Str8ts: Basic and Advanced Strategies
- 1. Str8ts Strategies V1.1 (2011-07-09) by SlowThinker
- 2. Introduction This text describes basic and advanced strategies for solving Str8ts puzzles. I hope you find it helpful. With the strategies discussed here, you’re able to solve most Weekly Extreme puzzles of www.str8ts.com. I’d like to hear from you: if you have comments, corrections or criticism either post to str8ts.com or to forum.str8ts.de. This text is licensed under Creative Commons Attribution- NonCommercial-ShareAlike 3.0 Unported License. Share & enjoy SlowThinker, 2011-07-09
- 3. Contents • Sure candidates • Singles • Compartment range check • Stranded digits • Split compartment • Mind the gap • Naked pair • Naked triple/quadruple/quintuple • Hidden pair • Hidden triple/quadruple/quintuple • Locked compartments
- 4. Contents • X-Wing • Swordfish • Jellyfish • Starfish • Setti’s rule • Unique solution constraint • Y-Wing
- 5. Sure Candidates The most important distinction in Str8ts is whether a candidate is a sure candidate or not. A sure candidate must be set in the compartment. Have a look at A123 on the right: it can be either 456 or 567. Both ranges contain 5 and 6 which is why 5 & 6 are sure candidates with regard to A123. Same is true for B123 (again 5 & 6). AB1 doesn’t have any sure candidates, as it could be either 45, 56 or 67, and the ranges do not share a common candidate. AB2 has 6 as the sure candidate, as it can either be 56 or 67.
- 6. Sure Candidates Here’s another way to look at it. Imagine the lowest and highest possible range inside a compartment. The intersection are those candidates that have to be set in any case these are sure candidates. 1 2 3 4 5 6 7 8 9 Numbers in A1234 Lowest possible range Highest possible range Intersection == Sure candidates Applying the same principle to B1234 we find 3 & 4 are sure candidates.
- 7. Implications Compartments with 5 or more fields have sure candidates, regardless of their range, even if the possible range is 1-9: Compartment with 5 fields 1 2 3 4 5 6 7 8 9 Lowest possible range Highest possible range Intersection == Sure candidates Mandatory sure candidates … 1 2 3 4 5 6 7 8 9 … of compartments with 6 fields … of compartments with 7 fields … of compartments with 8 fields
- 8. Application In this example, 5 is a sure candidate in A1234 and 3 and 4 are sure candidates in B1234. Thus we can remove 5 in A678 and 3 and 4 in B678, as those numbers must appear in A1234 and B1234. In this last example we can eliminate 34567 from A9, because they are sure candidates of A1..7.
- 9. Singles If a sure candidate appears in one field only, then you can set this field to the sure candidate. Single sure candidates are called singles and appear quite often in Str8ts. If a compartment is large, they are sometimes hard to spot. In the example given, 6 is a sure candidate which appears in one field only. Thus A3 can be set to 6. Although 8 appears only in A1, it is not a sure candidate and nothing can be said about it yet. 1 2 3 4 5 6 7 8 9 Numbers in A1234 Lowest possible range Highest possible range Intersection == Sure candidates
- 10. Compartment Range Checks As compartments must contain a continuous sequence of numbers, candidates which are out of reach even of a single field can be removed. In the example, one can remove 12 from A123, as they are out of reach from A2. In B1234 89 can be removed, because of B3. 1 2 3 4 5 6 7 8 9 Numbers in A2 Lowest possible range in A123 X X Numbers in B3 Highest possible range in B1234 X X
- 11. Compartment Range Checks Applying the same principle as before to this example, we can remove 1 from A3, because it is out of reach of A1 and A2. However, A1 and A2 share the same lowest candidate. Thus if one is 5 the other has to be at least 6. Therefore, we can limit the range as if 6 was the lowest number and thus remove 2 from A3 and A4 as well. The same principle can be applied to common highest candidates. 1 2 3 4 5 6 7 8 9 Numbers in A1234 Numbers in A1, A2 Lowest possible range (not quite) X Real lowest possible range X X
- 12. Stranded Digits Stranded digits are candidates that cannot be part of the solution, because their possible range is smaller than the size of the compartment. In the example, 12 and 9 are stranded digits and can be removed from A123, because they are not part of a continuous sequence of at least three numbers (size of the compartment.) 1 2 3 4 5 6 7 8 9 Numbers in A123 Impossible ranges X X Lowest possible range Highest possible range
- 13. Stranded Digits Stranded digits come in other forms as well. In this example 1 in A2 is a stranded digit, as it contains the only bridging digit 2 for a complete sequence of digits starting at 1. If A2=1 there would be no 2 left for the sequence. Thus 1 can be removed from A2. Same goes for 8 in A3, as it contains the bridging digit 7. This technique is especially useful for compartments of size 2. In the example on the right, A1 can be reduced to 1379 and A2 can be reduced to 2468, because the other candidates do not have corresponding candidates in the opposite field.
- 14. Split Compartments Split compartments are a powerful technique for eliminating candidates. In the example, we have two possible ranges in A1234: one is 1234, the other is 6789. Those ranges do not overlap, which is why this is called a split compartment. 1 2 3 4 5 6 7 8 9 Numbers in A1234 Impossible range (example) X Lowest possible range Highest possible range In such situations, you can analyse each range independently, as they do not influence each other
- 15. Split Compartments In the low range, we can remove 23 from A24, because of the naked pair in A13. A1234: Lower range We can also remove 678 from A3, as it contains the only 9 in the upper range (9 is a sure candidate in the upper range.) A1234: Upper range Thus A2=14678, A3=239, and A4=1467. No candidates are eliminated from A1. A1234: Combined result
- 16. Mind the Gap If a field has a large gap, defined as large distance between two candidates, as is the case here with A3=27, you can remove those candidates from all other fields. A gap is large, if the distance is equal or greater than the compartment size. In that case, both numbers cannot be part of a single range (as shown below) and therefore those numbers cannot appear in other fields, e.g. if A1=2 A3=7 or if A4=7 A3=2. Both cases are impossible. Therefore 27 can be removed from A124. 1 2 3 4 5 6 7 8 9 Numbers in A1234 Highest range containing 2 X Lowest range containing 7 X
- 17. Mind the Gap If the field with the large gap, has more than one candidate on a side, you can remove only the single candidate. In the example to the right, you can only remove 7 from A24. If you have a large gap with more than one candidate on both sides, no candidates can be removed.
- 18. Mind the Gap Large gaps can also span two fields. In this example, we find A1=35 and A3=58. The candidates 3 and 8 form a large gap (equal to the size of the compartment). Both share the same additional candidate 5. If one of A245 would be 5, A1 would be 3 and A3=8: an impossible range. Therefore, 5 must either be in A1 or A3 and can safely be removed from A245. Note that in this case we remove the bridging digit (5) and not one of the “gap digits” (3, 8).
- 19. Naked Pair A naked pair is a pair of candidates that appear in two fields and those fields do not contain any other candidates. In the example 45 is a naked pair, appearing in A2 and A4. The candidates of the naked pair can be removed from all other fields, because if A2=4 then A4=5 and vice versa. Thus, 4 and 5 are sure candidates of A1234 as well and we can remove 89 from A1/A3 (compartment range check). 1 2 3 4 5 6 7 8 9 Numbers in A1234 Sure candidates of naked pair Highest possible range X X
- 20. Naked Triple/Quadruple/Quintuple Whenever there are the same N candidates in N fields, we have a locked set of candidates. Those candidates get removed from all other fields and are sure candidates of the compartment. The naked pair was N=2. N=3: naked triple (467 in A246, as you can see not every candidate has to appear in every field.) N=4: naked quadruple (3467 in A2456) N=5: naked quintuple (34567 in A12456)
- 21. Cross-compartment Locked Sets A locked set of candidates may also occur across compartments, i.e. within the same row or column there are the same N candidates in N fields (again, not all candidates have to appear in all fields.) Here’s an example of a naked triple (345) across 3 compartments in row A. Because of this, 345 can be removed from the other cells (marked yellow) in row A.
- 22. Hidden Pair If two sure candidates appear in the same two fields but nowhere else, we call it a hidden pair. In that case, all other candidates in those two fields can be removed. In the example on the right, 4567 are sure candidates of A12345. 4 and 5 only appear in A1 and A3 and are a hidden pair. Thus we can set A1=45 and A3=45. Although, in both naked pairs and hidden pairs, two candidates appear in exactly two fields, there are differences. With naked pairs you remove the candidates of the pair in other fields, with hidden pairs you remove additional candidates from the fields of the pair. In both cases the candidates of the pair are sure candidates of the compartment.
- 23. Hidden Triple/Quadruple/Quintuple Whenever the same N sure candidates appear in exactly N fields, we have a hidden set of candidates. Note: not all N candidates have to appear in every of the N fields. Other candidates in those fields are removed. (N=1: singles, N=2: hidden pair) N=3: hidden triple (467 in A136 A1=46, A3=67, A6=467) N=4: hidden quadruple (3467 in A1367) N=5: hidden quintuple (23568 in A23578)
- 24. Cross-compartment Hidden Sets As with locked sets, hidden sets may cross compartment boundaries: whenever the same N sure candidates appear in exactly N fields, in a row or column, we have a hidden set of candidates. Here, candidates are sure with regard to the row or column in which they appear. In this example, we have the hidden quadruple 3478 in row A, which spans two compartments. 3478 are sure candidates with regard to row A we can remove all other candidates in those fields. This technique, is rarely used, as other rules (e.g. stranded digits) usually eliminate the same candidates.
- 25. Locked Compartments Locked compartments are compartments whose possible ranges are limited by other compartments. In this example, we have two compartments (A12 and A45) sharing the same range of 3..6. The compartments are interlocked as shown in this table: 1 2 3 4 5 6 7 8 9 Available range Possible arrangement 1 A12 A12 A45 A45 Possible arrangement 2 A45 A45 A12 A12 We can therefore apply the split compartment rules and remove 4 from A1 and 5 from A4:
- 26. Locked Compartments Even if there is some wiggle room, we can apply this strategy. In this example, we have 6 candidates in two compartments spanning 5 fields. The table shows all possible ranges for A12: as you can see, because of the interlock with A456, A12 is actually a split compartment! We can therefore remove 5 from A2 and 6 from A1. 1 2 3 4 5 6 7 8 9 Available range First possible range of A12 Second possible range of A12 Third possible range of A12 Fourth possible range of A12
- 27. Locked Compartments Here’s another example: with A1=67, neither of which is a sure candidate itself, we know that A3456 cannot include both 6 and 7 in its range, because then A1 would have no candidates left. Also, above 6 (i.e. 7..9) there is not enough room for A3456 (four fields.) Hence the range of A3456 gets limited to 2..6, because of the interlock with A1. Thus 3..5 become sure candidates of A3456. 1 2 3 4 5 6 7 8 9 Numbers in A3456 Numbers in A1 Example of impossible range Possible range left for A3456
- 28. X-Wing The X-Wing strategy can be applied, when a sure candidate appears in only two cells in two different columns, and these cells are in the same two rows. Note that the candidate has to be a sure candidate in both columns. The example on the right shows such a constellation: 3 is a sure candidate in column 2 and it is also a sure candidate in column 3 (marked blue.) In both columns 3 only appears in the same rows: row C and row D. There are no other 3s in the yellow columns.
- 29. X-Wing In an X-Wing constellation like this, we can remove all 3s from the red cells, i.e. from the rows where the X-Wing occurs. The reason is simple: if C2=3, then C3 can’t be 3 and as 3 is a sure candidate in ABCD3 D3 must be 3. It’s the same the other way around: if C3=3, then D2 must be 3. In either case, there’s a 3 in row C and D, hence we can remove 3 from these rows (red cells.)
- 30. X-Wing An X-Wing can also be built using rows: 5 is a sure candidate in rows A and B and only occurs in the same two columns (blue.) Thus we can remove the 5s in columns 4 and 5 (marked red.) HJ12 is not an X-Wing on 5. Checking the columns we find that 5 is not a sure candidate in column 1. Checking the rows we find that 5 is not a sure candidate in row J and 5 occurs in more than 2 cells in row H.
- 31. X-Wing Implications One of the important implications of an X-Wing is that the X-Wing candidate becomes a sure candidate in the columns and rows where the X-Wing occurs. On the right we have a row-based X-Wing on 5: in rows A and D 5 is a sure candidate that occurs in only two columns (1 and 2.) Thus the 5s in columns 1 and 2 are removed (marked red.) But, as 5 has to appear either in A1 or D1, 5 becomes a sure candidate in ABCD1! We therefore can remove 9 from B1 and D1 (range check.)
- 32. Swordfish A Swordfish is just like an X-Wing, but now in three columns and three rows. Again, when a sure candidate in three rows appears in exactly the same three columns, we can remove the candidate everywhere else in these three columns (or vice versa, i.e. swapping rows and columns.) The example on the right shows a Swordfish on 3: in columns 2, 3, and 5 we find that 3 is a sure candidate and the 3s only occur in three rows, namely row A, B, and C. As you can see, it is not necessary that the 3s appear in every of the three rows in the three columns.
- 33. Jellyfish A Jellyfish is just a 4-pronged X-Wing: a sure candidate in four rows appears in exactly the same four columns. Thus we can remove the candidate everywhere else in these four columns (or vice versa, i.e. swapping rows and columns.) The example shows a column- based Jellyfish on 4: in columns 1, 3, 4, and 6 we have 4 as sure candidate. 4 appears in the same rows (ABCD) and thus we can remove all other 4s in rows ABCD (marked red.)
- 34. Jellyfish The reason why a Jellyfish works is the same as why an X-Wing or Swordfish works: a number can appear in a column or row only once. Thus if we have four columns and four rows and the candidate must appear in every column (or row), we have to use all four rows (or columns) to place those four candidates. Here are three possible combinations from the previous example. Every time a 4 appears in rows A, B, C, and D.
- 35. Starfish Well, what works for 2 columns/rows (X-Wing), 3 columns/rows (Swordfish), or 4 columns/rows (Jellyfish) also works for five columns and rows: the Starfish. The example shows a row-based Starfish on 5: in rows ABCDE 5 is a sure candidate and occurs only in columns 12345. Thus we can remove 5 in every other cell in columns 12345 (marked red.)
- 36. Sea Creatures To recap: N=2: X-Wing, N=3: Swordfish, N=4: Jellyfish, and N=5: Starfish. N=6, N=7 or N=8 are possible as well, but hardly occur in Str8ts puzzles and thus have no special name. All these formations have in common that the candidate they are based on occurs in the same number of rows and columns and occurs exclusively in either those rows or those columns where the number must also be a sure candidate. In that case, the number can be removed from other cells in the same columns (row-based formation) or rows (column-based formation.) In addition, the candidate becomes a sure candidate in those columns and rows. Thus further reductions (compartment range) may be possible.
- 37. Cross-compartment Sea Creatures So far, we only looked at sea creatures within single compartments. Cross-compartment sea creatures are possible as well, if the candidate is a sure candidate with regard to that row (or column). I.e. the candidate is not necessarily a sure candidate of a single compartment. In this example, 2 is neither a sure candidate of AB1 nor of DE1. But 2 has to appear in either AB1 or DE1, because we only have four candidates (1234) for four cells (ABDE1.) Thus 2 is a sure candidate with regard to column 1 and we can use this to build an X-Wing (marked blue) with the 2s in column 2. Again the other 2s in rows B and D (marked red) are removed.
- 38. Setti’s Rule Setti’s rule, named after user Setti on str8ts.com, is a very powerful technique, based on a simple observation: in the final solution to a puzzle, a number occurs in exactly the same number of columns and rows. In the example puzzle below, 8 occurs in six rows and six columns, whereas 4 appears in eight rows and eight columns.
- 39. Setti’s Rule Due to the rules of Str8ts it is impossible that the number of rows and columns is different. If you place e.g. a single 7 on the grid, it occupies one row and one column. If you add another 7, they occupy two columns and two rows. Add yet another 7 and they use three columns and three rows. The same is true up to nine 7s in nine rows and nine columns, as a number may not appear twice in a row or column.
- 40. Application With Setti’s rule we can deduce certain properties about rows and columns. This example shows Setti’s rule applied to 4: 4 is a sure candidate in seven rows, and doesn’t occur in the other two rows 4 must occur in exactly seven columns as well. As there are only two columns (2 & 8) where 4 may be left out, these are the two columns where 4 must be removed 4 is removed from the fields marked red.
- 41. Application Here’s another puzzle, where we apply Setti’s rule on 4: the number 4 must occur in eight rows and is missing from row C. 4 does not appear in column 7 either. Thus we know that 4 must appear in all other columns, and therefore in column 4, where 4 was not yet a sure candidate. 4 becomes a sure candidate in ABCDE4 and we can remove 9 from that compartment (range check.)
- 42. Unique Solution Constraint Although not required by the Str8ts rules themselves, properly designed Str8ts puzzles have only one possible solution. This knowledge can be applied to remove certain possibilities that would result in two or more possible solutions the unique solution constraint is born. Have a look at the position on the right. While this is a perfectly valid position according to the rules, such a position always has two possible solutions: either A2/B3=4 and A3/B2=5 or A2/B3=5 and A3/B2=4. No other field can influence which solution is correct. If we know that we have a properly designed puzzle, then positions such as this cannot arise and must be avoided.
- 43. Application Therefore if we have a position like the one on the left, we know that B2 must be 6, because without 6, B2=45 and there would be two possible solutions. The unique solution constraint is also called unique rectangle rule (UR,) because most of the time the shape at hand is a rectangle (like above.) Here’s another example: CD89 (marked green) looks almost like a UR. To avoid two solutions, we therefore know that CD8 must contain a 5, because otherwise CD89=67 which produces two solutions. As we know CD8 has to include 5, we can remove 5 from F8 (marked red.)
- 44. Application Here’s an example with a split compartment which allows us to analyse the ranges separately J7 cannot be 78, as this would violate the unique solution constraint. Thus J7=2349 and H7=348. The example below shows an advanced application: If you compare A12345 with B12345, you’ll find that they only differ in that A1..5 has 4 as additional candidate. Thus if A1..5 would not contain 4, we could freely interchange A1..5 with B1..5, no other field can intervene. A1..5 must contain 4.
- 45. Application Be very careful when applying the unique solution constraint: the example on the right almost looks like the one before, but A3 and B3 differ. Here you cannot conclude that A1..5 must contain a 4. As long as numbers in other fields can intervene and influence the outcome, you cannot apply this strategy. Conversely, if certain fields are the last possible chance to intervene, we can apply the constraint: here A3..9 must contain a 9, because otherwise A1 C..J1 doesn‘t contain 1 nor 9 can be either 1 or 9. (not shown)
- 46. Y-Wing An Y-Wing is a two-pronged elimination technique, where three corners of a rectangle eliminate a candidate in the fourth corner. Have a look: C3 (blue) contains two candidates: 6 and 7. If C3=6, then C7 (green) must be 5 and therefore 5 is removed from A7 (red arrows.) On the other hand, if C3=7, then A3 (green) must be 5, and again A7 cannot be 5 (blue arrows.) Thus in either case, we can remove 5 from A7. Y-Wings do not often occur in Str8ts, but they provide insight into more complex chaining strategies.
- 47. Y-Wing In general, an Y-Wing has a base (marked blue) with XZ -Z two candidates, let’s call them X and Y, and two prongs which contain an XY YZ additional number Z, which gets removed in the target field (marked red.) The target field is at the intersection of the row and column (yellow) which contain the intermediate fields (marked green). The intermediate fields must be of the form XZ and YZ, which means that each candidate of the base cell is covered and leads to Z being removed from the target cell.
- 48. Str8ts Strategies by SlowThinker (with feedback from Auric, darktray, and John) This text is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.