### X tutorial (English)

Verfasst:

**Freitag 24. Februar 2023, 17:50**X Str8ts have an additional constraint: the diagonals A1-J9 and J1-A9 are treated like rows and columns, i.e., digits must not repeat within a diagonal, and digits must form straights within compartments separated by black cells. Numerous X Str8ts have been posted (including one weekly by kst on www.str8ts.com/weekly_str8ts.aspx), however I'm not aware of a tutorial on the extra solution strategies relevant for X puzzles.

This week's X puzzle by kst, is.gd/kst_661_X, requires practically all X techniques and has a purely analytic solution. It makes for a great textbook example. So, here goes the textbook!

Techniques:

* square rule, items 1,2,3

* skewed fish, items 2,3,4,7,8

* UR: when can and when can't they be used in X puzzles? 7,8,9

Solution:

1. X hi/lo 1/2 Just filling in numbers, so far. 23 in A12 due to gap or HP. We get our first two examples of

-- J9 != 8 (otherwise 8 eliminated from both HJ8)

-- B2 != 3 (otherwise eliminated from A12)

2. X hi/lo 2/2 First example of a simple

-- wing on 3, in diagonals /\. Eliminates 3 from the rest of columns 1 and 4 and makes 3 required in both cols (not decisive for this puzzle, so we're not filling it in).

We also see a beautiful example of

-- the NT 145 in col 3 makes 5 required on the two diagonal cells. This eliminates 5 from C7, G7 (and E5, if it weren't a black cell). The difference between this case of square rule and the above: both diagonals come into play here, where above we dealt with only one.

What really helps solve this puzzle, though, is the

Let's fill this in and work out immediate consequences. The skewed wing on 3 collapses: 3. No 3 in C! Here we can use the

It's tempting to look into Settis 23. But let's first simplify the cells we already have candidates for.

4. Fishing on the right-hand side * Must-use-intersection fish: H8=8 (CH\)

* Wing on 9 (C\) --> H4=1, 9 in /

5. Settis 23

2 and 3 must be absent from col 5 but present in G --> 3 in G2-7 --> no 9 in G2-7 --> F6=9 6. Wing on 7

* Regular

* Fill in some more cells: several singles, NPs, ... Settis 68 col 1, Setti 5 E. 7. Uniqueness and X

*

* If all involved cells are off-diagonal, standard URs can be used (see below).

* There are X-specific "skewed URs," though. See C34, G3F4, FG5. If FG5 were 45, the NPs on FG would make 45 ambiguous in row-space, column-space, and along the diagonal. We can't have that, FG5!=4 [I'm cheating, of course: we already know J5=4 due to the NP on J, so we don't need the skewed UR. It's valid, though.]

8. Generalized skewed wings

* The skewed wings so far were similar to regular fish in the sense that both sets of wingtips saw one another. That is not a requirement, though! Look at 6 in G (base), col 5 (antenna 1), \ (antenna 2). The two 'feelers' F5, J9 do not see each other. Still, we can eliminate 6 from all cells that see both feelers; in this case F9. Double-check: if F9=6, G5=6 via 5 but G7=6 via \. This is very useful to solve the puzzle. We've had another such skewed wing on the board for a while: F2 and B4 !=45 (base col 3, antennas \ and /)

* Regular skewed wing 5 (/5)

* UR EF89: F8!=34. This is valid because all involved cells are off-diagonal

* 3-fish 6 AH\: F7!=6. No intersections this time, so we can eliminate 6 from the rest of columns 679 (F7!=7 because of the wing on 7 above)

9. BUG We're now left with bi-value cells everywhere but H7. If H7!=6, we have two candidate cells for each unsolved digit in all rows, columns, and diagonals. That's a uniqueness problem called BUG, see www.sudokuwiki.org/BUG Hence H7=6, which solves.

If you're unsure/uncomfortable about BUG, it's not hard to show that H7!=5 (e.g., SI via col 7, G2, J2, J7=5). That, too, solves.

Thanks again, kst, for a great puzzle!

This week's X puzzle by kst, is.gd/kst_661_X, requires practically all X techniques and has a purely analytic solution. It makes for a great textbook example. So, here goes the textbook!

Techniques:

* square rule, items 1,2,3

* skewed fish, items 2,3,4,7,8

* UR: when can and when can't they be used in X puzzles? 7,8,9

Solution:

1. X hi/lo 1/2 Just filling in numbers, so far. 23 in A12 due to gap or HP. We get our first two examples of

**square rule**:-- J9 != 8 (otherwise 8 eliminated from both HJ8)

-- B2 != 3 (otherwise eliminated from A12)

2. X hi/lo 2/2 First example of a simple

**skewed fish**:-- wing on 3, in diagonals /\. Eliminates 3 from the rest of columns 1 and 4 and makes 3 required in both cols (not decisive for this puzzle, so we're not filling it in).

We also see a beautiful example of

**advanced square rule:**-- the NT 145 in col 3 makes 5 required on the two diagonal cells. This eliminates 5 from C7, G7 (and E5, if it weren't a black cell). The difference between this case of square rule and the above: both diagonals come into play here, where above we dealt with only one.

What really helps solve this puzzle, though, is the

**skewed 3-fish**on 2, H/\. It this weren't an X puzzle, we could eliminate 2 from the rest of columns 124. Not here, though: H and / intersect at H2, and 2 is a candidate in that cell (in 'regular' Str8ts, this can't happen). If H2=2, the fish logic collapses. We can think one notch deeper, though: if H2 != 2, we get a 3 fish of three non-intersecting compartments in 2 columns. That can't work. Hence H2=2. I call this (colloquially)**"must-use-intersection fish".**Let's fill this in and work out immediate consequences. The skewed wing on 3 collapses: 3. No 3 in C! Here we can use the

**square rule**again: C2!=45 (both required in \). Also, no 5 in the lower \ compartment (required in A1-D4). Plus, we get another nice**skewed wing**(C/): A3!=4.It's tempting to look into Settis 23. But let's first simplify the cells we already have candidates for.

4. Fishing on the right-hand side * Must-use-intersection fish: H8=8 (CH\)

* Wing on 9 (C\) --> H4=1, 9 in /

5. Settis 23

2 and 3 must be absent from col 5 but present in G --> 3 in G2-7 --> no 9 in G2-7 --> F6=9 6. Wing on 7

* Regular

**skewed wing**on 7, H\: eliminate 7 from rest of cols 79, no 1 in col 9 (7 required in D-J9). Solves B9, G9; single 7 in A.* Fill in some more cells: several singles, NPs, ... Settis 68 col 1, Setti 5 E. 7. Uniqueness and X

*

**"Standard" URs cannot be used if one (or more) cells involved are on a diagonal.**It's tempting to eliminate 67 from F9 for uniqueness reasons FJ39. However, that's not valid. The diagonal can break the ambiguity. 7 is already eliminated from F9 due to X power (skewed wing above).* If all involved cells are off-diagonal, standard URs can be used (see below).

* There are X-specific "skewed URs," though. See C34, G3F4, FG5. If FG5 were 45, the NPs on FG would make 45 ambiguous in row-space, column-space, and along the diagonal. We can't have that, FG5!=4 [I'm cheating, of course: we already know J5=4 due to the NP on J, so we don't need the skewed UR. It's valid, though.]

8. Generalized skewed wings

* The skewed wings so far were similar to regular fish in the sense that both sets of wingtips saw one another. That is not a requirement, though! Look at 6 in G (base), col 5 (antenna 1), \ (antenna 2). The two 'feelers' F5, J9 do not see each other. Still, we can eliminate 6 from all cells that see both feelers; in this case F9. Double-check: if F9=6, G5=6 via 5 but G7=6 via \. This is very useful to solve the puzzle. We've had another such skewed wing on the board for a while: F2 and B4 !=45 (base col 3, antennas \ and /)

* Regular skewed wing 5 (/5)

* UR EF89: F8!=34. This is valid because all involved cells are off-diagonal

* 3-fish 6 AH\: F7!=6. No intersections this time, so we can eliminate 6 from the rest of columns 679 (F7!=7 because of the wing on 7 above)

9. BUG We're now left with bi-value cells everywhere but H7. If H7!=6, we have two candidate cells for each unsolved digit in all rows, columns, and diagonals. That's a uniqueness problem called BUG, see www.sudokuwiki.org/BUG Hence H7=6, which solves.

If you're unsure/uncomfortable about BUG, it's not hard to show that H7!=5 (e.g., SI via col 7, G2, J2, J7=5). That, too, solves.

Thanks again, kst, for a great puzzle!