Extreme Str8ts Puzzle #67 was very difficult, requiring the use of advanced strategies. A key step was eliminating the candidate 3 from cell H9, revealing a 9 in E1. This opened up the puzzle, allowing hidden pairs/triples and X-wings to be found. XY-chains were also useful, removing candidates from cells like H4, J3, and H5. Once these strategies were applied, the puzzle became soluble, with the solution found through standard elimination techniques.

1. Weekly Extreme Str8ts Puzzle #67 October 2 – October 8, 2011www.str8ts.com Solution bySlowThinker

2. Start position Extreme Str8ts #67 is one of the toughest Extremes in a long time. In this solution I will only focus on the key points, not on easy removal of candidates. (Links to my strategy slides are on the last page)

3. Mind the gap After the first steps, B2 (marked blue) contains a large gap pair. Thus we can remove 19 from all other fields in column 2. This leads to D2=2 and D1=3.

4. Setti on 6 Because all rows contain a 6, all columns must contain a 6 too. Hence J7=6.

5. Setti on 8 Column 7 does not contain an 8  at least one row must not contain an 8 either. The only row where an 8 can be missing is row J  remove 89 from row J.

6. Hidden unique rectangle Because of F89 neither G89 nor H89 can contain a 9. If they would, the compartment would contain the candidates 89 and form a UR with F89. Thus 9 can be removed from GH89.

7. Hidden UR continued While it is obvious that GH89 cannot be both 78, the question arises, if the can be both 123. In that case one of GH9 must be 3, which in turn (due to C9) forces the other field to be 1  both GH8 would be 2, which is impossible.

8. Hidden UR continued We now know that GH89 do not have the same range: one is 78, one is 12(3). Together with F89 this forms a (hidden) X-Wing on 8 which removes the 8s from the yellow cells. Also, in one of the pink fields 89 gets removed thus forcing the whole compartment into the range 1234 (due to interlock with ABCD5).

9. Swordfish on 4 We now have a swordfish on at FHJ235 (marked blue) that removes the 4s from the red fields. Next we have a naked pair in EJ6 that forces C6=4.

10. Setti on 8 again Due to Setti’s rule an 8 is mandatory in column 3. So the 8 is either in A3 or in EFGHJ3. If it is in EFGHJ3, then A3=23. In the other case A3=8. In both cases, 67 does not appear in A3 and thus can be removed.

11. Setti on 7 As GH89 does not contain a 9 anymore, 7 becomes a sure candidate in rows G and H. Due to Setti’s rule, column 3 must contain a 7 too  we can remove 12 from EFGHJ3 (range check).

12. Setti on 2 This is one of the rare occasions where we know that a candidate can only appear at the intersection of an undecided row and column. As there are no 2s in EFGHJ3, either A3=2 or not, but there cannot be a 2 anywhere else in row A as this would violate Setti’s rule  remove 2s from the yellow cells.

13. Hidden X-Wing on 7 Remember that one of GH89 is 7 which removes 7s in its row and column.  No matter which field you take, we get two X-Wings on 7: one in GHJ34 (pink) and one at AD589 (orange). Both X-Wings remove the 7s in AD4 (yellow).

14. A simple chain:G4 ≠ 3 Let’s start with an easy removal: G4 cannot be 3: If G89=12  G5=3 If G89=78  G6=6, G2=5, G3=3 In both cases, 3 gets removed from G4.

15. Setti on 3 Let’s look at Setti on 3 again: either there’s a 3 in row H and column 3 or there is not. Let’s assume no 3: then H89=12  G89=78  G6=6, G2=5  no candidate left in G3! Thus there’s a 3 in row H & column 3  we can remove 2 from A3.

16. H9=3?No, H9 ≠ 3 Now that we know that row H has a 3, let’s test H9=3: a) H9=3  G89=78  G6=6, G2=5, G3=3 J5=3s  J6=2 b) H9=3  H5=4, H2=6  H3=7 (because G3=3)  J2=4 (because G2=5)  J3=5  J1=2 Contradiction! Hence H9!=3  E1=9s!

17. How it would look like Here is how the position would look like, if H9=3: a) H9=3  G89=78  G6=6, G2=5, G3=3 J5=3s  J6=2 b) H9=3  H5=4, H2=6  H3=7 (because G3=3)  J2=4 (because G2=5)  J3=5  J1=2 78 3 6 5 3 4 6 7 2 5 4 2 3

18. Swordfish on 5 After H9!=3 we know that 3 is in H1..6 and thus there’s no 9 there  E1=9s. And with Setti on 5 we get E6789=2345. Furthermore, there’s a swordfish on 5 (marked blue) that removes the 5s from FGJ4 (yellow fields).

19. Hidden triple That leaves us with the hidden triple 459 in ABD4.

20. Jellyfish on 6 With the 6s gone in ABD4 we now have a jellyfish on 6 (marked blue) that removes the 6 from G6  G6=8 With this we can solve quite a few cells.

21. X-Wing on 9 When we arrive at this position, we get an X-Wing on 9 at AD46 that removes the 9s from the yellow fields.

22. XY-Chain: H4 ≠ 6, F1 ≠ 2 Starting at F5 (blue) we can form an XY-chain that removes 6 from H4 (red): F5=2  F4=6 (green) F5=4  F3=5, G3=7, G4=6 (yellow) In both cases 6 is removed from H4. Using the same logic and again starting at F5, we can also remove 2 from F1.

23. More XY-Chains: J3 ≠ 4, H5 ≠ 4 Starting at G4 we can remove 4 from J3 and 4 from H5. In both cases we just follow the elimination for both possibilities of G4 (6, 7). For example, the removal of 4 from H5:Green arrows for G4=6 (lead to F5=4) and orange arrows for G4=7 (lead to H2=4)

24. More XY-Chains: H4 ≠ 3 Again starting at G4 we can remove 3 H4: Orange: G4=6  F4=2, C4=3 Green: G4=7  G3=5, F3=4, H3=3 With H4=12 we get H3=3s and H1=6p. From there on, the puzzle solves itself.

25. Solution The hardest step in this solution was eliminating 3 from H9. Once the 3 was gone, well known strategies and some XY-chains solved the puzzle. With a puzzle like this, there are of course many other ways to solve it.

26. Glossary Letters appended to steps indicate the last strategy used, just before filling in a field: No letter … number was last candidate in field s … single (last) candidate for that number in compartment c … compartment range check d … stranded (unreachable/impossible) digits removed h … high/low range check across compartments p/t/q … naked pair / naked triple / naked quadruple ph/th/qh … hidden pair / hidden triple / hidden quadruple x … X-wing (2 rows / 2 columns) w … Swordfish (3 rows / 3 columns) j … Jellyfish (4 rows / 4 columns) L … large gap field Sx … Setti’s rule (count the numbers rule) – ‘x’ is the analysed number u … unique rectangle y … Y-Wing or XY-chains

27. Weekly Extreme Str8ts Puzzle #67www.str8ts.com Solution by SlowThinker Note: there are other (maybe easier) ways to solve this puzzle. View & download my strategy slides from: http://slideshare.net/SlowThinker/str8ts-basic-and-advanced-strategies or from Google Docs: http://is.gd/slowthinker_str8ts_strategy