. 2,3,5 of {157/247/346} must be in R3C9 -> no 6,7 in R3C9

31. 45 rule on N3 3 innies R1C7 + R3C79 = 15 = {159/168/249/258/267/348/357/456}
31a. 9 of {159} must be in R1C7
31b. 8 of {168} must be in R3C7
31c. -> no 1 in R1C7

32. R258C3 (step 13) = {469/478/568}
32a. R5C3 = {45} -> no 4,5 in R28C3

33. 18(3) cage in N1 = {189/369/378/459/468/567}
33a. 1,3,4 of {189/378/468} must be in R2C2 -> no 8 in R2C2
33b. 8 on D\ locked in R1C1 + R8C8 -> no 8 in R1C8 + R8C1

34. R258C7 (step 16) = {179/269/278/359/368}
34a. 9 of {269} must be in R8C7 -> 2 of {269/278} must be in R5C7 -> no 2 in R2C7
34b. 1 of {179} must be in R2C7
34c. 9 of {269/359} must be in R8C7
34d. -> no 9 in R2C7

35. 17(3) cage in N3 (step 17) = {179/269/278/458/467}
35a. 2,4 of {278/458} must be in R2C8 -> no 8 in R2C8

36. 14(3) cage at R1C6 = {149/158/167/248/257/347/356} (cannot be {239} because 3,9 only in R1C7)
36a. 3,8,9 of {149/248/347} must be in R1C7 -> no 4 in R1C7

37. 45 rule on N7 3 innies R7C13 + R9C3 = 12 = {129/138/147/237/246/345} (cannot be {156} because 1,6 only in R9C3)
37a. 1 of {129/138} must be in R9C3 -> no 8,9 in R9C3
37b. 3 of {138} must be in R7C1 -> no 8 in R7C1

38. 2,3 in R6 locked in R6C123
38a. R6C12 cannot be {23} because max R7C1 = 5 -> R6C3 = {23}

39. 16(3) cage at R6C3 = {259/268/349/358/367} (cannot be {457} because R6C3 only contains 2,3)
39a. R6C3 = {23} -> no 2,3 in R7C34

40. 2,3 in R6 locked in R6C123 = {236/237/238}
40a. 45 rule on N7 6 outies R6C123 + R789C4 = 32
40b. R6C123 = 11,12,13 -> R789C4 = 19,20,21 = {389/469/489/568/569}
40c. Max R89C4 = 14 (because of 15(3) cage at R8C4) -> min R7C4 = 5

41. 12(3) cage at R3C6 = {147/156/246}, no 8
41a. 5 of {156} must be in R3C7 -> no 5 in R3C6

42. 9 in C8 locked in R237C8
42a. 45 rule on C89 4 innies R2378C8 = 1 outie R5C7 + 24 -> R2378C8 = 26,27
42b. R2378C8 = {2789/4589/4679/4689} (cannot be {5679} because 5,6 only in R3C8)
42c. 2,4 must be in R2C8 -> no 7,9 in R2C8

43. 17(3) cage in N3 (step 17) = {269/278/458/467} (cannot be {179} because R2C8 only contains 2,4), no 1

44. 9 on D/ locked in R7C3 + R8C2 + R9C1, locked for N7

45. 16(3) cage in N7 = {169/268/457} (cannot be {178} because R8C23 = {78} clashes with R8C8, cannot be {259} because R8C3 only contains 6,7,8)
45a. {457} must be [547]
45b. -> no 7 in R7C2, no 5,7 in R8C2

46. R7C258 (step 18) = {179/269/278/359/368}
46a. 1 of {179} must be in R7C2 -> no 1 in R7C5

47. R1C7 + R3C79 (step 29) = {159/348/357} (cannot be {168} because no 1,6,8 in R3C7, cannot be {249} which clashes with R2C8, cannot be {258/267/456} because there’s no place for 9 in N3), no 2,6
47a. 4 of {348} must be in R3C7 -> no 4 in R3C9

48. 12(3) cage at R3C6 (step 39) = {147/156/246}
48a. 6 of {156} must be in R4C7 (cannot be [651] which clashes with R3C258)
48b. 2 of {246} must be in R3C6
48c. -> no 6 in R3C6

49. R4C123 + R789C6 = 36 (step 19)
49a. R4C123 = 22,23,24 -> R789C6 = 12,13,14
49b. R789C6 = {147/156/246/157/247/256/167/257} [1/2]
49c. R789C5 = {129/138/237}
49d. Killer triple 1,2,3 in R789C56, locked for N8
49e. Min R89C4 = 9 -> max R9C3 = 6

50. 3 in N8 locked in R789C5, locked for C5
50a. R789C5 = {138/237}, no 9
50b. 9 in N8 locked in R789C4, locked for C4

51. R789C6 = 12,13,14 (step 47a) -> R789C4 = 19/20/21
51a. R789C4 = {469/569/489} [4/5]
51b. Hidden killer pair 4,5 in N8 -> R789C6 must contain 4/5 -> R789C6 not {167}

52. 45 rule on R89 4 innies R8C2378 = 1 outie R7C5 + 23
52a. Max R8C2378 = 30 -> no 8 in R7C5

53. R89C6 must contain 1/2 (step 49b)
53a. 10(3) cage at R8C6 = {127/136/145/235}
53b. 4 of {145} must be in R9C7 -> no 4 in R89C6

54. R7C258 (step 18) = {179/269/278/359/368} [8/9]
54a. Hidden killer pair 8,9 in R7 -> R7C34 must contain 8/9
54b. 16(3) cage at R6C3 (step 39) = {259/268/349/358} (cannot be {367}), no 7

55. R7C13 + R9C3 (step 37) = {129/138/246/345}
55a. 1,6 of {129/246} must be in R9C3 -> no 2 in R9C3
55b. R7C13 + R9C3 cannot be {246} because [246] clashes with R258C3 = {568}
55c. -> R7C13 + R9C3 = {129/138/345}, no 6
55d. 3 of {345} must be in R9C3 (cannot be in R7C1 because R79C3 = {45} clashes with R5C3) -> no 4,5 in R9C3
55e. {345} must be [453] (if [543] cannot make combination for 16(3) cage at R6C3) -> no 5 in R7C1, no 4 in R7C3
55f. R9C67 cannot be {13} -> no 6 in R8C6

56. R789C4 = {469/569/489}
56a. R9C3 = {13} -> R89C4 = {159/348} (cannot be {168} which clashes with R789C4), no 6
56b. R789C4 = {569/489} (cannot be {469} which clashes with R89C4) -> R7C4 = {69}

57. 16(3) cage at R6C3 (step 54b) = {259/268} (cannot be {358} because R7C4 only contains 6,9) -> R6C3 = 2, R7C34 = [59/86], no 9 in R7C3

58. 3 in N4 locked in R6C12, locked for 13(3) cage -> no 3 in R7C1

59. R7C13 + R9C3 = [453] (only remaining permutation), 5 locked for D/, R7C4 = 9 (step 57), R3C3 = 1, R5C3 = 4, R7C7 = 3, locked for D\, R5C78 = [23]

60. R7C79 + R9C7 (step 2) = [314] (only remaining permutation), R9C9 = 5, locked for D\

61. R2C2 = 4 (hidden single on D\), R2C8 = 2, locked for D/, R9C8 = 6, R8C9 = 2

62. Naked pair {78} in R78C8, locked for C8 and N9 -> R6C8 = 5, R8C7 = 9, R8C2 = 8, locked for D/, R8C8 = 7, locked for D\

and the rest is naked singles and cage sums, possibly only one cage sum if the right one is selected.
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