The New Sudoku Players' Forum

by **shye** » Wed Nov 03, 2021 2:18 am

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 - Qinlux skfr 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_al skfr 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 | '------------------------'--------------------'--------------------'

firework quadruple
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 - shye skfr 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 - shye skfr 8.6 9...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 - shye skfr 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 - shye skfr 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_al skfr 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

Website · by **denis_berthier** » Wed Nov 03, 2021 5:15 am

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 . . | . . . | . . . | +-------+-------+-------+

by **999_Springs** » Wed Nov 03, 2021 6:07 am

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

by **shye** » Wed Nov 03, 2021 6:29 am

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

Website · by **denis_berthier** » Wed Nov 03, 2021 6:40 am

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

OK, after seeing the logical fallacy, I hadn't read the examples.

[Edit:] I see it has been corrected.

by **shye** » Wed Nov 03, 2021 12:51 pm

[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 - RSP skfr 7.1 1...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

by **shye** » Thu Nov 04, 2021 5:00 pm

this deduction was moved to the main post

Website · by **denis_berthier** » Fri Nov 05, 2021 7:55 am

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≠9 stte

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≠3 stte

Website · by **denis_berthier** » Fri Nov 05, 2021 8:06 am

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≠3 stte

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≠2 stte

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≠2 stte

by **totuan** » Fri Nov 05, 2021 10:19 am

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

Website · by **denis_berthier** » Mon Nov 08, 2021 10:46 am

.
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

by **shye** » Mon Nov 08, 2021 5:22 pm

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(´▽`)/

Website · by **denis_berthier** » Tue Nov 09, 2021 4:17 am

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.

by **shye** » Wed Nov 10, 2021 2:27 pm

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

Website · by **denis_berthier** » Wed Nov 10, 2021 5:22 pm

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.

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