## Fireworks

Advanced methods and approaches for solving Sudoku puzzles

### Fireworks

hello all, wanted to talk about an area of sudoku logic im really interested in which i havent found much coverage about

its built around a very simple idea, that being when a candidate in an intersecting row and column is limited to the same box, it must be placed in the intersecting cell

Code: Select all
`+-------+-------+-------+| x x x | 1 2 3 | 4 5 . || x . . | . . . | . . . || x . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || . . . | . . . | . . 9 |+-------+-------+-------+`

in the diagram, 9 in r1 and c1 is restricted to the cells marked with an x, and therefore is placed in r1c1, which is easily done with locked candidates (9 in r1 lc elims 9b1p47, giving 9 in c1, or vice versa)
but, say the given 9 in r9c9 did not exist, we would still have an interesting pattern

Code: Select all
`+-------+-------+-------+| x . . | 1 2 3 | 4 5 x || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || x . . | . . . | . . . |+-------+-------+-------+`

our new conclusion is that a 9 has to appear within the x-marked cells, because either at least one exists outside the intersecting box, or they both take the intersecting cell. this is similar to (grouped) kites, except we have another extra position. i find this notable as it disregards up to 4 potential candidate placements from the intersecting box

at the moment im just calling this pattern a firework, it "explodes" outward from the intersection, in a way
on its own, a firework gives nothing; there is no common seen cell to eliminate a candidate. theyre useful when embedded into larger patterns, like strong links in AIC's, but not so useful when you only have a single one, this is because of the way you can more simply transform a kraken firework into a kraken line (for most cases) more talk about that and examples of it linked at the footnote of this post

the power of fireworks is when you have multiple, and they overlap on the same cell. to go back to the example above, youll see that there is a firework on 9s and another on 8s, both of which have r1c1 as the intersection cell. with this weak link between 8 and 9 in r1c1, you can conclude at least one of r1c9 & r9c1 is 8 or 9, which is not simple to prove otherwise i think

heres some examples ive gathered over time of interesting patterns using these fireworks:

Code: Select all
`+-------+-------+-------+| . 4 5 | . . . | . . . || . . . | 1 . . | . 7 . || 8 . . | . 2 3 | . . . |+-------+-------+-------+| . . . | 9 . 7 | 1 . . || . . . | . . . | 3 . . || . 8 . | 4 . 6 | . 2 . |+-------+-------+-------+| . . 3 | . . . | . . 5 || . 7 . | 8 . . | . . 6 || . . . | . . . | 9 . . |+-------+-------+-------+Triple Laser - Qinluxskfr 8.3.45.........1...7.8...23......9.71........3...8.4.6.2...3.....5.7.8....6......9...---------------------.-------------------.--------------------.|x123-679 4     5     | 67   6789   89    | 268   189   x123-89|| 2369    2369  269   | 1    45689  4589  | 2568  7      2389  || 8       169   1679  | 567  2      3     | 456   1459   149   |:---------------------+-------------------+--------------------:| 23456   2356  246   | 9    358    7     | 1     4568   48    || 145679  1569  14679 | 25   158    1258  | 3     45689  4789  || 13579   8     179   | 4    135    6     | 57    2      79    |:---------------------+-------------------+--------------------:| 12469   1269  3     | 267  14679  1249  | 2478  148    5     || 12459   7     1249  | 8    1459   12459 | 24    3      6     || 12456   1256  8     | 3    14567  1245  | 9     14    x12-47 |'---------------------'-------------------'--------------------'`

firework triple

123 in r1 and c9 limited to r1c1, r9c9, and box 3, and therefore limited to r1c1, r9c9 and r1c9
three cells for three candidates, all others removed
stte

and similar:

Code: Select all
`+-------+-------+-------+| . . 2 | 3 . . | 5 . . || . 1 . | . 4 . | . 9 . || . . . | 5 . . | . . 6 |+-------+-------+-------+| . 7 6 | . . . | . . . || 8 . . | . 2 . | . 4 . || 9 . . | . . . | 8 . 3 |+-------+-------+-------+| . . . | . . 5 | . . 2 || . . . | . . 6 | . 1 . || . . . | 8 7 . | . . . |+-------+-------+-------+Cobra Roll - jovi_alskfr 8.5..23..5...1..4..9....5....6.76......8...2..4.9.....8.3.....5..2.....6.1....87.....------------------------.--------------------.--------------------.| 467     4689    2      | 3     1689   1789  | 5      78    14    || 3567    1       3578   | 267   4      278   | 23     9     78    || 347     3489    34789  | 5     189    12789 | 14     23    6     |:------------------------+--------------------+--------------------:|x1234-5  7       6      | 149   13589 x34-189| 129    25    159   || 8       35      135    | 1679  2      1379  | 1679   4     1579  || 9       245     145    | 1467  156    147   | 8      2567  3     |:------------------------+--------------------+--------------------:| 13467   34689   134789 | 149   139    5     | 34679  3678  2     || 23457   234589  345789 | 249   39     6     | 3479   1     45789 ||x12-3456 234569  13459  | 8     7     x1234-9| 3469   356   459   |'------------------------'--------------------'--------------------'`

original post

12 in r9 and c1 limited to r4c1, r9c6 or b7 and therefore r4c1, r9c6, r9c1
34 in r4 and c6 limited to r4c1, r9c6 or b5 and therefore r4c1, r9c6, r4c6
four cells for four candidates, all others removed
stte (after naked pairs at the start)

moving away from tuples:

Code: Select all
`+-------+-------+-------+| . 8 5 | 7 . . | 6 . . || 3 . . | . 4 . | . 1 . || 2 . . | . . . | . . 8 |+-------+-------+-------+| 5 . 4 | 8 . . | . . . || 6 . . | . 2 . | . 5 . || . 9 . | . . 1 | . . 3 |+-------+-------+-------+| . . . | . . 9 | . . 4 || . . . | 1 . . | . . 7 || . . . | . 3 7 | 2 8 . |+-------+-------+-------+Pear and Rocket - shyeskfr 7.1.857..6..3...4..1.2.......85.48.....6...2..5..9...1..3.....9..4...1....7....3728..---------------------.------------------.-------------------.|x149  8       5      | 7    #19    23   | 6      2349  2-9  || 3    67      679    | 2569  4     8    | 579    1     259  || 2    1467    1679   | 3569  1569  356  | 34579  3479  8    |:---------------------+------------------+-------------------:| 5    1237    4      | 8     679   36   | 179    2679  1269 || 6    137     1378   | 349   2     34   | 14789  5    #19   || 78   9       278    | 456   567   1    | 478    2467  3    |:---------------------+------------------+-------------------:| 178  123567  123678 | 256   568   9    | 135    36    4    || 489  23456   23689  | 1     568   2456 | 359    369   7    ||x149  1456    169    | 456   3     7    | 2      8    x1569 |'---------------------'------------------'-------------------'`

dual firework w-wing

fireworks on 1&9 r9c1b7
either a naked 19 pair in r1c15 or r59c9 exists
=> -9r1c9 stte

this one was the first example i made of a non-rank0 deduction. it uses the deduction mentioned earlier at the top of the post, at least one of the 1 or 9 candidates outside the intersecting box has to be true (more generally, we can express it as: (x|y)cell1 = (x|y)cell2 for any two perfectly overlapping 3-pos fireworks)

Code: Select all
`+-------+-------+-------+| 9 . . | . 3 7 | . 8 . || . 7 . | 1 . . | . . 9 || 5 . . | 2 . . | . . . |+-------+-------+-------+| . 6 . | . . 5 | . . 8 || 4 . . | . . . | . . 7 || . . . | . . . | 2 3 . |+-------+-------+-------+| 7 . 5 | . . . | . . . || . 2 . | . . 6 | . 9 . || 6 . 9 | . 4 . | 5 . 1 |+-------+-------+-------+Heartline Roll - shyeskfr 8.69...37.8..7.1....95..2......6...5..84.......7......23.7.5.......2...6.9.6.9.4.5.1.------------------.----------------------.-------------------.| 9    14    1246  |*x456    3      7     | 146    8    *25   || 238  7     23468 |  1      568    48    | 346    25    9    || 5    1348  13468 |  2      689    489   | 13467  1467  346  |:------------------+----------------------+-------------------:| 123  6     1237  |  3479   1279   5     | 149    14    8    || 4    59    1238  |  3689   12689  12389 | 169    156   7    || 18   59    178   | x46789  16789  1489  | 2      3    x46-5 |:------------------+----------------------+-------------------:| 7    1348  5     |  389    1289   12389 | 3468   246   2346 || 138  2     1348  |  3578   1578   6     | 3478   9     34   || 6    38    9     |  378    4      238   | 5      27    1    |'------------------'----------------------'-------------------'`

dual firework l-wing

fireworks on 4&6 in r6c4b5
5r1c9 = (5-4|6)r1c4 = 46r6c49
-5r6c9 stte

perhaps easiest to explain in words: the 5 in r1 is either in c9 (-5r6c9) or c4 turning the firework into a hidden pair (-5r6c9)
you can also use 5 in c9 instead for this puzzle specifically
this is the most bare-bones deduction i've come across using fireworks, it's only 5 truths which is less than everything else so far!

Code: Select all
`+-------+-------+-------+| . . . | 6 . 1 | . . . || . . 2 | . 3 . | 7 . . || . 8 . | . . . | . 4 . |+-------+-------+-------+| 1 . . | . . 4 | . . 5 || . 7 . | 1 . . | . 9 . || 3 . . | 9 6 . | . . 1 |+-------+-------+-------+| . 6 . | . . . | . 8 . || . . 9 | . 5 . | 2 . . || . . . | 7 . 6 | . . . |+-------+-------+-------+Takabisha - shyeskfr 8.3...6.1.....2.3.7...8.....4.1....4..5.7.1...9.3..96...1.6.....8...9.5.2.....7.6....---------------------.------------------.--------------------.| 4579   345    3457  | 6   x24789  1    | 3589   235  x28-39 || 4569   145    2     | 458  3      89   | 7      156   689   || 5679   8      13567 | 25   79-2   279  | 13569  4     2369  |:---------------------+------------------+--------------------:| 1      9      68    | 238  7-28   4    | 368    2367  5     ||x24568  7      456-8 | 1   #28     35-28| 346-8  9    x23468 || 3      245    458   | 9    6      2578 | 48     27    1     |:---------------------+------------------+--------------------:| 2457   6      13457 | 234  149-2  239  | 13459  8     3479  || 478    134    9     | 348  5      38   | 2      136   3467  ||x28-45  12345  13458 | 7   x12489  6    | 13459  135   349   |'---------------------'------------------'--------------------'`

dual firework ALP
original post

like the above this uses the (x|y)cell1 = (x|y)cell2 idea, but for something a bit fancier

fireworks on 2s and 8s in both r1c9b3 & r9c1b7
(2|8)r1c5 = (2|8)r5c9
(2|8)r5c1 = (2|8)r9c5
combined with r5c5 limited to only [28], at most one of 2 or 8 can be in r5c19 and at most one in r19c5. almost locked pairs in r5 and c5
stte

also worth mentioning is r1c9 and r9c1 become limited to [28], much like how in a regular almost locked pair you get limitations on a cell
here are some more resisitant puzzles which have the same setup as this

this next example is actually the first puzzle i made that had these ideas, back then i didnt view them the same way though, and its pretty complex
it features a firework with 5 positions (9s) being useful

Code: Select all
`+-------+-------+-------+| . 1 2 | . 7 . | 3 5 . || 3 . . | 2 . 1 | 4 . 7 || 4 . . | 5 . . | . 1 6 |+-------+-------+-------+| 2 . . | . . . | . 7 . || 1 6 . | . 8 . | . . 2 || . 4 8 | . . . | 6 3 . |+-------+-------+-------+| . . . | 8 . . | . . 4 || . . . | 1 5 . | . . 3 || . . . | . 4 3 | 1 2 . |+-------+-------+-------+Hanabi - shyeskfr 8.0.12.7.35.3..2.14.74..5...162......7.16..8...2.48...63....8....4...15...3....4312..---------------------.-----------------.----------------.| x689     1     2    | 4-69  7    4689 | 3    5   B89   ||  3       589   569  | 2     69   1    | 4    89   7    ||  4       789   79   | 5     3    89   | 2    1    6    |:---------------------+-----------------+----------------:|  2       59    359  | 3469  169  4569 | 589  7    1589 ||  1       6     3579 | 379   8    579  | 59   4    2    || x579     4     8    |B79    129  2579 | 6    3    1-59 |:---------------------+-----------------+----------------:|  5679    3     1    | 8     269  2679 | 579  69   4    ||  6789    2     4    | 1     5    679  | 789  689  3    ||xT789-56  5789  5679 |x679   4    3    | 1    2   x589  |'---------------------'-----------------'----------------'`

firework exocet
original post

fireworks on 56789 in r9c1b7
base set: r1c9 & r6c4 - target set: r9c1
non-base candidates removed from target cell
fireworks on 5 and 6 become two-string kites, giving elims in r1c4 & r6c9
compatibility testing, 9 must be limited to base cells, giving elims in r1c4 & r6c9 (resulting in naked singles)
solves with a turbot fish

this pattern is very complex and rare
here i wrote a less jargon-y way to view the deduction (as well as provide a easier example of it)

the last puzzle is probably the most interesting application

Code: Select all
`+-------+-------+-------+| . . 9 | 8 . . | 7 . . || . . . | . 6 . | 5 . . || 4 . . | . 5 . | . 9 3 |+-------+-------+-------+| 7 . . | . . . | . . . || . 3 5 | . . . | 2 8 . || . . . | . . . | . . 6 |+-------+-------+-------+| 8 . . | . 2 . | . . 4 || . 2 . | . 7 . | . . . || 1 . 6 | . . 3 | 9 . . |+-------+-------+-------+Roman Candle V2 - shye & jovi_alskfr 9.0..98..7......6.5..4...5..937.........35...28.........68...2...4.2..7....1.6..39...--------------------.----------------------.-------------------.|*36-25 156    9     | 8      x*134   124    | 7     1246   12   || 23    178    12378 | 123479   6     12479  | 5     124    128  || 4     1678   1278  | 127      5     127    | 168   9      3    |:--------------------+----------------------+-------------------:| 7     14689  1248  | 2356-1   1389  2568-1 | 134   135    159  ||*69    3      5     | 1467   x*149   1467   | 2     8    x*179  || 29    1489   1248  | 2357-1   1389  2578-1 | 134   1357   6    |:--------------------+----------------------+-------------------:| 8     579    37    | 1569     2     1569   | 136   13567  4    || 359   2      34    | 14569    7     145689 | 1368  1356   158  || 1     457    6     | 45      *48    3      | 9     257   *78-25|'--------------------'----------------------'-------------------'`

MSHS with firework

seven cells for the following placements: a firework on 1s in r5c5b5 along with positions of 3r1, 4c5, 6c1, 7c9, 8r9, 9r5. all other candidates removed
stte
(big thanks to jovi for helping find harder grids with this pattern!)

it may be worth noting here (and this also applies for Triple Laser and Cobra Roll mentioned earlier) that any remaining positions for the firework candidate in the intersecting box, here being b5p1379, can be eliminated. these tuples/multifish are rank0 and fireworks embedded into a rank0 pattern will have this property. in most cases ive found, this is not very useful

ill share more in this thread if ever i find new discoveries, and link to it below (or edit it into this post if its a more notable find) :>

new resources:
why using kraken fireworks is most often an overcomplication
inverting sue de coq and other ALS based techniques into firework MSHS's
Last edited by shye on Sun Nov 20, 2022 7:05 am, edited 14 times in total.

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

shye wrote: an interesting pattern
Code: Select all
`+-------+-------+-------+| x . . | 1 2 3 | 4 5 x || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || x . . | . . . | . . . |+-------+-------+-------+`

our new conclusion is that 9 is limited to the x-marked cells,

Hi shye,
How do you exclude this possibility:

Code: Select all
`+-------+-------+-------+| . . 9 | 1 2 3 | 4 5 . || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || 9 . . | . . . | . . . |+-------+-------+-------+`
denis_berthier
2010 Supporter

Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

denis_berthier, the conclusion is that at least one of the cells marked x (r1c1, r1c9, r9c1) must be 9. "limited to" might not be the clearest way of describing it

seems like a nice technique after following the examples
999_Springs

Posts: 591
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: Fireworks

poor wording yes, sorry. updated post ^^

another edit to the main post, just tidying up a few more parts that might be confusing/not written too well
Last edited by shye on Wed Nov 03, 2021 11:45 am, edited 1 time in total.

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

999_Springs wrote:denis_berthier, the conclusion is that at least one of the cells marked x (r1c1, r1c9, r9c1) must be 9. "limited to" might not be the clearest way of describing it

[Edit:] I see it has been corrected.
denis_berthier
2010 Supporter

Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

[edited post]
this is about kraken fireworks, the idea of chaining off each candidate to find a common elimination/assignment. these i would say are not as much worth looking for, even though relatively common
but first an example to show the concept

Code: Select all
`+-------+-------+-------+| 1 . . | . 8 . | . . 9 || . . 2 | . . . | 3 . . || . 9 . | 6 . 5 | . 1 . |+-------+-------+-------+| . 7 3 | . . . | 4 6 . || . . . | . . . | . . . || . . 4 | . 9 . | 7 . . |+-------+-------+-------+| . 5 . | . 7 . | . 4 . || . . . | 1 . 9 | . . . || 8 . . | . 6 . | . . 5 |+-------+-------+-------+memori_al - RSPskfr 7.11...8...9..2...3...9.6.5.1..73...46............4.9.7...5..7..4....1.9...8...6...5.--------------------.----------------------.--------------------.|  1     *346   567  |*2347   8    *2347    | 256    27    9     ||  4567   468   2    | 9      1-4   147     | 3      578  *4678  ||x*347    9     78   | 6      234   5       | 28     1   x*2478  |:--------------------+----------------------+--------------------:|  9      7     3    | 5      12    128     | 4      6     128   ||  26     1268  1568 | 23478  1234  1234678 | 12589  2389  1238  ||  256    1268  4    | 238    9     12368   | 7      2358  1238  |:--------------------+----------------------+--------------------:|  236    5     169  | 238    7     238     | 1269   4     1236  ||x*23467 *2346  67   | 1      5     9       | 268    2378  23678 ||  8      123   179  | 234    6     234     | 129    2379  5     |'--------------------'----------------------'--------------------'`

kraken firework 7r3c1b1
original post
(note: not the intended solution)

||(7-3)r3c1 = (3-4)r1c2 = 4r1c46
||(7-4)r3c9 = 4r2c9
||(7-4)r8c1 = 4r8c2 - 4r1c2 = 4r1c46
stte (after x-wing on 5s at the start)

and another example of a kraken firework, this time with four candidates, found in a more resistant generated puzzle:

Code: Select all
`+-------+-------+-------+| . . . | . 5 . | . 4 . || 7 . . | 6 . 4 | . . 8 || . . . | 9 . . | 1 . . |+-------+-------+-------+| 4 5 . | 8 . . | . 6 . || 9 . . | . 2 . | . . 5 || 1 . . | . . 5 | 4 9 . |+-------+-------+-------+| 8 . . | . . . | . . . || . 7 . | . . 6 | . . . || . . 5 | 7 9 3 | . 1 . |+-------+-------+-------+skfr 7.1....5..4.7..6.4..8...9..1..45.8...6.9...2...51....549.8.........7...6.....5793.1..------------------.-----------.------------------.|*6-3 13689  13689 | 2   5  78 | 679   4    x3679 || 7   239    239   | 6   1  4  | 259   235   8    || 5   2468   2468  | 9   3  78 | 1     27    267  |:------------------+-----------+------------------:| 4   5     #23    | 8   7  9  |*23    6     1    || 9   36     367   | 4   2  1  | 37    8     5    || 1   28     278   | 3   6  5  | 4     9    *27   |:------------------+-----------+------------------:| 8   1369   1369  | 15  4  2  | 5679  357   3679 ||x23  7     x1349  | 15  8  6  | 259   235 x*2349 ||*26  46     5     | 7   9  3  | 8     1    *246  |'------------------'-----------'------------------'`

kraken firework 3r8c9b9
|| 3r1c9
|| 3r8c1
|| 3r8c3 - (3=2)r4c3 - 2b6p1 = 2b6p9 - 2r9c9 = (2-6)r9c1 = 6r1c1
||(3-4)r8c9 = (4-2)r9c9 = (2-6)r9c1 = 6r1c1
-3r1c1 stte

these are viewable with regular krakens in lines, as long as one of the lines of the firework has one position outside the intersecting box, then you can use the other line for the strong link and the box positions previously discarded will link to the original one
so the above becomes:

kraken row (3)r8c1389
|| 3r8c1
|| 3r8c3 - (3=2)r4c3 - 2b6p1 = 2b6p9 - 2r9c9 = (2-6)r9c1 = 6r1c1
|| 3r8c8 - 3r78c9 = 3r1c9
||(3-4)r8c9 = (4-2)r9c9 = (2-6)r9c1 = 6r1c1

which when written out is one more strong link compared to the previous, but i think we would all agree is simpler (you do not have to prove or think about the firework at all)

if however, both lines have at least two positions with separate chains outside the intersecting box, then the alternative chain will become dynamic/nested, and then the firework may be considered simpler. ive yet to find a good example of this though
Last edited by shye on Wed Nov 10, 2021 3:44 pm, edited 1 time in total.

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

this deduction was moved to the main post
Last edited by shye on Wed Nov 10, 2021 3:48 pm, edited 1 time in total.

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

shye wrote:not sure what to call it, but its similar to the dual ALP in takabisha, in that it uses the (x|y)cell1 = (x|y)cell2 idea. this one isnt rank0 though
fireworks on 1&9 r9c1b7
either a naked 19 pair in r1c15 or r59c9 exists
=> -9r1c9 stte

Starting from the same PM:
Code: Select all
`   +----------------------+----------------------+----------------------+    ! 149    8      5      ! 7      19     23     ! 6      2349   29     !    ! 3      67     679    ! 2569   4      8      ! 579    1      259    !    ! 2      1467   1679   ! 3569   1569   356    ! 34579  3479   8      !    +----------------------+----------------------+----------------------+    ! 5      1237   4      ! 8      679    36     ! 179    2679   1269   !    ! 6      137    1378   ! 349    2      34     ! 14789  5      19     !    ! 78     9      278    ! 456    567    1      ! 478    2467   3      !    +----------------------+----------------------+----------------------+    ! 178    123567 123678 ! 256    568    9      ! 135    36     4      !    ! 489    23456  23689  ! 1      568    2456   ! 359    369    7      !    ! 149    1456   169    ! 456    3      7      ! 2      8      1569   !    +----------------------+----------------------+----------------------+ 182 candidates.`

Trying to eliminate the same candidate:
Code: Select all
`whip[9]: r5c9{n9 n1} - r4n1{c9 c2} - r4n3{c2 c6} - r1n3{c6 c8} - r1n4{c8 c1} - b1n1{r1c1 r3c3} - b1n9{r3c3 r2c3} - r9n9{c3 c1} - r9n1{c1 .} ==> r1c9≠9stte`

However, there's also a simpler 1-step solution in W8:
Code: Select all
`whip[8]: r1n4{c8 c1} - r1n1{c1 c5} - r1n9{c5 c9} - r5c9{n9 n1} - c7n1{r5 r7} - c1n1{r7 r9} - r9n9{c1 c3} - b1n9{r2c3 .} ==> r1c8≠3stte`
denis_berthier
2010 Supporter

Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

shye wrote:a better example of a kraken firework, this time with four candidates, found in a more resistant generated puzzle:
Code: Select all
`.------------------.-----------.------------------.|*6-3 13689  13689 | 2   5  78 | 679   4    x3679 || 7   239    239   | 6   1  4  | 259   235   8    || 5   2468   2468  | 9   3  78 | 1     27    267  |:------------------+-----------+------------------:| 4   5     #23    | 8   7  9  |*23    6     1    || 9   36     367   | 4   2  1  | 37    8     5    || 1   28     278   | 3   6  5  | 4     9    *27   |:------------------+-----------+------------------:| 8   1369   1369  | 15  4  2  | 5679  357   3679 ||x23  7     x1349  | 15  8  6  | 259   235 x*2349 ||*26  46     5     | 7   9  3  | 8     1    *246  |'------------------'-----------'------------------'`

kraken firework 3r8c9b9
|| 3r1c9
|| 3r8c1
|| 3r8c3 - (3=2)r4c3 - 2b6p1 = 2b6p9 - 2r9c9 = (2-6)r9c1 = 6r1c1
||(3-4)r8c9 = (4-2)r9c9 = (2-6)r9c1 = 6r1c1
-3r1c1 stte

Starting from the same PM:
Code: Select all
`   +-------------------+-------------------+-------------------+    ! 36    13689 13689 ! 2     5     78    ! 679   4     3679  !    ! 7     239   239   ! 6     1     4     ! 259   235   8     !    ! 5     2468  2468  ! 9     3     78    ! 1     27    267   !    +-------------------+-------------------+-------------------+    ! 4     5     23    ! 8     7     9     ! 23    6     1     !    ! 9     36    367   ! 4     2     1     ! 37    8     5     !    ! 1     28    278   ! 3     6     5     ! 4     9     27    !    +-------------------+-------------------+-------------------+    ! 8     1369  1369  ! 15    4     2     ! 5679  357   3679  !    ! 23    7     1349  ! 15    8     6     ! 259   235   2349  !    ! 26    46    5     ! 7     9     3     ! 8     1     246   !    +-------------------+-------------------+-------------------+ 112 candidates.`

and trying to eliminate the same candidate:
Code: Select all
`whip[7]: r8c1{n3 n2} - r9n2{c1 c9} - r6c9{n2 n7} - r5c7{n7 n3} - c2n3{r5 r7} - c9n3{r7 r8} - c9n4{r8 .} ==> r1c1≠3stte`

Also found two more 1-step solutions of same complexity:
Code: Select all
`whip[7]: r9n2{c1 c9} - b6n2{r6c9 r4c7} - r4c3{n2 n3} - b7n3{r8c3 r7c2} - c1n3{r8 r1} - c9n3{r1 r8} - c9n4{r8 .} ==> r8c1≠2stte`

Code: Select all
`whip[7]: c9n4{r9 r8} - r8n2{c9 c1} - c1n3{r8 r1} - r2n3{c3 c8} - r8n3{c8 c3} - r4c3{n3 n2} - b6n2{r4c7 .} ==> r9c9≠2stte`
denis_berthier
2010 Supporter

Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

Hi shye,
Very nice find!
Someone can find many way to present the same deductions, but based on your pattern I can find deductions quite easier and faster. Your examples are very nice and I’m especially like Cobra Roll, Roman Candle V2, Pear and Rocket.
I’ll study more and thank you for your sharing!

totuan
totuan

Posts: 231
Joined: 25 May 2010
Location: vietnam

### Re: Fireworks

.
Generally speaking, I don't like very much patterns/procedures of the forcing type, i.e. starting from an exclusive OR relation, such as forcing-whips, forcing-braids, forcing-T&E, the forcing-chains of Sudoku Explainer, Robert's conjugated-tracks, Kraken or kite patterns...
Why I don't like them is simple: they require following several streams of reasoning at the same time; from a maths PoV, they are instances of "reasoning by cases", which is considered as inelegant.
The way Shye's pattern has been used until now falls in this global "forcing" category.

However, Shye's pattern itself (with conclusion the OR of 3 candidates) is very general and it may have some other uses yet to be found. I therefore tried to estimate its frequency.
For this I used my cbg000 collection of the 21375 first controlled-bias generated puzzles.

Estimating the frequency requires inserting the pattern somewhere into my hierarchy. I decided to put it just before any pattern of size 6.
Why? Because the pattern itself has size 3, but it requires at least 3 branches to provide any useful elimination. So putting it first before any pattern of size 6 gives it the best chances to be useful. Notice that my program does not say that it is really useful, it only says that it is potentially useful.

I found 237 (1.1%) puzzles with this partial pattern. I think that's quite a lot for experimenting.

list: Show
puzzle#36: 123...7.9.5.7..1...8.....6..............7.9.28.7.6..5...56..........2..1.1.84..3. ; SER=8.3 ; or3 = n7{r9c6 v r9c9 v r4c9}
puzzle#71: ..345.78.4....9.2..8..2..6...16.5..7......2..9...4..3...4......5.2.1.....1....3.6 ; SER=8.4 ; or3 = n5{r9c4 v r9c8 v r5c8}
puzzle#109: .23....8.4....9..378....5......75.3.......215...61...7.6.5.1....42.3....9....4..8 ; SER=9.0 ; or3 = n3{r9c7 v r9c2 v r6c2}
puzzle#111: .2.4.6.8...6.8.1.......3..4..5...6......9..4589.......3....49.25..932.7.....1.... ; SER=8.4 ; or3 = n4{r8c7 v r8c3 v r6c3}
puzzle#119: .2.....8...678.........35.4.1....4....527.....9..6.237......6.1.68....4.9...3..7. ; SER=8.4 ; or3 = n4{r9c2 v r9c4 v r1c4}
puzzle#278: .2...67..45......3..91..5...3.5.1.9.8...9.47....24.......9.5.1..9.3....7....1.... ; SER=8.4 ; or3 = n5{r9c1 v r9c9 v r5c9}
puzzle#394: .....6....5.7...2...9.3...6..4....97..82...5.5...9......7..4.1.81..6.2.49..8...6. ; SER=8.4 ; or3 = n3{r9c3 v r9c7 v r6c7}
puzzle#400: .2.4..........91.......25.6...5.18...3.24...............7...6.8.128.4.9.8.56....2 ; SER=8.9 ; or3 = n3{r9c8 v r9c6 v r6c6}
puzzle#407: .2.....894.6...1.......2.....4...8...3..4..9..1567...4..85...6.5.23.7......8..3.5 ; SER=8.4 ; or3 = n2{r9c5 v r9c8 v r2c8}
puzzle#836: .2.4.678...6.8...37.....5...91..8..5.3.....1.....132...1.26..5...8......9...4.6.. ; SER=9.0 ; or3 = n2{r9c3 v r9c9 v r3c9}
puzzle#949: .....6.8945..8.....8...1..4.1...5...5..34...19.4........2..76.564...3..7....9.3.. ; SER=8.5 ; or3 = n1{r9c1 v r9c8 v r2c8}
puzzle#990: ..34..7..4567..........1...2....465.6.....9.1.15.....2.946.5.......1...68..3...9. ; SER=8.9 ; or3 = n7{r9c6 v r9c9 v r4c9}
puzzle#1022: ..34.6..9....8..23.8....64..3.8.5.1.6.......8..5.1.........85........9329.2.73... ; SER=8.7 ; or3 = n5{r9c2 v r9c4 v r2c4}
puzzle#1034: .2......9..678....7..1...65....6.9...8.5.4......9.3..15.....347..7...5...42....1. ; SER=8.4 ; or3 = n8{r8c1 v r8c6 v r4c6}
puzzle#1038: .2..5...94.6..9...7....24......65.4...98..23.....2..16.8...16..67.5.......2...3.. ; SER=8.9 ; or3 = n3{r6c6 v r6c2 v r2c2}
puzzle#1203: 12..56.8..........7.8...5.6......39....97...4..5.2.....7...84..5....36..68......1 ; SER=8.4 ; or3 = n5{r9c4 v r9c8 v r5c8}
puzzle#1330: 12.45...9..67.9....98.32.....1..7..5......3.....32...6.....4.57..4...9..8..5732.. ; SER=9.0 ; or3 = n1{r9c2 v r9c9 v r3c9}
puzzle#1366: ...4..7..456.8.....9....5..2..69...1..7...9..9......3..4126...........12.....36.5 ; SER=8.4 ; or3 = n4{r9c5 v r9c8 v r3c8}
puzzle#1440: ...........67...237...3.5.4.4..6....5.73.....6..2.4.9..129..6..8....7..2.......1. ; SER=8.4 ; or3 = n4{r9c7 v r9c4 v r1c4}
puzzle#1636: .......89......1....8.13.6......84..5.4.......19.4..37..58...91......37.9...75... ; SER=9.0 ; or3 = n3{r9c2 v r9c4 v r5c4}
puzzle#1884: 12.........6...1...98.3.....1.8..6.5........7..56.3..2..7...29156..9......2.47... ; SER=8.4 ; or3 = n8{r9c2 v r9c9 v r1c9}
puzzle#1908: .2.4.......6....2.79..31.6.......69.......8.1....95.3.3.7..4....4.97...89....8... ; SER=9.0 ; or3 = n5{r8c8 v r8c3 v r1c3}
puzzle#1978: 1.3..678..........79.2.15....5...8.6..1....4.8.9...3......9....6..8.79.1...3..6.. ; SER=8.4 ; or3 = n4{r9c3 v r9c6 v r6c6}
puzzle#2182: ....56.8.4...8.1..7....1..523..7...8.......6...1..8.3....9.....87..2......5.6..72 ; SER=9.0 ; or3 = n1{r8c4 v r8c9 v r5c9}
puzzle#2303: .....6.8....7...2.897.....52.4.......856....1.7.3.....5.....2.7....9.3....1..7.6. ; SER=8.5 ; or3 = n8{r9c9 v r9c4 v r4c4}
puzzle#2412: 12..5.....5.7.9.....9.2.5...17.48...3..6..........24.15....4....78...95......781. ; SER=9.0 ; or3 = n2{r9c9 v r9c3 v r5c3}
puzzle#2415: .23..6.....67...3278..2......83.49...7...5..8.......2.5.2.1...4.......91.9....35. ; SER=8.3 ; or3 = n6{r9c4 v r9c9 v r3c9}
puzzle#2464: .23.....94....9....8912.5....4.186.........91.6.5...2.......2.4.9..743....8....7. ; SER=9.0 ; or3 = n1{r9c6 v r9c1 v r1c1}
puzzle#2467: ..3.....94...8.1..78.....4...4.71.6..1......3.....5......8..29.....92..4.6.5.43.. ; SER=7.4 ; or3 = n1{r8c3 v r8c8 v r6c8}
puzzle#2478: ...45..8....7.9.......2...6.3.5..4...1....9....8.7..1.39.....5...5918..4.4.2..... ; SER=8.5 ; or3 = n1{r9c3 v r9c9 v r2c9}
puzzle#2524: .23.5.7....6....3....1.....2.7.15.6.........1.3...8.7.3..8...........9.8.75..2... ; SER=8.9 ; or3 = n6{r9c7 v r9c5 v r6c5}
puzzle#2651: .2.4.6....5...9...7..13......4.97.6..1.....2....8..4...9...5..7......3..6....3.58 ; SER=8.9 ; or3 = n4{r9c5 v r9c2 v r3c2}
puzzle#2679: .....6......7.913..8.1...462.....9...61...3..97......5...82......73.52....2.6.45. ; SER=8.5 ; or3 = n7{r9c6 v r9c9 v r5c9}
puzzle#2897: .....6.89.5..8.1.2..91325..2...9.....642.7......8...4.3......7..4....8....1.4...5 ; SER=8.9 ; or3 = n3{r9c7 v r9c4 v r4c4}
puzzle#2903: ..34...8945.........91........8...4.....9.3.......562736......1.94.1....8..523... ; SER=8.4 ; or3 = n2{r8c1 v r8c9 v r2c9}
puzzle#2929: ......7.9.......3.7..13.56.2.8..59..5..8.3.7.61........4.......86159........486.. ; SER=8.4 ; or3 = n7{r9c4 v r9c3 v r6c3}
puzzle#2997: ....5.......7.9..278......4.4.6....5..5.1...3.3...72....8...65..6.3....8.12...3.7 ; SER=8.5 ; or3 = n4{r9c8 v r9c1 v r2c1}
puzzle#3014: 1....67..4....91...8...2..4.4..9...6......9..6....182..1...4..3.6.....5.932..5..8 ; SER=9.0 ; or3 = n1{r9c8 v r9c5 v r3c5}
puzzle#3506: 12.....8...6..9......2.36...1..943....4......9..1...2.3..84..97..7.....8.41.7.2.. ; SER=8.7 ; or3 = n3{r6c5 v r6c3 v r1c3}
puzzle#3546: .........45..8.......2136....4..8.....1..237.97.3..2.8......9....8.67..36..13.... ; SER=7.3 ; or3 = n5{r9c6 v r9c8 v r3c8}
puzzle#3551: .......8...67...3....2..6.52.4...5..8.532...1..1..5..6.1.6.....5.8.7.2.......89.. ; SER=9.0 ; or3 = n9{r7c5 v r7c3 v r3c3}
puzzle#3704: ..3.......5...91.2..92..6..2....897.....74....4.6.......5.....781....5....23.5.6. ; SER=7.3 ; or3 = n1{r9c5 v r9c9 v r6c9}
puzzle#3801: ...45....4..7.91........6.5.74.9..1.36..7..2......4.....2...8...9.8..45.....65.93 ; SER=7.3 ; or3 = n2{r9c4 v r9c7 v r1c7}
puzzle#3841: ..3.56...45.7........2.3..5.7...1.96....3.2..9.1..7.......9582.5.........9.3.2.7. ; SER=8.5 ; or3 = n3{r8c9 v r8c2 v r6c2}
puzzle#3993: ..3......4..78....7.9....5.2....89...6...4..3..51...6...467..9....8.1.7.6....5..1 ; SER=8.6 ; or3 = n5{r8c1 v r8c9 v r4c9}
puzzle#4000: ..3....894....91...8.2........9....6.....2395.6583.....9...7....1.64..7...7...5.. ; SER=8.4 ; or3 = n6{r9c1 v r9c8 v r3c8}
puzzle#4041: ..34........7..1.2.8..1...........6...5..8.9794...7.....1...84.57.....16.64..5... ; SER=8.4 ; or3 = n9{r9c5 v r9c9 v r1c9}
puzzle#4047: ...4.6..9.56....3.7..2.....2...4..67..7...3.8.6.....1....1.589.6...984....1....7. ; SER=7.3 ; or3 = n6{r9c4 v r9c9 v r3c9}
puzzle#4196: .2.4.......6....3....231....1.....9.3...7...55.8..32.....9..4..8...2.....71.6.8.. ; SER=8.4 ; or3 = n9{r9c1 v r9c9 v r1c9}
puzzle#4266: .2...6...4...8......92.....2..1..6....86.39....1.4.3.8.74.6.8...15....9......5..7 ; SER=8.4 ; or3 = n4{r9c7 v r9c4 v r1c4}
puzzle#4568: .....6.894.....1...89....4...5.......37.4........653.7..8..4.75.6..1...8....2.6.. ; SER=9.0 ; or3 = n4{r9c9 v r9c3 v r6c3}
puzzle#4592: .2...6...4.....1....9...54.....1.8.....3...748....7..5..2.....8.7.6.3...9.1.4..5. ; SER=8.5 ; or3 = n7{r9c4 v r9c7 v r1c7}
puzzle#4788: 1...5....4.......2.89...5..26.1..3.8..4...69.5...4....6...18.....13.2...9.......3 ; SER=8.5 ; or3 = n7{r9c5 v r9c2 v r1c2}
puzzle#4918: ....5.....567....28..23..6......78.....3...2...1.2.3.734.9...1...21...7....8...9. ; SER=8.3 ; or3 = n6{r7c5 v r7c7 v r5c7}
puzzle#5004: .2...6..94......3.......5.4.6.3..87...7......5.16........51.3..7...62...91.8.3.2. ; SER=8.4 ; or3 = n7{r9c9 v r9c5 v r6c5}
puzzle#5005: .2....7..4.67..1..8...3..6......8.9..3..9.2.1..7...8.63..17....6..9.5......8....5 ; SER=8.3 ; or3 = n6{r9c7 v r9c5 v r1c5}
puzzle#5035: ..3....8...6..91...7......428.97.4.............76.8.2.6.....27.....6...3..512.... ; SER=8.8 ; or3 = n8{r9c1 v r9c9 v r5c9}
puzzle#5056: ..34.6.8.....8...17...236...1..3.9....8..2.1..7........91...4.764...789....2.4... ; SER=8.6 ; or3 = n5{r9c2 v r9c9 v r3c9}
puzzle#5189: ...4..........92..78...2.4.2.....65..9.....1...43659..3......2.57..13...9...7..6. ; SER=7.3 ; or3 = n2{r9c4 v r9c2 v r1c2}
puzzle#5236: 1.......9.56.8......91.....2483.51.....2...6.....4.....65..74.2.7...4.......236.. ; SER=8.3 ; or3 = n5{r9c4 v r9c9 v r6c9}
puzzle#5536: 1...5..8.45..89......2.71..27......43.4..582.......9......68...8..9..6.3.62..3... ; SER=8.4 ; or3 = n9{r9c1 v r9c8 v r3c8}
puzzle#5660: 12..5....45.1.92....923....2..8..96...596...7....2.5......4..72.......18...3..... ; SER=8.5 ; or3 = n4{r9c7 v r9c3 v r4c3}
puzzle#5715: ..3.5..8.4....923......7...2.....3...14..2.687..6....1..1.2.54....7.36..9........ ; SER=8.5 ; or3 = n1{r9c8 v r9c6 v r4c6}
puzzle#5720: 1.3..67...57.8.2.6...2......98.71.4.....6...7...9.4..5.1.7....88.4........2...3.. ; SER=9.0 ; or3 = n4{r9c9 v r9c5 v r3c5}
puzzle#5757: 1...5......71.923..8.2...15..1...67.84..2.........3.....86..9.7.........79...5..4 ; SER=9.0 ; or3 = n6{r9c3 v r9c8 v r1c8}
puzzle#5796: ........94.7...2...89..7.1...8..5...59....6.3.4.3.2.....5.2.8..81..9.5......4..6. ; SER=8.5 ; or3 = n9{r9c1 v r9c7 v r6c7}
puzzle#5895: 1.....7.94....9....8.23...5.7...8..15...2.6.........27...6..1......14.6...6..3... ; SER=7.4 ; or3 = n2{r9c9 v r9c1 v r4c1}
puzzle#5931: ...4....94.71...3..8..3...5...673.5.3..8.....87..9..4......2.6.......5..9.2...8.4 ; SER=8.9 ; or3 = n7{r9c4 v r9c8 v r3c8}
puzzle#6009: ....5.7....71.9.3.6..23...1.3.9.4..7......1..9.1.7......4........26....38...915.4 ; SER=8.3 ; or3 = n3{r9c3 v r9c4 v r5c4}
puzzle#6168: ..3.56.8.4.7.89..6....37....91....4.3......1.8...2............4.1...8.63..6...52. ; SER=8.4 ; or3 = n1{r9c5 v r9c9 v r3c9}
puzzle#6371: .2.4.6.8....1.9..668.......21....9...98.7..5.5..9.3.....23.58...4.7...........1.7 ; SER=8.5 ; or3 = n8{r9c1 v r9c6 v r4c6}
puzzle#6419: 1..4...89.5............75..23...8...5.8.6.3...7....6..3..8...5.71...4..3..5.2.1.4 ; SER=8.4 ; or3 = n6{r8c4 v r8c8 v r3c8}
puzzle#6427: 1...5..8...7.....668.....4.2..79..54......3....4..39.8.....48728.........7.32...5 ; SER=6.6 ; or3 = n6{r9c3 v r9c7 v r4c7}
puzzle#6658: ...4.6..94...8.2..6....3..5.3...85....8....7..4..9..2.3.5.......1....9...6...7..4 ; SER=8.5 ; or3 = n9{r9c4 v r9c3 v r3c3}
puzzle#6762: 1...5.78....18...6..9......278....5.5......1..4...8.673...6.5....2.15..8...9...2. ; SER=8.5 ; or3 = n6{r9c7 v r9c1 v r3c1}
puzzle#6926: .......8..5.1....6.....3....6.5...7..7......891..345...9286.3.1..6..1.......25... ; SER=8.5 ; or3 = n5{r7c1 v r7c8 v r3c8}
puzzle#7035: 12.....8..5......6..9...4....5...6.8.....2.5.7..81..42...3.8....3..27..19..5...7. ; SER=8.3 ; or3 = n6{r9c2 v r9c5 v r5c5}
puzzle#7240: 1...5.7.9.57......6....3.15........83468.........243...3.9...5...5.479..9...1...3 ; SER=9.0 ; or3 = n4{r7c1 v r7c9 v r2c9}
puzzle#7259: .....67...571...3.6........29....8..734....92.1...2..4...6....1....9..489....53.. ; SER=8.5 ; or3 = n6{r9c2 v r9c8 v r4c8}
puzzle#7273: ....5....45.1...36......41.2.8......5..7...2..3..921..3456....2..29.4...97......1 ; SER=8.4 ; or3 = n8{r6c4 v r6c9 v r1c9}
puzzle#7344: ....5...94.7....3..8....4..24.6......98.27..4.659..3.....8.1..38....291..1..6..7. ; SER=8.5 ; or3 = n4{r9c3 v r9c4 v r1c4}
puzzle#7354: ..345.........9...6...7...12.5....7........62..653.....9.8..3..7...4...8.3..27.4. ; SER=8.3 ; or3 = n5{r9c1 v r9c9 v r2c9}
puzzle#7523: 12....7..4....9..6.8....5.426...4...8.1..7.52.7..1.......6.5.7...2.986...9.3..... ; SER=9.0 ; or3 = n1{r9c9 v r9c6 v r3c6}
puzzle#7582: ....5.7...5.1.9..6.89......2..73......6..1..2...9..4..3...17..85...........8...47 ; SER=8.4 ; or3 = n5{r9c7 v r9c6 v r4c6}
puzzle#7625: ...4.6.8...7..9.........5.12.6.3...4.3...8......5..6....1.6....7....136..6...2..7 ; SER=9.0 ; or3 = n5{r9c8 v r9c3 v r5c3}
puzzle#7656: 1.....7.9.5.......6.92.354.2...1....7.6...3....86........8.56......6..1...23..9.. ; SER=8.4 ; or3 = n2{r8c6 v r8c7 v r2c7}
puzzle#7678: .2.4......57.89.....9.7..4.....97......3..1.......46.2.1..6.9...92.....58.5.4..1. ; SER=8.9 ; or3 = n2{r9c7 v r9c6 v r3c6}
puzzle#7772: 1.34....9.....92......7.5...74.3.....6....42.8...........942.1.7...683....63....2 ; SER=9.0 ; or3 = n1{r9c2 v r9c5 v r2c5}
puzzle#8131: ..3......4..189.3...9.2.1.4...9..5...7...8..2..8..5.1...5........4.9.3...6...3.47 ; SER=8.4 ; or3 = n1{r9c1 v r9c5 v r4c5}
puzzle#8313: .2.4.6.8...71.923.6.......52..61.....3...4......2....739.8......1...2.9...2.4.5.. ; SER=8.4 ; or3 = n3{r9c6 v r9c9 v r4c9}
puzzle#8340: ...4....9.5.1.92...8..2...5..6..13.....8...5..7.94.1..7.1....2...2..5....6..14.7. ; SER=8.5 ; or3 = n9{r9c1 v r9c7 v r5c7}
puzzle#8346: 1...567...5.1..2.6.89..74.5.9....6......1....5....48..7....2...8.....9....4.651.2 ; SER=8.5 ; or3 = n8{r9c4 v r9c8 v r1c8}
puzzle#8382: ..3456.....7....36.8.....5....9.8.74....31....9.........2..516.5..2..94..7..1.... ; SER=8.5 ; or3 = n3{r9c1 v r9c9 v r6c9}
puzzle#8422: ...4.6.89.5...9.3..8.3..4...4...8.6.......1..9..5....25...6..2..6...4..8..1.7.... ; SER=8.5 ; or3 = n4{r9c1 v r9c9 v r5c9}
puzzle#8430: ..3..6.8945.1.9....8...74....6.....3.......2.7....59..3.......7.1.73.......5...1. ; SER=8.4 ; or3 = n6{r9c2 v r9c5 v r6c5}
puzzle#8671: ...4567.......92.6...3.....294...1..5......4.7.6.....5..5..89....2....619..27..5. ; SER=8.3 ; or3 = n8{r9c3 v r9c7 v r6c7}
puzzle#8865: .2.4.......71..2.6.8...75.12.......8.4..9.....1.6.8..3..1.7.8.4.........9....56.. ; SER=8.4 ; or3 = n2{r9c3 v r9c9 v r5c9}
puzzle#8894: 1..4..........9.3...93.2.4..1.7....3..5..862..4..6...1...5...6.5..8..9...7......2 ; SER=8.5 ; or3 = n9{r6c1 v r6c8 v r1c8}
puzzle#9043: 1..4.6..9..7.8.2...8......5.....73..7...95..8.4.6.....31.....5...6....2.87....9.4 ; SER=9.0 ; or3 = n5{r9c3 v r9c5 v r1c5}
puzzle#9102: ...4.........892...8......5..8..591.394..1..85......7..3......18..2.3.6.965...32. ; SER=8.4 ; or3 = n4{r9c6 v r9c9 v r2c9}
puzzle#9117: .....67..45..8..3........45.91..4..737......8....6.31.5..7..........86..9.2.....4 ; SER=8.4 ; or3 = n8{r9c2 v r9c7 v r3c7}
puzzle#9240: ...4...8.....892.....3.2..42....7...39.5.8..7.6..2.5.1.3.........42.597........65 ; SER=8.4 ; or3 = n4{r9c7 v r9c6 v r6c6}
puzzle#9271: ..3.......571....66...7.1...6...8....3..1.92...82.35.........9...29.1..59..7..8.2 ; SER=8.4 ; or3 = n1{r9c3 v r9c8 v r4c8}
puzzle#9455: .......89...1..2.6....724....5...9.376...5...9..2.45....6.4..2..1.72.6.......1.5. ; SER=8.5 ; or3 = n3{r8c3 v r8c6 v r2c6}
puzzle#9588: .23......4....9..668.3......1....6....67...289......1.3..6.48..7.......4...92.5.. ; SER=8.9 ; or3 = n1{r9c3 v r9c9 v r3c9}
puzzle#9595: 1...5.7.94..18.....89.724..2..9.5....3..2.1...4......8.......1..9....3.78..23.... ; SER=8.5 ; or3 = n1{r9c6 v r9c3 v r6c3}
puzzle#9663: ..3.........18.2.6.8...2.5....79......5..4....1.5.36...9.......7..96.8...46...1.7 ; SER=9.0 ; or3 = n8{r9c1 v r9c6 v r4c6}
puzzle#9683: 1.34.6.....7....3.6.....45.2....491...1..7..8.6.5.......2.95...7...6.1.....2...7. ; SER=8.4 ; or3 = n9{r8c2 v r8c8 v r1c8}
puzzle#9824: .234.6...4.........8.3..5......6.........715.7.1.4.8......9862...6....159.....4.8 ; SER=9.0 ; or3 = n7{r9c5 v r9c8 v r4c8}
puzzle#9835: 12..56..9..7.8.2.....3...41...8.....3.8....1..9..2.4........8.3...9...5.....68.27 ; SER=8.3 ; or3 = n9{r9c7 v r9c3 v r3c3}
puzzle#9899: ....5..89..718.........25..2....4..38..5...2.9.1.2.....1....46...274.......6.8.5. ; SER=9.0 ; or3 = n8{r8c9 v r8c2 v r3c2}
puzzle#9902: ..34............36....725...7...31.8.4..18.7.8....53..5....762..6.....9..32..4... ; SER=8.4 ; or3 = n5{r9c4 v r9c9 v r5c9}
puzzle#9923: .23.56..94....92..6.......1.9.537.....1.6..52.....1.....2.4..7.7.......5.34...6.. ; SER=7.4 ; or3 = n9{r9c1 v r9c8 v r6c8}
puzzle#9952: .2...6.894......3..89.2.1..2.........7..419...4..62...56..........8.465.8...3...7 ; SER=8.4 ; or3 = n2{r9c3 v r9c8 v r5c8}
puzzle#9978: ..34....9....8........231.5...9.4..8.948...277...3.4...........81...2..494.....53 ; SER=7.4 ; or3 = n8{r9c6 v r9c7 v r1c7}
puzzle#10086: ...45...94.7.8.......7.3...2...1......4...62.9..6....7..2..157.......9..815.....2 ; SER=8.4 ; or3 = n6{r9c6 v r9c8 v r2c8}
puzzle#10140: 1.......9.5718...6.8...3......6...5353..4.8....8....9....2.7..8....1.9....5.6.31. ; SER=8.8 ; or3 = n5{r8c6 v r8c9 v r3c9}
puzzle#10288: .2...6...4...8..3...9...1......4.3...9....47...12.8.9.5.6.9...7.325.4......8..... ; SER=8.6 ; or3 = n6{r9c5 v r9c8 v r3c8}
puzzle#10292: 1....678.4571....6..9......2..9.5...53.84......4.......76.1.4.2...6...1.......6.8 ; SER=8.4 ; or3 = n7{r9c8 v r9c4 v r3c4}
puzzle#10296: .2.4...8..57.892.66.........71.....53..59.........7318.......9..6...2..3..4.7.8.. ; SER=8.5 ; or3 = n2{r9c1 v r9c9 v r5c9}
puzzle#10368: 1.......9.5..89......72.4......7894.3.....8..9.....56...63.7.2.79.........2.4.... ; SER=8.4 ; or3 = n8{r8c4 v r8c8 v r1c8}
puzzle#10375: 1.......94..18.2...897.........7.3.....5.29.4...3...7.762........1..5.4.9.......1 ; SER=8.3 ; or3 = n3{r9c8 v r9c5 v r1c5}
puzzle#10426: .2.4.6...45........89.2....2...1.9..3......4..7..6.3.......7..3......8.2831..56.. ; SER=9.0 ; or3 = n4{r9c5 v r9c9 v r3c9}
puzzle#10580: ...........718.2.668..2...42.45..9...1......7...34...53.......1.4.....9...126.4.. ; SER=9.0 ; or3 = n3{r9c6 v r9c9 v r4c9}
puzzle#10614: ..34.......7..9...68....5..2.1....6.....6.49......48..3..61.......3.5.72.1..4.6.. ; SER=8.9 ; or3 = n7{r9c1 v r9c6 v r3c6}
puzzle#10634: ......78.45...92..6.....5.......16..3..5....2965..4...5...92..77328.......46..... ; SER=8.4 ; or3 = n8{r9c7 v r9c1 v r4c1}
puzzle#10720: ...4......5.....36...7..5.42......57...9.2..8.3.56.9..36..9..7.51.........8.31... ; SER=7.2 ; or3 = n6{r9c4 v r9c8 v r5c8}
puzzle#10756: ...4...8.4....923.6..72...12....81..5...3..6..1....3..34........9.56......2....5. ; SER=9.0 ; or3 = n1{r9c1 v r9c6 v r5c6}
puzzle#10971: ..34....9.57......6......4.2..6....7....9.8...1.2...54...92.4............943..56. ; SER=7.3 ; or3 = n7{r9c1 v r9c6 v r6c6}
puzzle#11042: ...45...9..7........9..214....8....7..5.1.8..8..6..95.34.......516..3...9......23 ; SER=8.4 ; or3 = n6{r9c7 v r9c5 v r2c5}
puzzle#11192: 1..4...8...7.89...6..7.2.5.2.......1.....736..963.......56......4....69.9..8....7 ; SER=8.5 ; or3 = n3{r9c6 v r9c8 v r2c8}
puzzle#11350: ...4.6..9.5.1.......9.324...463..5..5.....9....8....23......85.7.......281.5..3.. ; SER=8.5 ; or3 = n2{r9c5 v r9c3 v r2c3}
puzzle#11437: 12..5.......1...36......41......4..7.1...8...8..6..5....29......7...3.9.96.2....4 ; SER=8.4 ; or3 = n7{r9c5 v r9c7 v r1c7}
puzzle#11521: .2.4...89...1.....68...251..4......7.3..6.9..9...4.32........7...1.2.49.8..3....2 ; SER=8.5 ; or3 = n6{r9c2 v r9c8 v r4c8}
puzzle#11550: .2..5..894.7.8.........2.....4.1...5..19..46..9......37..8....1.4....3.2..52.3... ; SER=8.5 ; or3 = n4{r9c5 v r9c9 v r3c9}
puzzle#11622: ...4.....4.7...2.66...32....9......75.....8....1...94.3...75..8..5..1.7.9.62..15. ; SER=8.4 ; or3 = n9{r8c5 v r8c9 v r1c9}
puzzle#11798: .2.4....94.7..9236..9...5.....3..618...96..52.....1...3.........61..7.9..7.8..... ; SER=9.0 ; or3 = n1{r9c5 v r9c9 v r3c9}
puzzle#11841: 1....67..45.1.9.....8...1.5......4....5.489.2...6.2..3.....1.2.5...2...79.27..... ; SER=8.3 ; or3 = n5{r9c6 v r9c7 v r6c7}
puzzle#12117: .2345...94...8.....98..7.1.2..3....1..6...5...7....39..3.7....8.6...395...4..21.. ; SER=8.4 ; or3 = n9{r9c1 v r9c5 v r5c5}
puzzle#12143: 12...6...4....9....98.....5...37..5.5...6..2..31.2...8......8...15..3.6.9..8...4. ; SER=8.5 ; or3 = n2{r9c3 v r9c6 v r3c6}
puzzle#12176: ...4...894...8...6..8....1......53....9..4..2.1..6....3.5..1.2.76........8..2.57. ; SER=8.9 ; or3 = n4{r9c3 v r9c9 v r6c9}
puzzle#12257: ......7.9...18.23.6.....51.2.9.7.16.5.43.....78.........28.5...8...2169.....4...2 ; SER=8.4 ; or3 = n5{r9c8 v r9c3 v r2c3}
puzzle#12290: ...4.678...7.......9....514.....4...3....5......81..6..3....9.1.46...3.2..27..... ; SER=8.4 ; or3 = n1{r8c1 v r8c6 v r2c6}
puzzle#12689: 1.3.5.7.94...8.2...9..7.....3....4.5..5....6.84.....1.....62..8...9........73.5.1 ; SER=8.9 ; or3 = n8{r9c6 v r9c3 v r3c3}
puzzle#12851: 1....6.8..5.1....6...27......47......3.6..54.8.5.3....3..5....4......97.9....18.2 ; SER=9.0 ; or3 = n6{r9c5 v r9c3 v r3c3}
puzzle#12859: 12.........7..9..66....3.......4.39.3....5....71.....8...9..82..4..1895..3.5..... ; SER=8.4 ; or3 = n4{r9c6 v r9c9 v r3c9}
puzzle#12896: 1..4..7..45...9.3...8.....5..5.4..7..........93..2.8....49..1......34.2.86.5..... ; SER=8.4 ; or3 = n9{r9c3 v r9c8 v r3c8}
puzzle#12934: .....67..4..18...669.2.....2.5.9.4.1......35.84...5.7.3.47..8..5.........8..3..2. ; SER=8.4 ; or3 = n5{r9c7 v r9c4 v r1c4}
puzzle#12976: 1..4.........8...6...27.5..2.....3.871.....6...9.45....8...7...5...1.8..9.2...6.1 ; SER=9.0 ; or3 = n7{r9c2 v r9c8 v r6c8}
puzzle#13269: .2.4.6....5..8.....9......4.19.4.3...8..7...5...9...2.5....36....6.1.5........87. ; SER=9.0 ; or3 = n9{r9c1 v r9c9 v r1c9}
puzzle#13274: ..34..7..45...9..66.....15.....6..97......4..9.67....137.....18.6...1.....42..... ; SER=9.1 ; or3 = n9{r9c2 v r9c5 v r5c5}
puzzle#13337: 1234..........9.......2..542.....3..3.5.6492......8..55.....6.38.1.9.....76....1. ; SER=8.9 ; or3 = n9{r9c1 v r9c9 v r1c9}
puzzle#13397: 1...5.7..4..1....6..8..7..........93.1...5....462....5....42...8..93....9.2.7.34. ; SER=8.9 ; or3 = n1{r9c6 v r9c9 v r3c9}
puzzle#13420: .2.4.6..9.......366.8...4.5.1.9.8....36..4.9.......12.3..7......7.....4.98.5.1.6. ; SER=8.4 ; or3 = n2{r9c5 v r9c3 v r4c3}
puzzle#13421: ..3..6..945..8.2.........1..1.9.8...38......497.5........2..87.....3.6..8....5..1 ; SER=8.4 ; or3 = n3{r9c2 v r9c7 v r4c7}
puzzle#13471: .2......94...8.........7.1.2.9....715....48......3.6..36.8.1....726...4......3..8 ; SER=8.4 ; or3 = n7{r9c7 v r9c5 v r5c5}
puzzle#13554: .2..5.78.4...8...6........1..5..4...3...1.97.7.9.......3...1...87.9...249..2..6.. ; SER=8.3 ; or3 = n3{r9c5 v r9c9 v r4c9}
puzzle#13574: ..3...7..4..1....66...27........5.7...9..8...7.4....15...5..8...1.6....29.68....7 ; SER=8.4 ; or3 = n1{r9c6 v r9c7 v r3c7}
puzzle#13678: .2.4.........8..366....7..1....7.9..3...9...89..8.14..51.2...6...2.6.....6.7....2 ; SER=8.4 ; or3 = n1{r9c7 v r9c5 v r1c5}
puzzle#13841: 1...5..89...1...3.6.8..21..2..8..6...84.....5......3..7...9......17.4.....5.....3 ; SER=9.0 ; or3 = n3{r8c5 v r8c1 v r5c1}
puzzle#13905: .234....9.5.1..2..69..7.....649.75.....5.......5...64....7...2.....43..18...9.... ; SER=8.5 ; or3 = n5{r9c6 v r9c9 v r3c9}
puzzle#14180: .2...6..945.1.......837....2..7..1..3..948.25...........6.94..........548....73.. ; SER=8.9 ; or3 = n5{r9c3 v r9c4 v r6c4}
puzzle#14267: ....5.7..4.......6....23.4...98...5..8...2....34.1.....75...9..8..96.5..9....8.2. ; SER=8.5 ; or3 = n7{r9c9 v r9c5 v r4c5}
puzzle#14426: 12..5....4..1.9..6.98....5...1..84.334..7....8.9.41......6....1....159.........75 ; SER=8.8 ; or3 = n6{r9c3 v r9c7 v r5c7}
puzzle#14434: 1.3...7....7...2.6....2..5..7............4.789.6...34.5...4.....4...592...2.17..5 ; SER=8.5 ; or3 = n9{r9c4 v r9c2 v r3c2}
puzzle#14566: .234...89.....9.3.69.......27.....4.8..2...57.16.4.8......7..64...81.......3..5.. ; SER=8.4 ; or3 = n1{r9c3 v r9c8 v r3c8}
puzzle#14598: ...4.....4....9.3669..2.4..284..197.31...764......81...6.8.....7.2......8...7..9. ; SER=8.5 ; or3 = n2{r9c6 v r9c9 v r6c9}
puzzle#14638: .2..56..9.5.1.....6.87.3.................46.3.64..2.7.3.....4.75..9.7..29.....3.. ; SER=8.5 ; or3 = n6{r8c3 v r8c8 v r2c8}
puzzle#14817: .2.....89..71..2..6..73..4.........33..2.4.........56.5..62..1......5...98...1... ; SER=8.4 ; or3 = n7{r9c5 v r9c9 v r5c9}
puzzle#14831: .2..5..8...718...66....2.4.2..3....1....6.........73..3.9........2...8...8...4913 ; SER=8.4 ; or3 = n6{r9c4 v r9c3 v r4c3}
puzzle#14939: 1......8...7...2.6.9...2........8....8.69..7.....4.59..14...9..8....436...23....7 ; SER=8.4 ; or3 = n9{r9c1 v r9c6 v r2c6}
puzzle#15001: .2....7.9...1892....87..4....63...9.3.5.......7..2....5.......7..9..562...2.....3 ; SER=8.4 ; or3 = n7{r9c1 v r9c6 v r4c6}
puzzle#15007: .2....7..4...8...6...73..1..1....854....9....8...149....59......62.....1.4...75.. ; SER=8.4 ; or3 = n9{r9c8 v r9c3 v r4c3}
puzzle#15146: ..3..6.8.4.....2.6.9.7.2..42.95......46..8..5..5....4..6...59......1.8......63... ; SER=9.0 ; or3 = n9{r9c1 v r9c4 v r1c4}
puzzle#15150: ..345.....5..89.3.6..7.......9.7..4...4...69.8.......7...8...2..125.3...9...6.4.. ; SER=8.9 ; or3 = n7{r8c1 v r8c8 v r1c8}
puzzle#15187: ......7..4.7....36..8.32.1....64..5..8.9....7......8.1.1......2..62.4...9..56.... ; SER=8.5 ; or3 = n1{r9c7 v r9c6 v r5c6}
puzzle#15280: 1.......94...8.2.....2.7..5.7...1....4.328.....1..5..838.....6.6...7.5.3..4..3.9. ; SER=8.4 ; or3 = n2{r9c2 v r9c9 v r4c9}
puzzle#15304: 1..45......7..92..86...7.4.2..5..8.33......9..7.....2.6....8.....89....1.4...5.6. ; SER=8.4 ; or3 = n1{r9c3 v r9c5 v r3c5}
puzzle#15543: 1...56...45...9...8.92.........6.97.......3.....3.8.42..1...4.7.4.....2.97..3.56. ; SER=8.4 ; or3 = n9{r8c9 v r8c4 v r5c4}
puzzle#15672: ....5.7..4....9....6.2..41...8....7.3........97....6.15...6..2..345.......2..1..8 ; SER=8.9 ; or3 = n7{r8c9 v r8c1 v r3c1}
puzzle#15719: ..3.....94..18...6.6....51...83..67.39..4.12.5...........9..3.7....2....9.5..7.4. ; SER=9.0 ; or3 = n3{r9c2 v r9c5 v r3c5}
puzzle#15873: .....6..9.5.1.9.3.....2.41.24..7.8...........7.86.5....8.........6...3.7..2..364. ; SER=8.9 ; or3 = n9{r9c1 v r9c4 v r4c4}
puzzle#15920: ..3......4..1892..8...2..15..5.3.....1...2.7..7...43..5....39......4..53..6....4. ; SER=8.4 ; or3 = n7{r9c5 v r9c9 v r1c9}
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puzzle#16032: ...4.67......8.2..86...2....859...1.3.16.....7....8........3..16..74.8.39.4....7. ; SER=8.5 ; or3 = n6{r9c5 v r9c7 v r4c7}
puzzle#16090: ......78...71....6.6.3..1.4.45......38.2.5...7.6..3.........4926....8..5.3.5.4... ; SER=9.0 ; or3 = n7{r9c5 v r9c9 v r5c9}
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puzzle#16309: 1..4...89....8..3..6.7.3...27..6.....94.12.........3..6.5.......3.6...91.....78.. ; SER=8.5 ; or3 = n1{r9c4 v r9c3 v r4c3}
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puzzle#16410: ..3..6...45...9...8.97......8....31....8..5425...4..6....2.7...7...9.8..94..1..7. ; SER=8.9 ; or3 = n3{r9c4 v r9c9 v r3c9}
puzzle#16483: 1...5...945.1......6...3...2....81...1......2.35...4...8467.......8....7.......23 ; SER=8.5 ; or3 = n1{r9c3 v r9c6 v r6c6}
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puzzle#16630: .2....78.4....9.3..6..32........85....1...6...9.51..2...56.....6...21.....2.4.3.1 ; SER=8.9 ; or3 = n9{r9c1 v r9c4 v r4c4}
puzzle#16635: .2.....8....1.9....6..3.15.2..9....7..4..8.9...16..52..4...1..56..8.........643.. ; SER=8.4 ; or3 = n5{r9c1 v r9c4 v r5c4}
puzzle#16845: 1......89.57......8..2...5...5..39.1.1...4....6.51.......3..62.64..9...37........ ; SER=9.1 ; or3 = n1{r8c4 v r8c8 v r2c8}
puzzle#16868: ..3..6.8.4......3.8..2...54.8....5..6....4.....482...13.29..6......13....4.5...7. ; SER=8.5 ; or3 = n6{r9c3 v r9c5 v r4c5}
puzzle#17022: ..3..6.8.4.7.....68..27..5.26..4...1.3......5..8...62....7.5......3..9.26.29..... ; SER=8.5 ; or3 = n4{r9c6 v r9c8 v r5c8}
puzzle#17106: ..3.5..8...71..2.6.9......4.6..1...5....2.9.1...8..3..5.2....98.8..3..5....5..4.. ; SER=8.3 ; or3 = n7{r9c8 v r9c2 v r5c2}
puzzle#17311: ........9..71..2..8.6...15....53.8.....9...6..1...8..3641....9......5....3524.... ; SER=8.4 ; or3 = n1{r9c6 v r9c8 v r4c8}
puzzle#17505: .2..5...94.7..9.....63....42......1.......4.374.8.1.6.3....75.....94...1.62...... ; SER=8.9 ; or3 = n3{r9c7 v r9c6 v r4c6}
puzzle#17598: .2.45.........9.3.8....31.4.......67..9...5..5..6.2....8...731.7412..6........4.. ; SER=8.3 ; or3 = n8{r9c9 v r9c6 v r5c6}
puzzle#17600: 12..56....57..92.6..6.....42.......853...79...6.8.......5.1...7.....2...9.....8.5 ; SER=9.0 ; or3 = n7{r9c2 v r9c5 v r3c5}
puzzle#17728: .2....7...5.1.9.3.8..7.....2....1.....9...4.871..6..9......85.1...3.5947...6..... ; SER=8.9 ; or3 = n7{r9c2 v r9c6 v r5c6}
puzzle#17929: 12..5...9........6.682.......46.1.7339..4....7....3...5.....8....28......7..6..15 ; SER=8.4 ; or3 = n2{r9c6 v r9c7 v r2c7}
puzzle#18112: 123.......5.....369.8.2.1...8..41.6....6..49....9..5.....7...4.74.8.5..38...6.... ; SER=8.4 ; or3 = n9{r9c2 v r9c9 v r1c9}
puzzle#18340: .2..56.8.4..1.....9.8.....4.7..4..6......85.76..3.5.....1.9.4........3....25...91 ; SER=9.0 ; or3 = n8{r9c7 v r9c2 v r6c2}
puzzle#18363: .2.4..7.......9....68....5.29...831...1...6.8....4......98.2.6.7..56........3.5.. ; SER=8.4 ; or3 = n7{r9c9 v r9c6 v r3c6}
puzzle#18494: ...4...8......9......2.3.542..6..3....4.......6.8.5.1.34.7..5...95.1.82...1...... ; SER=9.0 ; or3 = n2{r9c2 v r9c6 v r5c6}
puzzle#18605: ...45.........9.36....7.1....57....8.9.....4.8.4......3.98......1.6...72.6251.... ; SER=8.4 ; or3 = n7{r9c6 v r9c1 v r5c1}
puzzle#18608: ...45...94.71.......6....5..3..1...76...2...5.....49...7..........2..89...2.936.. ; SER=8.5 ; or3 = n4{r9c2 v r9c9 v r3c9}
puzzle#18658: .2.45..8..5....2..6.9.3.......5..6983...6.45.........17....18....6.....4...7.8... ; SER=8.4 ; or3 = n5{r9c1 v r9c7 v r3c7}
puzzle#18723: 1.3.5......71......89...4.5.9157.6..3....4.....8.....2....6.1..87..1...4..2..5.3. ; SER=8.4 ; or3 = n7{r9c4 v r9c9 v r1c9}
puzzle#18740: .234....945.189...6...3.4..2..........8..1....3.874...........5.92....3..415.39.. ; SER=8.5 ; or3 = n8{r9c8 v r9c1 v r1c1}
puzzle#18843: ...4...8...7.892.........5..4.....1.7....89...35.....63.25.1....1.8.76.2.7..2.... ; SER=8.6 ; or3 = n4{r9c6 v r9c3 v r3c3}
puzzle#19114: 1...5...94....9..3.8.3....5....4..9..6...1..8..586.....9...48..57.6...34...27.... ; SER=8.4 ; or3 = n5{r9c8 v r9c6 v r4c6}
puzzle#19482: ..3..67..45...9..3........1..6..51.8.1.9.......8..3...7.1.6.5...6..9..4..4.2....6 ; SER=8.4 ; or3 = n8{r9c1 v r9c7 v r2c7}
puzzle#19550: ..34.6..94.7....6.....2...5.4...8.1.7.....8....5....923...97...5.8........28..3.1 ; SER=9.0 ; or3 = n7{r8c2 v r8c9 v r4c9}
puzzle#19600: ..34..7.........6..8.7..1...3..9....5...72..8..4....2.....18.76...93.....91...5.. ; SER=8.4 ; or3 = n8{r9c1 v r9c8 v r1c8}
puzzle#19700: .2.4...8.4....92.3..9...1.....8..5...35.17.2....5...3139.........42.1.7.....4...6 ; SER=8.4 ; or3 = n9{r9c7 v r9c4 v r5c4}
puzzle#19909: .234..7..4....9..36.97....12.19..6.8...8....4....63........4..........128..31..9. ; SER=9.0 ; or3 = n4{r9c7 v r9c3 v r6c3}
puzzle#19990: 1.34...8.4.7......69..3......196..48...74.1.2.......57..68...2.....9...6.....25.. ; SER=8.5 ; or3 = n4{r9c9 v r9c3 v r6c3}
puzzle#20081: ......7..45..89.6...82...1.......4.73.45.....56..2....7....1..2...67...19..3..... ; SER=8.4 ; or3 = n6{r9c3 v r9c9 v r5c9}
puzzle#20108: ...4.6.....71...63.9..731..2......9.....4..275....2.36..6..8....3......4..2...67. ; SER=8.9 ; or3 = n2{r8c5 v r8c7 v r2c7}
puzzle#20232: 1.3.....9.5...9.6.6....7154...7..8..3...6..25..5.1...7516..3.....4..1...9.25..... ; SER=8.5 ; or3 = n6{r9c6 v r9c9 v r4c9}
puzzle#20293: ...4..78..5...92.....3.....2.9.3.67...1.9.......7.4.92.7.....1.5..6...28.8...5... ; SER=8.9 ; or3 = n2{r9c3 v r9c4 v r5c4}
puzzle#20528: ..345.7.......9...6..73.....8.5..6.......8.9257....1.8.4.....2..3......596.8..3.7 ; SER=8.3 ; or3 = n2{r9c3 v r9c5 v r6c5}
puzzle#20530: ...4.6.8.....8..6...92...5.2..96.5.7.....23..94......6..2.......1..23...6..7.54.. ; SER=8.4 ; or3 = n8{r8c4 v r8c9 v r5c9}
puzzle#20553: ..34.6...4..1....38....7.1.2.8....9..4..12..85.1.7....6....38.7....2..5.......6.. ; SER=8.4 ; or3 = n1{r9c6 v r9c9 v r4c9}
puzzle#20559: 12..5.....57..9.6......7..42...4..75...61.....8.3...2................3.7678.....2 ; SER=9.0 ; or3 = n4{r9c6 v r9c8 v r5c8}
puzzle#20680: ...4....9....8.2..8....2154..69183.......7948.....4....34..18.66......3.7.2...... ; SER=8.9 ; or3 = n8{r9c4 v r9c2 v r6c2}
puzzle#20810: .2.456.8.........389.32..5...9...4..5......266..8.2....1.9.4.....8.3.....6.5..... ; SER=8.4 ; or3 = n9{r9c1 v r9c8 v r6c8}
puzzle#21053: ..3.5.......1893...8..7......4...8...9.....52.1..42....4..9...75....196....7....8 ; SER=8.9 ; or3 = n5{r9c7 v r9c6 v r4c6}
puzzle#21056: ..3.5......71.9.2..8..7..4..7.....9.516.......3...2.57...7....8....9526.....61... ; SER=8.4 ; or3 = n8{r9c4 v r9c3 v r6c3}
puzzle#21061: 1.3..6.......8..2.6....34.123.5.89....1.......4..62....1.6...97..293.1....4.....5 ; SER=7.3 ; or3 = n6{r9c2 v r9c7 v r2c7}
puzzle#21151: 12..5.7.....1.93.....7..5......9.8.77..84..3..91.....43....7..85..........6...2.. ; SER=7.3 ; or3 = n6{r8c8 v r8c5 v r2c5}
puzzle#21158: 1.3..67...5.18..2...9.......945.3..8...2..9.4..5......5..9....2.....56...71....3. ; SER=8.5 ; or3 = n9{r9c1 v r9c9 v r1c9}
Last edited by denis_berthier on Tue Nov 09, 2021 4:13 am, edited 1 time in total.
denis_berthier
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Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

denis_berthier wrote:Shye's pattern itself (with conclusion the OR of 3 candidates) is very general and it may have some other uses yet to be found. I therefore tried to estimate its frequency.
For this I used my cbg000 collection of the 21375 first controlled-bias generated puzzles.
[...]
I found 237 (1.1%) puzzles with this partial pattern. I think that's quite a lot for experimenting.

thats quite a surprise to me also, considering i've only ever seen (with exception to the krakens) these in hand-crafted puzzles. i'll have a manual sift through some of these later and see if anything interesting comes up

999_Springs wrote:seems like a nice technique after following the examples

totuan wrote:Very nice find!
Someone can find many way to present the same deductions, but based on your pattern I can find deductions quite easier and faster. Your examples are very nice and I’m especially like Cobra Roll, Roman Candle V2, Pear and Rocket.

thanks for the kind words! appreciate it a lot ヽ(´▽`)/

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

shye wrote:
denis_berthier wrote:Shye's pattern itself (with conclusion the OR of 3 candidates) is very general and it may have some other uses yet to be found. I therefore tried to estimate its frequency.
For this I used my cbg000 collection of the 21375 first controlled-bias generated puzzles.
[...]
I found 237 (1.1%) puzzles with this partial pattern. I think that's quite a lot for experimenting.

thats quite a surprise to me also, considering i've only ever seen (with exception to the krakens) these in hand-crafted puzzles. i'll have a manual sift through some of these later and see if anything interesting comes up

In order to make tests easier, I've added the "or3" relation obtained in each case.
denis_berthier
2010 Supporter

Posts: 4072
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

denis_berthier wrote:In order to make tests easier, I've added the "or3" relation obtained in each case.

thanks for adding it! makes things much more convenient

i've gone through down to puzzle #6168 had for each puzzle highlighted the detected 3-pos firework and looked for potential patterns to build off them. a lot of them had 1 or 2 positions in the intersecting box and would therefore likely be replaceable by easier chains, so i discarded those. the ones that weren't like this had little ability to chain from the intersection cell, so i looked for other overlapping fireworks. i think thats the real power of these because it deals with the link you need to draw from the intersection cell, in every example i had (bar the krakens) this property existed. some of these i could get very close to a deduction, but just not enough. perhaps i missed things though

another thing, i noticed in many of the puzzles there were other available 3-pos fireworks in different areas of the grid to what was detected, and they were sometimes more powerful too. i assume the search was looking through columns 9 to 1 then rows 9 to 1 (pretty much all of them were in row 9 and a high value column), i would have suspected it to be a bit more sporadic otherwise. if so then i would also expect a higher result of puzzles with the pattern, even if theyre just as useless

so not very exciting results at the moment unfortunately, sorry to report. but its helped me realise why certain firework techniques are useful yet most occurrences of the pattern are not, i'll edit the main post eventually with this information

shye

Posts: 297
Joined: 12 June 2021

### Re: Fireworks

shye wrote:i've gone through down to puzzle #6168 had for each puzzle highlighted the detected 3-pos firework and looked for potential patterns to build off them. a lot of them had 1 or 2 positions in the intersecting box and would therefore likely be replaceable by easier chains, so i discarded those.

You mean there were only 1 or 2 candidates with the right digit in block 1? If that's it, I could easily eliminate these cases from my list.

shye wrote:another thing, i noticed in many of the puzzles there were other available 3-pos fireworks in different areas of the grid to what was detected, and they were sometimes more powerful too.

I stopped the search in a puzzle as soon as the pattern was detected once. I have no control on the order of the search. I could probably do a full search as easily.

shye wrote: its helped me realise why certain firework techniques are useful yet most occurrences of the pattern are not, i'll edit the main post eventually with this information

If you can refine the conditions, I can re-launch the search.
denis_berthier
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Joined: 19 June 2007
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