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Str8ts: Basic and Advanced Strategies


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Description of basic and advanced strategies for solving Str8ts puzzles.

  • Jeff, you're absolutely right. Didn't occur to me, but it's obvious :) I'll correct this in the next version of the slides. Thanks for pointing this out.
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  • Thanks for this!
    On slide 18, 'Mind the Gap' spanning two fields, if any of A245 were a 3, then A1=5 and A3=8, and we'd have an impossible range. Similarly if any of A245 were an 8. So can't we remove 3, 5 & 8 from A245? So it's not just the bridging digit, but the gap digits as well?
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  • @Ganger Sterc: yes, it does count as naked triple. Same for quadruple, quintuple, etc. Whenever the number of distinct candidates equals the number of fields they are in, you have a naked tuple. It is not relevant if the candidates appear in all fields or (as in your example) just in 2 out of 3.
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  • Does an array of, say 46, 47, 67 in three different fiends in a compartment (N>=4) count as a naked triple? Same question apply to naked quadruple and quintuple.
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  • does an array of 46, 67, 47, in 3 fields in a compartment (N>3) count as a naked triple?
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Str8ts: Basic and Advanced Strategies

  1. 1. Str8ts Strategies V1.1 (2011-07-09) by SlowThinker
  2. 2. IntroductionThis text describes basic and advanced strategies for solvingStr8ts puzzles. I hope you find it helpful.With the strategies discussed here, you’re able to solve mostWeekly Extreme puzzles of’d like to hear from you: if you have comments, corrections orcriticism either post to or to text is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.Share & enjoySlowThinker, 2011-07-09
  3. 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. 4. Contents• X-Wing• Swordfish• Jellyfish• Starfish• Setti’s rule• Unique solution constraint• Y-Wing
  5. 5. Sure CandidatesThe most important distinction in Str8ts is whether a candidateis a sure candidate or not. A sure candidate must be set in thecompartment.Have a look at A123 on the right: it can beeither 456 or 567. Both ranges contain 5and 6 which is why 5 & 6 are surecandidates 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. 6. Sure CandidatesHere’s another way to look at it.Imagine the lowest and highest possiblerange inside a compartment. Theintersection are those candidates thathave to be set in any case  these are sure candidates. 1 2 3 4 5 6 7 8 9Numbers in A1234Lowest possible rangeHighest possible rangeIntersection == Sure candidatesApplying the same principle to B1234 we find 3 & 4 are surecandidates.
  7. 7. ImplicationsCompartments 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 9Lowest possible rangeHighest possible rangeIntersection == Sure candidatesMandatory 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. 8. ApplicationIn this example, 5 is asure candidate inA1234 and 3 and 4 aresure candidates inB1234.Thus we can remove5 in A678 and 3 and 4in B678, as thosenumbers must appearin A1234 and B1234.In this last example we can eliminate 34567 from A9,because they are surecandidates of A1..7.
  9. 9. SinglesIf a sure candidate appears in one fieldonly, then you can set this field to thesure candidate. Single sure candidatesare called singles and appear quite often in Str8ts. If acompartment is large, they are sometimes hard to spot.In the example given, 6 is a sure candidate which appears in onefield 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 9Numbers in A1234Lowest possible rangeHighest possible rangeIntersection == Sure candidates
  10. 10. Compartment Range ChecksAs compartments must contain acontinuous sequence of numbers,candidates which are out of reach evenof a single field can be removed.In the example, one can remove 12 from A123, as they are out ofreach from A2. In B1234 89 can be removed, because of B3. 1 2 3 4 5 6 7 8 9Numbers in A2Lowest possible range in A123 X XNumbers in B3Highest possible range in B1234 X X
  11. 11. Compartment Range ChecksApplying the same principle as before tothis 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 ifone is 5 the other has to be at least 6. Therefore, we can limitthe range as if 6 was the lowest number and thus remove 2 fromA3 and A4 as well. The same principle can be applied to commonhighest candidates. 1 2 3 4 5 6 7 8 9Numbers in A1234Numbers in A1, A2Lowest possible range (not quite) XReal lowest possible range X X 
  12. 12. Stranded DigitsStranded digits are candidates thatcannot be part of the solution, becausetheir possible range is smaller than thesize of the compartment.In the example, 12 and 9 are stranded digits and can be removedfrom A123, because they are not part of a continuous sequenceof at least three numbers (size of the compartment.) 1 2 3 4 5 6 7 8 9Numbers in A123Impossible ranges X XLowest possible rangeHighest possible range
  13. 13. Stranded DigitsStranded digits come in other forms aswell. In this example 1 in A2 is a strandeddigit, as it contains the only bridgingdigit 2 for a complete sequence of digits starting at 1. If A2=1there would be no 2 left for the sequence. Thus 1 can beremoved from A2.Same goes for 8 in A3, as it contains the bridging digit 7.This technique is especially useful forcompartments of size 2. In the example on theright, A1 can be reduced to 1379 and A2 canbe reduced to 2468, because the other candidates do not havecorresponding candidates in the opposite field.
  14. 14. Split CompartmentsSplit compartments are a powerfultechnique for eliminating candidates.In the example, we have two possibleranges in A1234: one is 1234, the other is 6789. Those ranges donot overlap, which is why this is called a split compartment. 1 2 3 4 5 6 7 8 9Numbers in A1234Impossible range (example) XLowest possible rangeHighest possible rangeIn such situations, you can analyse each range independently, asthey do not influence each other 
  15. 15. Split Compartments In the low range, we can remove 23from A24, because of the naked pairin A13. A1234: Lower rangeWe can also remove 678 from A3, as itcontains the only 9 in the upper range(9 is a sure candidate in the upper range.) A1234: Upper rangeThus A2=14678, A3=239, and A4=1467.No candidates are eliminated from A1. A1234: Combined result
  16. 16. Mind the GapIf a field has a large gap, defined as largedistance between two candidates, as isthe case here with A3=27, you canremove those candidates from all other fields. A gap is large, ifthe distance is equal or greater than the compartment size.In that case, both numbers cannot be part of a single range (asshown below) and therefore those numbers cannot appear inother fields, e.g. if A1=2  A3=7 or if A4=7  A3=2. Both casesare impossible. Therefore 27 can be removed from A124. 1 2 3 4 5 6 7 8 9Numbers in A1234Highest range containing 2  XLowest range containing 7 X 
  17. 17. Mind the GapIf the field with the large gap, has morethan one candidate on a side, you canremove only the single candidate. In theexample to the right, you can only remove 7 from A24.If you have a large gap with more thanone candidate on both sides, nocandidates can be removed.
  18. 18. Mind the GapLarge gaps can also span two fields.In this example, we find A1=35and A3=58. The candidates 3 and 8form a large gap (equal to the size of the compartment). Bothshare 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 andcan safely be removed from A245.Note that in this case we remove the bridging digit (5) and notone of the “gap digits” (3, 8).
  19. 19. Naked PairA naked pair is a pair of candidates thatappear in two fields and those fieldsdo not contain any other candidates.In the example 45 is a naked pair, appearing in A2 and A4.The candidates of the naked pair canbe removed from all other fields,because if A2=4 then A4=5 andvice versa. Thus, 4 and 5 are sure candidates of A1234 as welland we can remove 89 from A1/A3 (compartment range check). 1 2 3 4 5 6 7 8 9Numbers in A1234Sure candidates of naked pairHighest possible range X X
  20. 20. Naked Triple/Quadruple/QuintupleWhenever there are the same N candidates in N fields, we havea locked set of candidates. Those candidates get removed fromall other fields and are sure candidates of the compartment. Thenaked pair was N=2.N=3: naked triple(467 in A246, as you can seenot every candidate has toappear in every field.)N=4: naked quadruple(3467 in A2456)N=5: naked quintuple(34567 in A12456)
  21. 21. Cross-compartment Locked SetsA locked set of candidates may also occur across compartments,i.e. within the same row or column there are the same Ncandidates in N fields (again, not all candidates have to appear inall fields.)Here’s an example ofa naked triple (345)across 3 compartmentsin row A.Because of this, 345can be removed fromthe other cells (markedyellow) in row A.
  22. 22. Hidden PairIf two sure candidates appear in the same two fields butnowhere else, we call it a hidden pair. In that case, allother candidates in those two fields can be removed.In the example on the right, 4567are sure candidates of A12345.4 and 5 only appear in A1 and A3and are a hidden pair. Thus we can set A1=45 and A3=45.Although, in both naked pairs and hidden pairs, two candidatesappear in exactly two fields, there are differences. With nakedpairs you remove the candidates of the pair in other fields, withhidden pairs you remove additional candidates from the fields ofthe pair. In both cases the candidates of the pair are surecandidates of the compartment.
  23. 23. Hidden Triple/Quadruple/QuintupleWhenever the same N sure candidates appear in exactly N fields,we have a hidden set of candidates. Note: not all N candidateshave to appear in every of the N fields. Other candidates in thosefields 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. 24. Cross-compartment Hidden SetsAs with locked sets, hidden sets may cross compartmentboundaries: whenever the same N sure candidates appear inexactly N fields, in a row or column, we have a hidden set ofcandidates. Here, candidates are sure with regard to the row orcolumn in which they appear.In this example, wehave the hiddenquadruple 3478 in rowA, which spans two compartments. 3478 are sure candidates withregard to row A  we can remove all other candidates in thosefields.This technique, israrely used, as otherrules (e.g. stranded digits) usually eliminate the same candidates.
  25. 25. Locked CompartmentsLocked compartments are compartments whose possible rangesare limited by other compartments.In this example, we have twocompartments (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 9Available rangePossible arrangement 1 A12 A12 A45 A45Possible arrangement 2 A45 A45 A12 A12We can therefore apply the splitcompartment rules and remove4 from A1 and 5 from A4:
  26. 26. Locked CompartmentsEven if there is some wiggle room,we can apply this strategy. In thisexample, we have 6 candidatesin two compartments spanning 5 fields. The table shows allpossible ranges for A12: as you can see, because of the interlockwith 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 9Available rangeFirst possible range of A12Second possible range of A12Third possible range of A12Fourth possible range of A12
  27. 27. Locked CompartmentsHere’s another example: withA1=67, neither of which is a surecandidate itself, we know thatA3456 cannot include both 6 and 7 in its range, because then A1would have no candidates left. Also, above 6 (i.e. 7..9) there isnot enough room for A3456 (four fields.) Hence the range ofA3456 gets limited to 2..6, because of the interlock with A1. Thus3..5 become sure candidates of A3456. 1 2 3 4 5 6 7 8 9Numbers in A3456Numbers in A1Example of impossible rangePossible range left for A3456
  28. 28. X-WingThe X-Wing strategy can be applied, whena sure candidate appears in only two cellsin two different columns, and these cellsare in the same two rows. Note that thecandidate has to be a sure candidate inboth columns.The example on the right shows such aconstellation: 3 is a sure candidate incolumn 2 and it is also a sure candidatein column 3 (marked blue.) In both columns3 only appears in the same rows:row C and row D.There are no other 3s in the yellow columns.
  29. 29. X-WingIn an X-Wing constellation like this, we canremove all 3s from the red cells, i.e. fromthe rows where the X-Wing occurs.The reason is simple: if C2=3, then C3 can’tbe 3 and as 3 is a sure candidate in ABCD3D3 must be 3. It’s the same the other wayaround: if C3=3, then D2 must be 3. In eithercase, there’s a 3 in row C and D, hence wecan remove 3 from these rows (red cells.)
  30. 30. X-WingAn X-Wing can also be builtusing rows: 5 is a sure candidatein rows A and B and only occursin the same two columns (blue.)Thus we can remove the 5s incolumns 4 and 5 (marked red.)HJ12 is not an X-Wing on 5.Checking the columns we findthat 5 is not a sure candidate incolumn 1. Checking the rows wefind that 5 is not a sure candidatein row J and 5 occurs in morethan 2 cells in row H.
  31. 31. X-Wing ImplicationsOne of the important implications of anX-Wing is that the X-Wing candidatebecomes a sure candidate in the columnsand rows where the X-Wing occurs.On the right we have a row-based X-Wingon 5: in rows A and D 5 is a sure candidatethat occurs in only two columns (1 and 2.)Thus the 5s in columns 1 and 2 areremoved (marked red.)But, as 5 has to appear either in A1 or D1,5 becomes a sure candidate in ABCD1! Wetherefore can remove 9 from B1 and D1(range check.)
  32. 32. SwordfishA Swordfish is just like an X-Wing, but now in three columns andthree rows. Again, when a sure candidate in three rows appearsin exactly the same three columns, we can remove the candidateeverywhere else in these three columns (or vice versa, i.e.swapping rows and columns.)The example on the right showsa Swordfish on 3: in columns 2, 3,and 5 we find that 3 is a surecandidate and the 3s only occur inthree rows, namely row A, B, and C.As you can see, it is not necessarythat the 3s appear in every of thethree rows in the three columns.
  33. 33. JellyfishA Jellyfish is just a 4-pronged X-Wing: a sure candidate in fourrows appears in exactly the same four columns. Thus we canremove the candidate everywhere else in these four columns (orvice versa, i.e. swapping rows andcolumns.)The example shows a column-based Jellyfish on 4: in columns1, 3, 4, and 6 we have 4 as surecandidate. 4 appears in the samerows (ABCD) and thus we canremove all other 4s in rows ABCD(marked red.)
  34. 34. JellyfishThe reason why a Jellyfish works is the same as why an X-Wingor Swordfish works: a number can appear in a column or rowonly once. Thus if we have four columns and four rows and thecandidate must appear in every column (or row), we have to useall four rows (or columns) to place those four candidates.Here are three possible combinations from the previousexample. Every time a 4 appears in rows A, B, C, and D.
  35. 35. StarfishWell, what works for 2 columns/rows (X-Wing), 3 columns/rows(Swordfish), or 4 columns/rows (Jellyfish) also works forfive columns and rows: the Starfish.The example shows arow-based Starfish on5: in rows ABCDE 5is a sure candidateand occurs only incolumns 12345.Thus we can remove 5in every other cell incolumns 12345(marked red.)
  36. 36. Sea CreaturesTo 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 occurin Str8ts puzzles and thus have no special name.All these formations have in common that the candidate they arebased on occurs in the same number of rows and columns andoccurs exclusively in either those rows or those columns wherethe number must also be a sure candidate.In that case, the number can be removed from other cells in thesame columns (row-based formation) or rows (column-basedformation.)In addition, the candidate becomes a sure candidate in thosecolumns and rows. Thus further reductions (compartment range)may be possible.
  37. 37. Cross-compartment Sea CreaturesSo far, we only looked at sea creatures within singlecompartments. Cross-compartment sea creatures are possible aswell, if the candidate is a sure candidate with regard to that row(or column). I.e. the candidate is not necessarilya sure candidate of a single compartment.In this example, 2 is neither a sure candidateof AB1 nor of DE1. But 2 has to appear in eitherAB1 or DE1, because we only have fourcandidates (1234) for four cells (ABDE1.)Thus 2 is a sure candidate with regard to column1 and we can use this to build an X-Wing (markedblue) with the 2s in column 2. Again the other 2sin rows B and D (marked red) are removed.
  38. 38. Setti’s RuleSetti’s rule, named after user Setti on, is a verypowerful technique, based on a simple observation: in the finalsolution to a puzzle, a number occurs in exactly the samenumber of columns and rows.In the example puzzle below, 8 occurs in six rows and sixcolumns,whereas 4appears ineight rowsand eightcolumns.
  39. 39. Setti’s RuleDue to the rules of Str8ts it is impossible that the number ofrows and columns is different.If you place e.g. a single 7 onthe grid, it occupies one rowand one column. If you addanother 7, they occupy twocolumns and two rows. Add yetanother 7 and they use threecolumns and three rows.The same is true up to nine 7sin nine rows and nine columns,as a number may not appeartwice in a row or column.
  40. 40. ApplicationWith Setti’s rule we can deduce certain properties about rowsand columns.This example shows Setti’s ruleapplied to 4: 4 is a sure candidatein seven rows, and doesn’t occurin the other two rows 4 must occur in exactly sevencolumns as well. As there areonly two columns (2 & 8) where4 may be left out, these are thetwo columns where 4 must beremoved  4 is removed fromthe fields marked red.
  41. 41. ApplicationHere’s another puzzle, where we apply Setti’s rule on 4:the number 4 must occur ineight rows and is missing fromrow C. 4 does not appear incolumn 7 either. Thus we knowthat 4 must appear in all othercolumns, and therefore incolumn 4, where 4 was not yeta sure candidate. 4 becomes a sure candidatein ABCDE4 and we can remove9 from that compartment(range check.)
  42. 42. Unique Solution ConstraintAlthough not required by the Str8ts rules themselves, properlydesigned Str8ts puzzles have only one possible solution. Thisknowledge can be applied to remove certain possibilities thatwould result in two or more possible solutions  the uniquesolution constraint is born.Have a look at the position on the right.While this is a perfectly valid positionaccording to the rules, such a position alwayshas two possible solutions: either A2/B3=4and A3/B2=5 or A2/B3=5 and A3/B2=4. Noother field can influence which solution is correct.If we know that we have a properly designed puzzle, thenpositions such as this cannot arise and must be avoided.
  43. 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 calledunique rectangle rule (UR,) because most of thetime 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, wetherefore know that CD8 must contain a 5, becauseotherwise CD89=67 which produces two solutions.As we know CD8 has to include 5, we can remove5 from F8 (marked red.)
  44. 44. ApplicationHere’s an example with a splitcompartment which allows us to analysethe ranges separately  J7 cannot be 78,as this would violate the unique solutionconstraint. Thus J7=2349 and H7=348.The example below shows an advancedapplication: If you compare A12345 withB12345, you’ll find that they only differ in that A1..5 has 4 asadditional candidate. Thus if A1..5 would not contain 4, we couldfreely interchange A1..5with B1..5, no otherfield can intervene. A1..5 must contain 4.
  45. 45. ApplicationBe very careful when applying theunique solution constraint: theexample on the right almost looks likethe 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 influencethe outcome, you cannot apply this strategy.Conversely, if certain fields are the last possible chance tointervene, we can applythe constraint: hereA3..9 must contain a 9,because otherwise A1 C..J1 doesn‘t contain 1 nor 9can be either 1 or 9. (not shown)
  46. 46. Y-WingAn Y-Wing is a two-prongedelimination technique,where three corners of arectangle eliminate acandidate in the fourthcorner. Have a look:C3 (blue) contains two candidates: 6 and 7.If C3=6, then C7 (green) must be 5 and therefore 5 is removedfrom A7 (red arrows.) On the other hand, if C3=7, then A3(green) must be 5, and again A7 cannot be 5 (blue arrows.) Thusin either case, we can remove 5 from A7.Y-Wings do not often occur in Str8ts, but they provide insightinto more complex chaining strategies.
  47. 47. Y-WingIn general, an Y-Wing hasa base (marked blue) with XZ -Ztwo candidates, let’s callthem X and Y, and twoprongs which contain an XY YZadditional number Z, whichgets removed in the targetfield (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, whichmeans that each candidate of the base cell is covered and leadsto Z being removed from the target cell.
  48. 48. Str8ts Strategiesby SlowThinker(with feedback from Auric, darktray, and John)This text is licensed under Creative CommonsAttribution-NonCommercial-ShareAlike 3.0Unported License.