Description of basic and advanced strategies for solving Str8ts puzzles.

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.