## Fireworks

Advanced methods and approaches for solving Sudoku puzzles

### Re: Fireworks

denis_berthier wrote:
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.

that would be wonderful! i think searching for the following will yield the most interesting results:

for a row and column, (at least) two candidates are limited to one position per line outside the intersecting box, and within the box have at least two positions per line

Code: Select all
`+-------+-------+-------+| . . . | 1 2 . | 4 5 . || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || . . . | . . . | . . 9 |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || . . . | . . 8 | . . . |+-------+-------+-------+`

example
8 and 9 in r1/c1/b1
two positions for each 8 and 9 outside b1, one per line (8r1c9, 8r6c1 & 9r1c6, 9r9c1)
and two or more positions inside b1, per line (8r123c1, 8r1c123 & 9r123c1, 9r1c123)

the more they overlap externally the more likely it will have better results but this is already definitely refined enough, so i think the slight wiggle room will be fine ^^

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

Let's make sure we understand the same things.

As before: a row r1, a column c1 and the block b1 containing their intersection.
New: two digits n1 and n2 instead of one..

For each digit ni, similar pattern as before:
- only one occurrence in r1 outside b1 (not necessarily at the same places for n1 and n2)
- only one occurrence in c1 outside b1 (not necessarily at the same places for n1 and n2)
but in b1 refined to:
- at least two occurrences in r1 (not necessarily at the same places for n1 and n2)
- at least two occurrences in c1 (not necessarily at the same places for n1 and n2)
What's new is:
- n1 or n2 may but don't have to be candidates in r1c1 - which I assumed in my previous search
- there must at least 2 occurrences in b1 (at least one in r1, at least one in c1);note that this is not necessary to prove the OR3 conclusion.

As there are more CSP-Variables to define the pattern, its size becomes 6. It's difficult to know how many branches one would have to add to make it useful, but I'll assume that its 3 as before.
That puts it just before patterns of length 9.
The problem with this is, this will exclude all the puzzles in W8 or less. After checking the W ratings, this will leave only 7 potential puzzles.

In a first stage it'd probably be more instructive to consider only 1 digit with the new conditions.

As I have no idea how you plan to use the version with two digits, I let you decide what you're interested in.
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

denis_berthier wrote:- n1 or n2 may but don't have to be candidates in r1c1 - which I assumed in my previous search

sorry i failed to mention that, they should each appear in the intersection

denis_berthier wrote:As I have no idea how you plan to use the version with two digits, I let you decide what you're interested in.

the property i have in mind is by creating a weak link in the intersection (r1c1 in the example) between the two candidates, the only chain ends we need to worry about are outside the intersecting box (b1)

denis_berthier wrote:As there are more CSP-Variables to define the pattern, its size becomes 6. It's difficult to know how many branches one would have to add to make it useful, but I'll assume that its 3 as before.
That puts it just before patterns of length 9.
The problem with this is, this will exclude all the puzzles in W8 or less. After checking the W ratings, this will leave only 7 potential puzzles.

In a first stage it'd probably be more instructive to consider only 1 digit with the new conditions.

i hadnt considered the complexity scaling, instead of 3 branches i'd think of it as 2; since theyre overlapping at the intersection that deals with one branch already. does that open up more options?

thanks for running a clarity check

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

shye wrote:
denis_berthier wrote:- n1 or n2 may but don't have to be candidates in r1c1 - which I assumed in my previous search

sorry i failed to mention that, they should each appear in the intersection

OK

shye wrote:
denis_berthier wrote:As I have no idea how you plan to use the version with two digits, I let you decide what you're interested in.

the property i have in mind is by creating a weak link in the intersection (r1c1 in the example) between the two candidates, the only chain ends we need to worry about are outside the intersecting box (b1)

OK, so that can make as little as 2 more "strong links" or maybe even 1 in some cases.

shye wrote:
denis_berthier wrote:As there are more CSP-Variables to define the pattern, its size becomes 6. It's difficult to know how many branches one would have to add to make it useful, but I'll assume that its 3 as before.
That puts it just before patterns of length 9.
The problem with this is, this will exclude all the puzzles in W8 or less. After checking the W ratings, this will leave only 7 potential puzzles.
In a first stage it'd probably be more instructive to consider only 1 digit with the new conditions.

i hadnt considered the complexity scaling, instead of 3 branches i'd think of it as 2; since theyre overlapping at the intersection that deals with one branch already. does that open up more options?

Yes.
If we consider it will be used with only 2 more "strong links", that makes 52 more potential puzzles.
If we consider it will be used with only 1 more "strong links", that makes 52+168 more potential puzzles.

I don't have time today, but I'll try to write the rule and run the calculations tomorrow.
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

marvelous. no rush of course, thank you for your interest and cooperation

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

.
It didn't take long to adapt the code for one or3 pattern to a double one in the same row and column. I let the computations run during the night.
The results are quite surprising: there are more such double patterns than I expected.
I used the same convention as before: the detection occurs after all the rules with 5 or less CSP-Variables have been fired (which a priori eliminates all the puzzles in W5 or less).
Among the 21375 puzzles in cbg-000, 147 (0.7%) have this pattern (not necessarily at the start, but it appears during resolution). In many cases, it appears several times.
Here is the full list:
Hidden Text: 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 ; W=6; double-or3: n5{r9c6!r9c7!r3c7}+n6{r9c1!r9c7!r4c7} ; n2{r9c7!r9c1!r4c1}+n9{r9c6!r9c1!r2c1} ; n3{r4c2!r4c4!r8c4}+n4{r4c2!r4c4!r1c4} ;
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 ; W=6; double-or3: n1{r5c6!r5c9!r1c9}+n5{r5c3!r5c9!r3c9} ;
puzzle#109: .23....8.4....9..378....5......75.3.......215...61...7.6.5.1....42.3....9....4..8 ; SER=9.0 ; W=9; double-or3: n7{r7c3!r7c8!r2c8}+n9{r7c5!r7c8!r6c8} ;
puzzle#119: .2.....8...678.........35.4.1....4....527.....9..6.237......6.1.68....4.9...3..7. ; SER=8.4 ; W=6; double-or3: n3{r2c1!r2c9!r8c9}+n9{r2c6!r2c9!r8c9} ; n1{r2c6!r2c8!r5c8}+n2{r2c6!r2c8!r7c8} ;
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 ; W=6; double-or3: n4{r6c4!r6c7!r2c7}+n8{r6c6!r6c7!r7c7} ; n7{r1c1!r1c8!r8c8}+n8{r1c2!r1c8!r6c8} ;
puzzle#400: .2.4..........91.......25.6...5.18...3.24...............7...6.8.128.4.9.8.56....2 ; SER=8.9 ; W=6; double-or3: n4{r4c3!r4c8!r3c8}+n7{r4c2!r4c8!r9c8} ; n4{r2c9!r2c1!r6c1}+n5{r2c5!r2c1!r5c1} ; n4{r3c8!r3c1!r6c1}+n7{r3c5!r3c1!r5c1} ; n1{r6c9!r6c1!r1c1}+n9{r6c5!r6c1!r3c1} ; n1{r6c9!r6c3!r1c3}+n9{r6c5!r6c3!r3c3} ;
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 ; W=7; double-or3: n1{r8c6!r8c9!r1c9}+n2{r8c1!r8c9!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 ; W=6; double-or3: n3{r6c2!r6c9!r2c9}+n8{r6c6!r6c9!r9c9} ;
puzzle#1022: ..34.6..9....8..23.8....64..3.8.5.1.6.......8..5.1.........85........9329.2.73... ; SER=8.7 ; W=6; double-or3: n1{r1c7!r1c1!r8c1}+n2{r1c5!r1c1!r4c1} ;
puzzle#1034: .2......9..678....7..1...65....6.9...8.5.4......9.3..15.....347..7...5...42....1. ; SER=8.4 ; W=6; double-or3: n1{r8c1!r8c6!r4c6}+n8{r8c1!r8c6!r4c6} ;
puzzle#1203: 12..56.8..........7.8...5.6......39....97...4..5.2.....7...84..5....36..68......1 ; SER=8.4 ; W=6; double-or3: n4{r8c5!r8c3!r1c3}+n9{r8c5!r8c3!r1c3} ; n2{r3c8!r3c6!r9c6}+n9{r3c2!r3c6!r9c6} ;
puzzle#1330: 12.45...9..67.9....98.32.....1..7..5......3.....32...6.....4.57..4...9..8..5732.. ; SER=9.0 ; W=6; double-or3: n4{r5c5!r5c2!r2c2}+n7{r5c8!r5c2!r8c2} ; n6{r3c4!r3c7!r7c7}+n7{r3c1!r3c7!r6c7} ; n6{r1c6!r1c7!r7c7}+n7{r1c3!r1c7!r6c7} ;
puzzle#1366: ...4..7..456.8.....9....5..2..69...1..7...9..9......3..4126...........12.....36.5 ; SER=8.4 ; W=6; double-or3: n4{r8c7!r8c5!r5c5}+n7{r8c2!r8c5!r6c5} ;
puzzle#1884: 12.........6...1...98.3.....1.8..6.5........7..56.3..2..7...29156..9......2.47... ; SER=8.4 ; W=6; double-or3: n3{r7c4!r7c2!r5c2}+n8{r7c6!r7c2!r5c2} ; n3{r5c7!r5c3!r1c3}+n4{r5c4!r5c3!r1c3} ; n4{r2c1!r2c4!r5c4}+n9{r2c9!r2c4!r5c4} ;
puzzle#1908: .2.4.......6....2.79..31.6.......69.......8.1....95.3.3.7..4....4.97...89....8... ; SER=9.0 ; W=7; double-or3: n2{r5c1!r5c6!r8c6}+n3{r5c2!r5c6!r8c6} ; n7{r4c2!r4c4!r2c4}+n8{r4c1!r4c4!r2c4} ; n3{r2c2!r2c9!r9c9}+n9{r2c6!r2c9!r7c9} ; n3{r1c3!r1c9!r9c9}+n9{r1c6!r1c9!r7c9} ;
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 ; W=6; double-or3: n3{r2c9!r2c5!r4c5}+n8{r2c2!r2c5!r5c5} ;
puzzle#2303: .....6.8....7...2.897.....52.4.......856....1.7.3.....5.....2.7....9.3....1..7.6. ; SER=8.5 ; W=6; double-or3: n2{r9c2!r9c4!r3c4}+n8{r9c9!r9c4!r4c4} ; n4{r7c2!r7c4!r1c4}+n8{r7c3!r7c4!r4c4} ; n1{r7c8!r7c4!r3c4}+n8{r7c3!r7c4!r4c4} ; n1{r7c8!r7c4!r3c4}+n4{r7c2!r7c4!r1c4} ; n5{r4c7!r4c6!r8c6}+n9{r4c7!r4c6!r2c6} ; n1{r8c8!r8c4!r3c4}+n5{r8c8!r8c4!r4c4} ;
puzzle#2412: 12..5.....5.7.9.....9.2.5...17.48...3..6..........24.15....4....78...95......781. ; SER=9.0 ; W=7; double-or3: n2{r9c9!r9c1!r4c1}+n4{r9c9!r9c1!r2c1} ;
puzzle#2467: ..3.....94...8.1..78.....4...4.71.6..1......3.....5......8..29.....92..4.6.5.43.. ; SER=7.4 ; W=6; double-or3: n4{r6c7!r6c5!r1c5}+n6{r6c1!r6c5!r7c5} ; n4{r6c7!r6c4!r1c4}+n6{r6c1!r6c4!r8c4} ; n4{r5c7!r5c5!r1c5}+n6{r5c1!r5c5!r7c5} ; n4{r5c7!r5c4!r1c4}+n6{r5c1!r5c4!r8c4} ;
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 ; W=6; double-or3: n1{r2c1!r2c7!r9c7}+n3{r2c3!r2c7!r8c7} ; n4{r2c8!r2c1!r5c1}+n5{r2c7!r2c1!r5c1} ;
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 ; W=6; double-or3: n2{r3c5!r3c9!r6c9}+n5{r3c2!r3c9!r6c9} ; n4{r1c9!r1c6!r5c6}+n9{r1c9!r1c6!r5c6} ;
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 ; W=6; double-or3: n4{r2c1!r2c9!r8c9}+n6{r2c3!r2c9!r8c9} ; n5{r6c9!r6c3!r8c3}+n7{r6c8!r6c3!r8c3} ;
puzzle#2679: .....6......7.913..8.1...462.....9...61...3..97......5...82......73.52....2.6.45. ; SER=8.5 ; W=6; double-or3: n7{r7c6!r7c9!r5c9}+n9{r7c2!r7c9!r1c9} ;
puzzle#2903: ..34...8945.........91........8...4.....9.3.......562736......1.94.1....8..523... ; SER=8.4 ; W=6; double-or3: n5{r7c3!r7c8!r3c8}+n7{r7c5!r7c8!r3c8} ;
puzzle#2929: ......7.9.......3.7..13.56.2.8..59..5..8.3.7.61........4.......86159........486.. ; SER=8.4 ; W=6; double-or3: n2{r6c7!r6c5!r7c5}+n7{r6c3!r6c5!r7c5} ;
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 ; W=6; double-or3: n1{r9c8!r9c5!r3c5}+n7{r9c8!r9c5!r3c5} ;
puzzle#3546: .........45..8.......2136....4..8.....1..237.97.3..2.8......9....8.67..36..13.... ; SER=7.3 ; W=6; double-or3: n2{r2c3!r2c9!r9c9}+n9{r2c6!r2c9!r4c9} ;
puzzle#3704: ..3.......5...91.2..92..6..2....897.....74....4.6.......5.....781....5....23.5.6. ; SER=7.3 ; W=6; double-or3: n2{r8c8!r8c5!r6c5}+n9{r8c8!r8c5!r6c5} ;
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 ; W=6; double-or3: n2{r1c7!r1c2!r6c2}+n8{r1c8!r1c2!r6c2} ;
puzzle#4000: ..3....894....91...8.2........9....6.....2395.6583.....9...7....1.64..7...7...5.. ; SER=8.4 ; W=6; double-or3: n1{r3c1!r3c5!r4c5}+n5{r3c8!r3c5!r4c5} ;
puzzle#4041: ..34........7..1.2.8..1...........6...5..8.9794...7.....1...84.57.....16.64..5... ; SER=8.4 ; W=6; double-or3: n3{r3c6!r3c8!r6c8}+n5{r3c4!r3c8!r6c8} ;
puzzle#4196: .2.4.......6....3....231....1.....9.3...7...55.8..32.....9..4..8...2.....71.6.8.. ; SER=8.4 ; W=6; double-or3: n5{r1c5!r1c7!r8c7}+n7{r1c1!r1c7!r8c7} ;
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 ; W=6; double-or3: n6{r9c8!r9c1!r6c1}+n9{r9c5!r9c1!r6c1} ;
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 ; W=6; double-or3: n2{r6c5!r6c7!r2c7}+n9{r6c2!r6c7!r8c7} ; n4{r4c3!r4c6!r7c6}+n5{r4c9!r4c6!r2c6} ;
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 ; W=6; double-or3: n4{r1c9!r1c4!r6c4}+n5{r1c8!r1c4!r4c4} ;
puzzle#5035: ..3....8...6..91...7......428.97.4.............76.8.2.6.....27.....6...3..512.... ; SER=8.8 ; W=6; double-or3: n3{r7c2!r7c6!r4c6}+n5{r7c9!r7c6!r4c6} ;
puzzle#5056: ..34.6.8.....8...17...236...1..3.9....8..2.1..7........91...4.764...789....2.4... ; SER=8.6 ; W=6; double-or3: n2{r2c2!r2c7!r6c7}+n7{r2c4!r2c7!r5c7} ;
puzzle#5236: 1.......9.56.8......91.....2483.51.....2...6.....4.....65..74.2.7...4.......236.. ; SER=8.3 ; W=6; double-or3: n5{r8c5!r8c9!r6c9}+n8{r8c1!r8c9!r6c9} ; n1{r8c5!r8c8!r2c8}+n8{r8c1!r8c8!r1c8} ; n1{r8c5!r8c8!r2c8}+n3{r8c1!r8c8!r2c8} ; n3{r8c1!r8c8!r2c8}+n8{r8c1!r8c8!r1c8} ;
puzzle#5660: 12..5....45.1.92....923....2..8..96...596...7....2.5......4..72.......18...3..... ; SER=8.5 ; W=6; double-or3: n1{r9c6!r9c3!r6c3}+n4{r9c7!r9c3!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 ; W=6; double-or3: n3{r5c1!r5c5!r3c5}+n9{r5c7!r5c5!r8c5} ;
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 ; W=6; double-or3: n1{r9c9!r9c5!r3c5}+n4{r9c9!r9c5!r3c5} ;
puzzle#5757: 1...5......71.923..8.2...15..1...67.84..2.........3.....86..9.7.........79...5..4 ; SER=9.0 ; W=6; double-or3: n3{r5c3!r5c9!r8c9}+n9{r5c3!r5c9!r1c9} ; n6{r3c1!r3c6!r5c6}+n7{r3c7!r3c6!r8c6} ;
puzzle#5895: 1.....7.94....9....8.23...5.7...8..15...2.6.........27...6..1......14.6...6..3... ; SER=7.4 ; W=6; double-or3: n2{r8c9!r8c1!r4c1}+n7{r8c4!r8c1!r3c1} ;
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 ; W=7; double-or3: n2{r4c7!r4c2!r1c2}+n9{r4c7!r4c2!r2c2} ; n7{r3c8!r3c6!r8c6}+n9{r3c3!r3c6!r8c6} ;
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 ; W=6; double-or3: n1{r8c8!r8c1!r1c1}+n7{r8c8!r8c1!r5c1} ;
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 ; W=6; double-or3: n4{r9c2!r9c5!r5c5}+n9{r9c1!r9c5!r5c5} ; n2{r4c1!r4c9!r3c9}+n8{r4c4!r4c9!r9c9} ;
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 ; W=6; double-or3: n2{r9c8!r9c6!r5c6}+n8{r9c1!r9c6!r4c6} ; n2{r6c5!r6c7!r2c7}+n4{r6c3!r6c7!r3c7} ; n1{r3c3!r3c9!r5c9}+n9{r3c3!r3c9!r7c9} ;
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 ; W=6; double-or3: n4{r3c8!r3c1!r6c1}+n8{r3c5!r3c1!r9c1} ;
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 ; W=6; double-or3: n1{r4c9!r4c1!r1c1}+n7{r4c4!r4c1!r8c1} ; n1{r6c6!r6c1!r1c1}+n7{r6c4!r6c1!r8c1} ;
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 ; W=6; double-or3: n1{r2c4!r2c8!r4c8}+n2{r2c4!r2c8!r7c8} ;
puzzle#7240: 1...5.7.9.57......6....3.15........83468.........243...3.9...5...5.479..9...1...3 ; SER=9.0 ; W=7; double-or3: n1{r5c6!r5c9!r8c9}+n7{r5c5!r5c9!r7c9} ;
puzzle#7259: .....67...571...3.6........29....8..734....92.1...2..4...6....1....9..489....53.. ; SER=8.5 ; W=7; double-or3: n3{r7c6!r7c1!r1c1}+n5{r7c8!r7c1!r6c1} ;
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 ; W=6; double-or3: n5{r7c7!r7c1!r3c1}+n9{r7c5!r7c1!r3c1} ; n1{r2c4!r2c9!r6c9}+n6{r2c6!r2c9!r8c9} ;
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 ; W=6; double-or3: n5{r9c9!r9c1!r6c1}+n7{r9c5!r9c1!r3c1} ; n1{r3c8!r3c4!r8c4}+n2{r3c8!r3c4!r6c4} ; n1{r3c8!r3c4!r8c4}+n7{r3c3!r3c4!r8c4} ; n2{r3c8!r3c4!r6c4}+n7{r3c3!r3c4!r8c4} ;
puzzle#7582: ....5.7...5.1.9..6.89......2..73......6..1..2...9..4..3...17..85...........8...47 ; SER=8.4 ; W=6; double-or3: n2{r9c5!r9c2!r1c2}+n6{r9c5!r9c2!r1c2} ;
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 ; W=7; double-or3: n1{r1c5!r1c2!r4c2}+n5{r1c5!r1c2!r4c2} ;
puzzle#7656: 1.....7.9.5.......6.92.354.2...1....7.6...3....86........8.56......6..1...23..9.. ; SER=8.4 ; W=7; double-or3: n3{r8c9!r8c1!r2c1}+n5{r8c9!r8c1!r6c1} ; n3{r2c1!r2c8!r7c8}+n6{r2c6!r2c8!r4c8} ; n4{r5c9!r5c4!r1c4}+n9{r5c8!r5c4!r8c4} ;
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 ; W=6; double-or3: n7{r5c2!r5c8!r1c8}+n9{r5c1!r5c8!r1c8} ;
puzzle#7772: 1.34....9.....92......7.5...74.3.....6....42.8...........942.1.7...683....63....2 ; SER=9.0 ; W=9; double-or3: n3{r6c2!r6c9!r2c9}+n7{r6c4!r6c9!r7c9} ;
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 ; W=6; double-or3: n3{r8c9!r8c4!r3c4}+n4{r8c9!r8c4!r1c4} ;
puzzle#8430: ..3..6.8945.1.9....8...74....6.....3.......2.7....59..3.......7.1.73.......5...1. ; SER=8.4 ; W=7; double-or3: n8{r4c1!r4c4!r7c4}+n9{r4c2!r4c4!r7c4} ;
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 ; W=7; double-or3: n7{r8c9!r8c4!r3c4}+n8{r8c7!r8c4!r4c4} ; n7{r7c9!r7c4!r3c4}+n8{r7c7!r7c4!r4c4} ;
puzzle#9102: ...4.........892...8......5..8..591.394..1..85......7..3......18..2.3.6.965...32. ; SER=8.4 ; W=6; double-or3: n5{r7c7!r7c5!r1c5}+n9{r7c8!r7c5!r6c5} ;
puzzle#9588: .23......4....9..668.3......1....6....67...289......1.3..6.48..7.......4...92.5.. ; SER=8.9 ; W=7; double-or3: n1{r3c5!r3c7!r8c7}+n2{r3c6!r3c7!r8c7} ;
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 ; W=9; double-or3: n1{r7c6!r7c3!r3c3}+n2{r7c9!r7c3!r4c3} ; n7{r5c2!r5c8!r1c8}+n8{r5c4!r5c8!r1c8} ; n7{r2c6!r2c3!r6c3}+n9{r2c6!r2c3!r6c3} ; n5{r2c6!r2c2!r8c2}+n7{r2c6!r2c2!r5c2} ;
puzzle#9824: .234.6...4.........8.3..5......6.........715.7.1.4.8......9862...6....159.....4.8 ; SER=9.0 ; W=8; double-or3: n4{r5c9!r5c2!r7c2}+n6{r5c9!r5c2!r2c2} ;
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 ; W=6; double-or3: n4{r5c2!r5c5!r7c5}+n9{r5c7!r5c5!r3c5} ;
puzzle#9923: .23.56..94....92..6.......1.9.537.....1.6..52.....1.....2.4..7.7.......5.34...6.. ; SER=7.4 ; W=6; double-or3: n4{r3c8!r3c4!r6c4}+n7{r3c2!r3c4!r9c4} ; n3{r3c8!r3c4!r8c4}+n7{r3c2!r3c4!r9c4} ; n3{r3c8!r3c4!r8c4}+n4{r3c8!r3c4!r6c4} ; n4{r1c4!r1c8!r8c8}+n8{r1c4!r1c8!r4c8} ;
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 ; W=7; double-or3: n1{r8c1!r8c8!r4c8}+n6{r8c5!r8c8!r3c8} ; n6{r8c5!r8c7!r6c7}+n8{r8c1!r8c7!r1c7} ;
puzzle#10292: 1....678.4571....6..9......2..9.5...53.84......4.......76.1.4.2...6...1.......6.8 ; SER=8.4 ; W=6; double-or3: n3{r9c1!r9c4!r6c4}+n5{r9c3!r9c4!r1c4} ;
puzzle#10296: .2.4...8..57.892.66.........71.....53..59.........7318.......9..6...2..3..4.7.8.. ; SER=8.5 ; W=7; double-or3: n2{r6c4!r6c1!r9c1}+n4{r6c5!r6c1!r2c1} ; n2{r5c3!r5c8!r9c8}+n6{r5c3!r5c8!r9c8} ;
puzzle#10368: 1.......9.5..89......72.4......7894.3.....8..9.....56...63.7.2.79.........2.4.... ; SER=8.4 ; W=6; double-or3: n1{r3c6!r3c8!r5c8}+n5{r3c6!r3c8!r8c8} ; n1{r2c4!r2c8!r5c8}+n3{r2c3!r2c8!r8c8} ;
puzzle#10375: 1.......94..18.2...897.........7.3.....5.29.4...3...7.762........1..5.4.9.......1 ; SER=8.3 ; W=6; double-or3: n3{r3c6!r3c8!r9c8}+n5{r3c5!r3c8!r4c8} ;
puzzle#10426: .2.4.6...45........89.2....2...1.9..3......4..7..6.3.......7..3......8.2831..56.. ; SER=9.0 ; W=7; double-or3: n4{r8c5!r8c2!r4c2}+n6{r8c4!r8c2!r4c2} ;
puzzle#10614: ..34.......7..9...68....5..2.1....6.....6.49......48..3..61.......3.5.72.1..4.6.. ; SER=8.9 ; W=6; double-or3: n2{r9c3!r9c6!r5c6}+n7{r9c1!r9c6!r3c6} ;
puzzle#10634: ......78.45...92..6.....5.......16..3..5....2965..4...5...92..77328.......46..... ; SER=8.4 ; W=6; double-or3: n1{r8c5!r8c7!r6c7}+n4{r8c5!r8c7!r5c7} ; n3{r7c4!r7c7!r6c7}+n4{r7c4!r7c7!r5c7} ;
puzzle#10720: ...4......5.....36...7..5.42......57...9.2..8.3.56.9..36..9..7.51.........8.31... ; SER=7.2 ; W=6; double-or3: n6{r1c6!r1c3!r4c3}+n7{r1c7!r1c3!r6c3} ;
puzzle#11042: ...45...9..7........9..214....8....7..5.1.8..8..6..95.34.......516..3...9......23 ; SER=8.4 ; W=6; double-or3: n2{r1c7!r1c2!r6c2}+n3{r1c7!r1c2!r6c2} ;
puzzle#11192: 1..4...8...7.89...6..7.2.5.2.......1.....736..963.......56......4....69.9..8....7 ; SER=8.5 ; W=6; double-or3: n4{r5c5!r5c3!r3c3}+n8{r5c9!r5c3!r8c3} ;
puzzle#11437: 12..5.......1...36......41......4..7.1...8...8..6..5....29......7...3.9.96.2....4 ; SER=8.4 ; W=6; double-or3: n1{r7c6!r7c7!r4c7}+n7{r7c5!r7c7!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 ; W=6; double-or3: n7{r9c5!r9c2!r2c2}+n9{r9c6!r9c2!r2c2} ; n5{r9c8!r9c2!r2c2}+n9{r9c6!r9c2!r2c2} ; n5{r9c8!r9c2!r2c2}+n7{r9c5!r9c2!r2c2} ;
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 ; W=6; double-or3: n5{r7c6!r7c8!r3c8}+n9{r7c5!r7c8!r4c8} ;
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 ; W=6; double-or3: n4{r4c3!r4c6!r9c6}+n6{r4c8!r4c6!r1c6} ;
puzzle#11798: .2.4....94.7..9236..9...5.....3..618...96..52.....1...3.........61..7.9..7.8..... ; SER=9.0 ; W=7; double-or3: n3{r8c5!r8c7!r5c7}+n4{r8c5!r8c7!r5c7} ;
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 ; W=7; double-or3: n5{r4c8!r4c5!r1c5}+n9{r4c3!r4c5!r7c5} ;
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 ; W=6; double-or3: n1{r8c5!r8c1!r1c1}+n7{r8c9!r8c1!r1c1} ; n1{r7c5!r7c1!r1c1}+n5{r7c6!r7c1!r3c1} ; n4{r6c9!r6c6!r7c6}+n5{r6c3!r6c6!r7c6} ;
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 ; W=6; double-or3: n2{r3c7!r3c4!r8c4}+n4{r3c7!r3c4!r8c4} ; n1{r2c4!r2c7!r4c7}+n2{r2c4!r2c7!r8c7} ; n1{r3c8!r3c4!r5c4}+n4{r3c7!r3c4!r8c4} ; n1{r3c8!r3c4!r5c4}+n2{r3c7!r3c4!r8c4} ;
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 ; W=7; double-or3: n4{r6c3!r6c8!r8c8}+n5{r6c1!r6c8!r2c8} ; n6{r3c1!r3c6!r9c6}+n7{r3c2!r3c6!r6c6} ;
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 ; W=7; double-or3: n5{r8c6!r8c2!r2c2}+n8{r8c6!r8c2!r1c2} ; n3{r7c1!r7c8!r2c8}+n9{r7c3!r7c8!r4c8} ;
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 ; W=7; double-or3: n1{r6c2!r6c9!r3c9}+n7{r6c2!r6c9!r1c9} ;
puzzle#12859: 12.........7..9..66....3.......4.39.3....5....71.....8...9..82..4..1895..3.5..... ; SER=8.4 ; W=6; double-or3: n2{r4c9!r4c1!r8c1}+n5{r4c9!r4c1!r7c1} ;
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 ; W=6; double-or3: n8{r1c3!r1c8!r4c8}+n9{r1c4!r1c8!r7c8} ;
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 ; W=9; double-or3: n7{r8c2!r8c4!r2c4}+n8{r8c1!r8c4!r4c4} ; n7{r1c3!r1c7!r6c7}+n9{r1c5!r1c7!r5c7} ;
puzzle#13274: ..34..7..45...9..66.....15.....6..97......4..9.67....137.....18.6...1.....42..... ; SER=9.1 ; W=7; double-or3: n1{r5c3!r5c4!r2c4}+n5{r5c3!r5c4!r8c4} ;
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 ; W=6; double-or3: n1{r8c9!r8c3!r2c3}+n2{r8c9!r8c3!r4c3} ; n1{r3c8!r3c5!r5c5}+n7{r3c8!r3c5!r6c5} ;
puzzle#13471: .2......94...8.........7.1.2.9....715....48......3.6..36.8.1....726...4......3..8 ; SER=8.4 ; W=7; double-or3: n2{r9c7!r9c5!r3c5}+n7{r9c7!r9c5!r5c5} ; n3{r2c4!r2c7!r4c7}+n7{r2c3!r2c7!r9c7} ;
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 ; W=6; double-or3: n3{r8c7!r8c5!r6c5}+n6{r8c3!r8c5!r6c5} ; n1{r4c2!r4c7!r8c7}+n8{r4c2!r4c7!r7c7} ;
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 ; W=6; double-or3: n7{r1c3!r1c7!r7c7}+n8{r1c1!r1c7!r7c7} ; n5{r2c6!r2c3!r4c3}+n7{r2c7!r2c3!r7c3} ; n5{r1c5!r1c7!r8c7}+n8{r1c1!r1c7!r7c7} ; n5{r1c5!r1c7!r8c7}+n7{r1c3!r1c7!r7c7} ;
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 ; W=8; double-or3: n1{r5c8!r5c5!r9c5}+n2{r5c8!r5c5!r9c5} ;
puzzle#13905: .234....9.5.1..2..69..7.....649.75.....5.......5...64....7...2.....43..18...9.... ; SER=8.5 ; W=6; double-or3: n4{r9c2!r9c9!r3c9}+n5{r9c6!r9c9!r3c9} ; n8{r8c4!r8c7!r1c7}+n9{r8c3!r8c7!r5c7} ;
puzzle#14180: .2...6..945.1.......837....2..7..1..3..948.25...........6.94..........548....73.. ; SER=8.9 ; W=7; double-or3: n3{r4c9!r4c6!r8c6}+n5{r4c3!r4c6!r3c6} ;
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 ; W=6; double-or3: n6{r5c3!r5c8!r7c8}+n7{r5c3!r5c8!r8c8} ;
puzzle#14426: 12..5....4..1.9..6.98....5...1..84.334..7....8.9.41......6....1....159.........75 ; SER=8.8 ; W=6; double-or3: n8{r9c7!r9c5!r2c5}+n9{r9c1!r9c5!r4c5} ; n3{r9c3!r9c5!r3c5}+n9{r9c1!r9c5!r4c5} ;
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 ; W=7; double-or3: n3{r2c8!r2c6!r4c6}+n9{r2c8!r2c6!r7c6} ;
puzzle#14566: .234...89.....9.3.69.......27.....4.8..2...57.16.4.8......7..64...81.......3..5.. ; SER=8.4 ; W=6; double-or3: n3{r4c7!r4c6!r3c6}+n5{r4c3!r4c6!r7c6} ; n3{r7c7!r7c1!r6c1}+n9{r7c4!r7c1!r6c1} ;
puzzle#14817: .2.....89..71..2..6..73..4.........33..2.4.........56.5..62..1......5...98...1... ; SER=8.4 ; W=6; double-or3: n1{r4c5!r4c2!r8c2}+n6{r4c5!r4c2!r8c2} ; n7{r5c2!r5c8!r8c8}+n9{r5c5!r5c8!r8c8} ; n8{r6c1!r6c4!r8c4}+n9{r6c3!r6c4!r8c4} ;
puzzle#14831: .2..5..8...718...66....2.4.2..3....1....6.........73..3.9........2...8...8...4913 ; SER=8.4 ; W=6; double-or3: n1{r8c5!r8c2!r3c2}+n4{r8c9!r8c2!r2c2} ; n4{r5c3!r5c7!r7c7}+n7{r5c1!r5c7!r1c7} ;
puzzle#14939: 1......8...7...2.6.9...2........8....8.69..7.....4.59..14...9..8....436...23....7 ; SER=8.4 ; W=6; double-or3: n3{r3c5!r3c8!r4c8}+n5{r3c1!r3c8!r7c8} ; n3{r1c3!r1c5!r4c5}+n6{r1c3!r1c5!r7c5} ;
puzzle#15001: .2....7.9...1892....87..4....63...9.3.5.......7..2....5.......7..9..562...2.....3 ; SER=8.4 ; W=6; double-or3: n7{r9c1!r9c5!r4c5}+n9{r9c7!r9c5!r5c5} ;
puzzle#15007: .2....7..4...8...6...73..1..1....854....9....8...149....59......62.....1.4...75.. ; SER=8.4 ; W=6; double-or3: n3{r9c4!r9c1!r1c1}+n9{r9c8!r9c1!r4c1} ; n1{r9c4!r9c1!r1c1}+n9{r9c8!r9c1!r4c1} ; n1{r9c4!r9c1!r1c1}+n3{r9c4!r9c1!r1c1} ;
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 ; W=6; double-or3: n4{r8c3!r8c4!r1c4}+n9{r8c1!r8c4!r1c4} ; n7{r6c2!r6c5!r7c5}+n9{r6c9!r6c5!r2c5} ;
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 ; W=7; double-or3: n3{r2c8!r2c3!r4c3}+n5{r2c4!r2c3!r7c3} ; n3{r2c8!r2c2!r6c2}+n5{r2c4!r2c2!r9c2} ;
puzzle#15672: ....5.7..4....9....6.2..41...8....7.3........97....6.15...6..2..345.......2..1..8 ; SER=8.9 ; W=8; double-or3: n2{r5c2!r5c7!r2c7}+n5{r5c6!r5c7!r9c7} ;
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 ; W=6; double-or3: n2{r8c8!r8c6!r5c6}+n4{r8c1!r8c6!r5c6} ;
puzzle#15920: ..3......4..1892..8...2..15..5.3.....1...2.7..7...43..5....39......4..53..6....4. ; SER=8.4 ; W=6; double-or3: n2{r4c1!r4c8!r7c8}+n6{r4c1!r4c8!r7c8} ;
puzzle#16004: 123........7.....6..93.2..5.4.7.59..39...48....5...4.2...6.......69385......41... ; SER=7.3 ; W=6; double-or3: n5{r9c4!r9c2!r2c2}+n7{r9c8!r9c2!r6c2} ; n5{r9c4!r9c1!r2c1}+n7{r9c8!r9c1!r6c1} ; n2{r7c5!r7c8!r2c8}+n3{r7c2!r7c8!r2c8} ;
puzzle#16032: ...4.67......8.2..86...2....859...1.3.16.....7....8........3..16..74.8.39.4....7. ; SER=8.5 ; W=6; double-or3: n4{r6c2!r6c7!r7c7}+n9{r6c2!r6c7!r7c7} ;
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 ; W=7; double-or3: n2{r6c2!r6c7!r2c7}+n8{r6c4!r6c7!r9c7} ; n1{r4c1!r4c5!r9c5}+n6{r4c8!r4c5!r1c5} ;
puzzle#16155: 1.3...........9...8..372..52...6....6.5..83.4.3...5..1.......4....8....77.65..8.3 ; SER=9.0 ; W=8; double-or3: n2{r1c2!r1c9!r7c9}+n8{r1c5!r1c9!r4c9} ;
puzzle#16325: ..34......5...9....697......9..3...7.7.2.1.98.....431.5......2...63...5..1...86.3 ; SER=8.4 ; W=6; double-or3: n4{r3c1!r3c9!r7c9}+n5{r3c5!r3c9!r6c9} ;
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 ; W=7; double-or3: n1{r3c2!r3c7!r7c7}+n4{r3c6!r3c7!r7c7} ;
puzzle#16579: .2.45..8....1....6..9....4.2.....5.....69..72.4...7..8..2.....4..43.1.5..8....9.. ; SER=8.9 ; W=8; double-or3: n1{r4c5!r4c8!r7c8}+n9{r4c2!r4c8!r2c8} ; n2{r3c7!r3c6!r9c6}+n6{r3c2!r3c6!r9c6} ;
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 ; W=7; double-or3: n7{r9c4!r9c2!r4c2}+n8{r9c4!r9c2!r5c2} ;
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 ; W=6; double-or3: n1{r1c7!r1c2!r5c2}+n2{r1c7!r1c2!r5c2} ;
puzzle#17311: ........9..71..2..8.6...15....53.8.....9...6..1...8..3641....9......5....3524.... ; SER=8.4 ; W=6; double-or3: n7{r3c9!r3c5!r7c5}+n9{r3c2!r3c5!r8c5} ; n2{r3c2!r3c5!r5c5}+n9{r3c2!r3c5!r8c5} ; n2{r3c2!r3c5!r5c5}+n7{r3c9!r3c5!r7c5} ; n2{r1c2!r1c5!r5c5}+n8{r1c8!r1c5!r7c5} ;
puzzle#17505: .2..5...94.7..9.....63....42......1.......4.374.8.1.6.3....75.....94...1.62...... ; SER=8.9 ; W=6; double-or3: n5{r5c2!r5c4!r9c4}+n7{r5c8!r5c4!r1c4} ; n3{r1c3!r1c8!r8c8}+n7{r1c4!r1c8!r5c8} ; n3{r2c2!r2c8!r8c8}+n8{r2c5!r2c8!r7c8} ; n1{r3c7!r3c2!r7c2}+n5{r3c8!r3c2!r4c2} ;
puzzle#17598: .2.45.........9.3.8....31.4.......67..9...5..5..6.2....8...731.7412..6........4.. ; SER=8.3 ; W=6; double-or3: n5{r8c6!r8c9!r2c9}+n8{r8c6!r8c9!r1c9} ; n2{r4c7!r4c1!r9c1}+n4{r4c6!r4c1!r2c1} ;
puzzle#17600: 12..56....57..92.6..6.....42.......853...79...6.8.......5.1...7.....2...9.....8.5 ; SER=9.0 ; W=6; double-or3: n7{r8c5!r8c2!r4c2}+n8{r8c5!r8c2!r3c2} ; n5{r6c6!r6c8!r3c8}+n7{r6c1!r6c8!r3c8} ; n5{r4c6!r4c8!r3c8}+n7{r4c2!r4c8!r3c8} ;
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 ; W=7; double-or3: n1{r9c5!r9c1!r1c1}+n9{r9c5!r9c1!r1c1} ;
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 ; W=7; double-or3: n4{r6c8!r6c3!r8c3}+n9{r6c7!r6c3!r8c3} ; n4{r5c8!r5c3!r8c3}+n9{r5c4!r5c3!r8c3} ; n8{r4c1!r4c7!r9c7}+n9{r4c4!r4c7!r1c7} ;
puzzle#18605: ...45.........9.36....7.1....57....8.9.....4.8.4......3.98......1.6...72.6251.... ; SER=8.4 ; W=6; double-or3: n3{r6c5!r6c9!r9c9}+n7{r6c2!r6c9!r1c9} ; n2{r6c2!r6c8!r1c8}+n6{r6c5!r6c8!r7c8} ; n6{r6c5!r6c7!r7c7}+n7{r6c2!r6c7!r1c7} ;
puzzle#18608: ...45...94.71.......6....5..3..1...76...2...5.....49...7..........2..89...2.936.. ; SER=8.5 ; W=7; double-or3: n5{r4c3!r4c4!r9c4}+n6{r4c8!r4c4!r7c4} ;
puzzle#18658: .2.45..8..5....2..6.9.3.......5..6983...6.45.........17....18....6.....4...7.8... ; SER=8.4 ; W=6; double-or3: n3{r2c3!r2c9!r9c9}+n6{r2c6!r2c9!r9c9} ; n1{r2c8!r2c4!r5c4}+n8{r2c1!r2c4!r5c4} ;
puzzle#18843: ...4...8...7.892.........5..4.....1.7....89...35.....63.25.1....1.8.76.2.7..2.... ; SER=8.6 ; W=6; double-or3: n2{r6c8!r6c4!r3c4}+n9{r6c1!r6c4!r9c4} ;
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 ; W=6; double-or3: n4{r6c2!r6c8!r3c8}+n7{r6c6!r6c8!r7c8} ;
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 ; W=6; double-or3: n4{r1c9!r1c4!r4c4}+n5{r1c8!r1c4!r8c4} ;
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 ; W=8; double-or3: n2{r7c4!r7c8!r1c8}+n5{r7c4!r7c8!r5c8} ;
puzzle#19600: ..34..7.........6..8.7..1...3..9....5...72..8..4....2.....18.76...93.....91...5.. ; SER=8.4 ; W=6; double-or3: n3{r3c6!r3c9!r9c9}+n4{r3c1!r3c9!r8c9} ;
puzzle#19909: .234..7..4....9..36.97....12.19..6.8...8....4....63........4..........128..31..9. ; SER=9.0 ; W=8; double-or3: n4{r8c7!r8c3!r6c3}+n6{r8c4!r8c3!r5c3} ; n6{r8c4!r8c2!r5c2}+n9{r8c5!r8c2!r5c2} ; n4{r8c7!r8c2!r6c2}+n9{r8c5!r8c2!r5c2} ; n4{r8c7!r8c2!r6c2}+n6{r8c4!r8c2!r5c2} ;
puzzle#20108: ...4.6.....71...63.9..731..2......9.....4..275....2.36..6..8....3......4..2...67. ; SER=8.9 ; W=8; double-or3: n3{r4c3!r4c4!r9c4}+n6{r4c2!r4c4!r8c4} ;
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 ; W=6; double-or3: n7{r7c5!r7c8!r1c8}+n9{r7c5!r7c8!r6c8} ;
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 ; W=6; double-or3: n1{r8c6!r8c9!r4c9}+n9{r8c2!r8c9!r1c9} ; n3{r6c2!r6c8!r9c8}+n6{r6c2!r6c8!r2c8} ; n2{r2c7!r2c2!r9c2}+n6{r2c8!r2c2!r6c2} ; n2{r1c2!r1c7!r6c7}+n9{r1c5!r1c7!r8c7} ;
puzzle#20559: 12..5.....57..9.6......7..42...4..75...61.....8.3...2................3.7678.....2 ; SER=9.0 ; W=7; double-or3: n1{r9c6!r9c8!r3c8}+n4{r9c6!r9c8!r5c8} ;
puzzle#20680: ...4....9....8.2..8....2154..69183.......7948.....4....34..18.66......3.7.2...... ; SER=8.9 ; W=6; double-or3: n2{r4c1!r4c9!r8c9}+n7{r4c2!r4c9!r8c9} ;
puzzle#21053: ..3.5.......1893...8..7......4...8...9.....52.1..42....4..9...75....196....7....8 ; SER=8.9 ; W=6; double-or3: n6{r2c9!r2c2!r9c2}+n7{r2c8!r2c2!r8c2} ; n7{r6c1!r6c8!r2c8}+n9{r6c4!r6c8!r3c8} ;
puzzle#21056: ..3.5......71.9.2..8..7..4..7.....9.516.......3...2.57...7....8....9526.....61... ; SER=8.4 ; W=6; double-or3: n1{r7c8!r7c3!r3c3}+n5{r7c7!r7c3!r3c3} ; n5{r7c7!r7c2!r2c2}+n9{r7c7!r7c2!r1c2} ; n4{r4c3!r4c7!r9c7}+n6{r4c6!r4c7!r3c7} ; n1{r4c5!r4c7!r3c7}+n6{r4c6!r4c7!r3c7} ; n1{r4c5!r4c7!r3c7}+n4{r4c3!r4c7!r9c7} ; n3{r4c7!r4c5!r2c5}+n8{r4c1!r4c5!r2c5} ;
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 ; W=6; double-or3: n3{r6c4!r6c7!r2c7}+n8{r6c1!r6c7!r7c7} ; n5{r5c7!r5c1!r8c1}+n8{r5c7!r5c1!r8c1} ;

- the presence of the pattern doesn't mean it can be used in any useful way;
- as it currently does nothing, it doesn't change the resolution path one would get without it.
More remarks:
- the percentage is 147/239 = 62% if one considers only the relevant puzzles (those not already solved in W5);
- if the pattern was looked for with higher priority than "just before L6", I think it'd appear in a large proportion of the puzzles;
Last edited by denis_berthier on Fri Nov 12, 2021 6:11 am, edited 1 time in total.
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

.
Out of sheer curiosity, I also tried to see what happened in the case that had the most occurrences of the pattern - the last but one in the list.
(solve "..3.5......71.9.2..8..7..4..7.....9.516.......3...2.57...7....8....9526.....61...")
Code: Select all
`Resolution state after Singles and whips[1]:   +-------------------+-------------------+-------------------+   ! 12469 269   3     ! 246   5     468   ! 16789 178   169   !   ! 46    56    7     ! 1     348   9     ! 3568  2     356   !   ! 1269  8     1259  ! 236   7     36    ! 13569 4     13569 !   +-------------------+-------------------+-------------------+   ! 28    7     248   ! 5     1348  3468  ! 13468 9     1346  !   ! 5     1     6     ! 9     348   7     ! 348   38    2     !   ! 89    3     489   ! 46    148   2     ! 1468  5     7     !   +-------------------+-------------------+-------------------+   ! 1369  569   159   ! 7     2     34    ! 13459 13    8     !   ! 7     4     18    ! 38    9     5     ! 2     6     13    !   ! 2389  259   2589  ! 348   6     1     ! 34579 37    3459  !   +-------------------+-------------------+-------------------+156 candidates.`

There first appears a series of relatively simple rules: only Subsets, bivalue-chains and z-chains (only reversible patterns, no rule with any t-candidate):
Hidden Text: Show
naked-pairs-in-a-block: b9{r7c8 r8c9}{n1 n3} ==> r9c9≠3, r9c8≠3, r9c7≠3, r7c7≠3, r7c7≠1
naked-single ==> r9c8=7
hidden-single-in-a-block ==> r1c7=7
finned-x-wing-in-rows: n4{r7 r5}{c7 c6} ==> r4c6≠4
finned-x-wing-in-rows: n8{r2 r5}{c5 c7} ==> r6c7≠8, r4c7≠8
whip[1]: b6n8{r5c8 .} ==> r5c5≠8
biv-chain[4]: r8c3{n8 n1} - b9n1{r8c9 r7c8} - r1c8{n1 n8} - c6n8{r1 r4} ==> r4c3≠8
z-chain[4]: r4c3{n4 n2} - r4c1{n2 n8} - b5n8{r4c6 r6c5} - c5n1{r6 .} ==> r4c5≠4
with z-candidates = n8r4c5 n1r4c5
z-chain[4]: c5n1{r6 r4} - c5n8{r4 r2} - c7n8{r2 r5} - r5n4{c7 .} ==> r6c5≠4
with z-candidates = n8r6c5 n4r5c5
z-chain[4]: r5n3{c8 c5} - r2n3{c5 c7} - b3n8{r2c7 r1c8} - r5c8{n8 .} ==> r4c9≠3
with z-candidates = n3r5c7 n3r2c9 n3r5c8
z-chain[4]: b6n3{r5c7 r5c8} - r5n8{c8 c7} - r2n8{c7 c5} - r2n3{c5 .} ==> r3c7≠3
with z-candidates = n3r4c7 n3r2c9 n3r2c7
whip[3]: c7n8{r2 r5} - r5c8{n8 n3} - c7n3{r5 .} ==> r2c7≠6
whip[3]: c7n8{r2 r5} - r5c8{n8 n3} - c7n3{r5 .} ==> r2c7≠5
biv-chain[5]: r4c1{n2 n8} - c6n8{r4 r1} - r1c8{n8 n1} - r7c8{n1 n3} - b7n3{r7c1 r9c1} ==> r9c1≠2
biv-chain[5]: c6n8{r1 r4} - r4c1{n8 n2} - r4c3{n2 n4} - c9n4{r4 r9} - b8n4{r9c4 r7c6} ==> r1c6≠4
hidden-single-in-a-column ==> r7c6=4
whip[1]: b8n3{r9c4 .} ==> r3c4≠3
biv-chain[4]: r4n2{c1 c3} - b4n4{r4c3 r6c3} - c4n4{r6 r1} - b2n2{r1c4 r3c4} ==> r3c1≠2

Code: Select all
`   +-------------------+-------------------+-------------------+   ! 12469 269   3     ! 246   5     68    ! 7     18    169   !   ! 46    56    7     ! 1     348   9     ! 38    2     356   !   ! 169   8     1259  ! 26    7     36    ! 1569  4     13569 !   +-------------------+-------------------+-------------------+   ! 28    7     24    ! 5     138   368   ! 1346  9     146   !   ! 5     1     6     ! 9     34    7     ! 348   38    2     !   ! 89    3     489   ! 46    18    2     ! 146   5     7     !   +-------------------+-------------------+-------------------+   ! 1369  569   159   ! 7     2     4     ! 59    13    8     !   ! 7     4     18    ! 38    9     5     ! 2     6     13    !   ! 389   259   2589  ! 38    6     1     ! 459   7     459   !   +-------------------+-------------------+-------------------+`

This is where the pattern appears, all the instances at the same time:
double-or3: n1[r7c8!r7c3!r3c3]+n5[r7c7!r7c3!r3c3] ;
double-or3: n5[r7c7!r7c2!r2c2]+n9[r7c7!r7c2!r1c2] ;
double-or3: n4[r4c3!r4c7!r9c7]+n6[r4c6!r4c7!r3c7] ;
double-or3: n1[r4c5!r4c7!r3c7]+n6[r4c6!r4c7!r3c7] ;
double-or3: n1[r4c5!r4c7!r3c7]+n4[r4c3!r4c7!r9c7] ;
double-or3: n3[r4c7!r4c5!r2c5]+n8[r4c1!r4c5!r2c5] ;

The end in Z6 had nothing noticeable:
Hidden Text: Show
z-chain[6]: r2c7{n3 n8} - r1c8{n8 n1} - c7n1{r3 r6} - r6n6{c7 c4} - b5n4{r6c4 r5c5} - r5n3{c5 .} ==> r4c7≠3
with z-candidates = n1r4c7 n3r5c8 n3r5c7
whip[1]: b6n3{r5c8 .} ==> r5c5≠3
singles ==> r5c5=4, r6c4=6, r3c4=2, r1c4=4, r2c1=4
naked-pairs-in-a-row: r2{c5 c7}{n3 n8} ==> r2c9≠3
biv-chain[4]: c9n3{r3 r8} - r7n3{c8 c1} - r7n6{c1 c2} - r2n6{c2 c9} ==> r3c9≠6
z-chain[4]: r6c1{n9 n8} - r4c1{n8 n2} - r1n2{c1 c2} - c2n9{r1 .} ==> r9c1≠9, r7c1≠9
with z-candidates = n9r9c2 n9r7c2
naked-pairs-in-a-row: r9{c1 c4}{n3 n8} ==> r9c3≠8
biv-chain[5]: r4c3{n4 n2} - r4c1{n2 n8} - c6n8{r4 r1} - b2n6{r1c6 r3c6} - c7n6{r3 r4} ==> r4c7≠4
biv-chain[5]: r2n6{c9 c2} - r7n6{c2 c1} - r7n3{c1 c8} - c9n3{r8 r3} - r3c6{n3 n6} ==> r3c7≠6
hidden-single-in-a-column ==> r4c7=6
biv-chain[3]: r4n1{c9 c5} - c5n3{r4 r2} - b3n3{r2c7 r3c9} ==> r3c9≠1
biv-chain[4]: c3n8{r6 r8} - r8n1{c3 c9} - b6n1{r4c9 r6c7} - r6n4{c7 c3} ==> r6c3≠9
hidden-single-in-a-block ==> r6c1=9
biv-chain[3]: c2n2{r9 r1} - b1n9{r1c2 r3c3} - b1n5{r3c3 r2c2} ==> r9c2≠5
biv-chain[4]: r3c1{n1 n6} - c6n6{r3 r1} - r1n8{c6 c8} - c8n1{r1 r7} ==> r7c1≠1
whip[1]: b7n1{r8c3 .} ==> r3c3≠1
biv-chain[3]: r3c3{n9 n5} - c2n5{r2 r7} - r7c7{n5 n9} ==> r7c3≠9, r3c7≠9
whip[1]: c7n9{r9 .} ==> r9c9≠9
biv-chain[3]: r9c9{n5 n4} - r4n4{c9 c3} - c3n2{r4 r9} ==> r9c3≠5
whip[1]: r9n5{c9 .} ==> r7c7≠5
naked-single ==> r7c7=9
naked-pairs-in-a-column: c2{r2 r7}{n5 n6} ==> r1c2≠6
biv-chain[3]: r3c7{n1 n5} - c3n5{r3 r7} - r7n1{c3 c8} ==> r1c8≠1
stte

If you're a user of CSP-Rules, I can post the additional code that would allow you to analyse each puzzle as above.
denis_berthier
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Posts: 3676
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Location: Paris

### Re: Fireworks

looking forward to going through these whenever i get the time moving from bottom to top this time around, just to start with stuff i havent looked at before

the first thing i've noticed isnt exactly a new technique, but something i hadn't fully appreciated yet. in puzzle #20680 there's an interesting sue de coq which has a small-scale firework MSHS as its inverse

Code: Select all
` Sue de Coq approach:                                             Firework MSHS approach:.--------------------.-------------------.-----------------.     .--------------------.-------------------.-----------------.| 123-5 12567  1357  | 4     3567   356  | 67   8     9    |     |x123-5 12567  1357  | 4     3567   356  | 67   8     9    ||#459   45679  579   | 1     8      569  | 2    67    3    |     | 459   45679  579   | 1     8      569  | 2    67    3    || 8     679    379   | 367   3679   2    | 1    5     4    |     | 8     679    379   | 367   3679   2    | 1    5     4    |:--------------------+-------------------+-----------------:     :--------------------+-------------------+-----------------:|#24-5  47-25  6     | 9     1      8    | 3    27    257  |     | 245   457-2  6     | 9     1      8    | 3    27    257  ||#123-5#125   #135   | 236-5 236-5  7    | 9    4     8    |     |x123-5 125    135   |x236-5x236-5  7    | 9    4     8    ||#239-5 789-25 789-35| 235   235    4    | 567  1267  1257 |     | 2359  5789-2 5789-3| 235   235    4    | 567  1267  1257 |:--------------------+-------------------+-----------------:     :--------------------+-------------------+-----------------:|#59    3      4     | 257   2579   1    | 8    279   6    |     | 59    3      4     | 257   2579   1    | 8    279   6    || 6     1589   1589  | 2578  24579  59   | 457  3     1257 |     | 6     1589   1589  | 2578  24579  59   | 457  3     1257 || 7     1589   2     | 3568  34569  3569 | 45   19    15   |     | 7     1589   2     | 3568  34569  3569 | 45   19    15   |'--------------------'-------------------'-----------------'     '--------------------'-------------------'-----------------'skfr 8.9...4....9....8.2..8....2154..69183.......7948.....4....34..18.66......3.7.2......`

fireworks on 2 and 3 in r5c1b4 along with 1c1 & 6r5 make four placements to go in four cells, all other candidates removed
additional positions for 2&3 in b4 also eliminated (pattern is rank0)

the SDC picks up some cannibalised elims on 5s in b4 which the alternative doesnt immediately get, but the basics that follow from the firework (locked 5s in r5) handle it
looking at this with T&L, the SDC is 7 cell truths, whereas the firework MSHS is 6 line truths (a firework always accounts for two line truths, one per row and column), but if we insist on accounting for the 5s eliminations in b4, we have to add 5r5 as a truth and then they become equivalent, 7 truths each for either perspective

so naturally i went to look for another generated puzzle where the inverse is less truths than the SDC (with the additional restraint that each technique must get the same eliminations), and it wasnt too hard to find (っ◔◡◔)っ

Code: Select all
` Sue de Coq:                                            Firework MSHS:.---------------.---------------.----------------.     .---------------.---------------.----------------.| 128  279  3   | 479  5   1479 | 6    149  289  |     | 128  279  3   | 479  5   1479 | 6    149  289  || 4    67   19  | 8    2   67   | 3    19   5    |     | 4    67   19  | 8    2   67   | 3    19   5    || 268  5    189 | 469  16  3    | 48-2 7    289  |     | 268  5    189 | 469  16  3    |*48-2 7    289  |:---------------+---------------+----------------:     :---------------+---------------+----------------:| 7    8    2   | 69   3   69   | 1    5    4    |     | 7    8    2   | 69   3   69   | 1    5    4    || 9    4    5   | 1    78  2    | 78   36   36   |     | 9    4    5   | 1    78  2    |x78   36   36   || 3    1    6   | 45   78  45   | 9    2    78   |     | 3    1    6   | 45   78  45   | 9    2    78   |:---------------+---------------+----------------:     :---------------+---------------+----------------:|#168  3   #189 | 2   #16  57-16|#457  #49 #679  |     | 168  3    189 | 2    16 x57-16|x457  49   679  || 5    26   4   | 367  9   8    |#27   36   1    |     | 5    26   4   | 367  9   8    | 27   36   1    || 126  269  7   | 356  4   156  |#25   8    369-2|     | 126  269  7   | 356  4   156  | 25   8    2369 |'---------------'---------------'----------------'     '---------------'---------------'----------------'skfr 7.1..3.5.6..4..82.3.5.5...3.7.782.3.1549451.2...316...92..3.2.....5.4.98..1..7.4..8.`

(note:both techniques dont appear until after locked 78 pair in b5)

firework on 7s in r7c7b9, along with 5r7, 4c7 & 8c7 make four placements to go in four cells, all other candidates removed
in T&L this is 5 line truths but accounting for the r9c9 elimination the SDC gets it becomes 6 truths (adding in 2c7)
the SDC in comparison is 8 cell truths

so for some cases it seems, you could use this pattern to more easily get eliminations than the equivalent SDC

furthermore, this i think extends beyond just sue de coq. any ALS rule/MSLS made of cells in a row and column and of which the intersection cell is also used, will have a firework tuple/MSHS inverse. to refer back to one of the initial example puzzles, Triple Laser by Qinlux:

Code: Select all
`.---------------------.-------------------.--------------------.| 123-679 4     5     |#67  #6789  #89    |#268  #189   #123-89|| 2369    2369  269   | 1    45689  4589  | 568-2 7     #2389  || 8       169   1679  | 56-7 2      3     | 456   459-1 #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     | 5-7   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     12-47 |'---------------------'-------------------'--------------------'skfr 8.3.45.........1...7.8...23......9.71........3...8.4.6.2...3.....5.7.8....6......9..`

the MSLS is made up of 11 cell truths
the firework MSHS instead made of 5 line truths (after basics removes 3 from r9c9 then we can use 3r1 rather than a firework on 3s)
accounting for elims the MSLS gets on 7s in b2&6, we bump up to 7 line truths, still well under the MSLS's 11

this might be worthwhile for the manual solver, as large-scale SDC's and related ALS/MSLS techniques can be difficult, and cannibal eliminations are tricky also

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

Hi shye
None of these 3 examples is very convincing:
Code: Select all
`skfr 8.9...4....9....8.2..8....2154..69183.......7948.....4....34..18.66......3.7.2......`

Can be solved in W6

Code: Select all
`skfr 7.1..3.5.6..4..82.3.5.5...3.7.782.3.1549451.2...316...92..3.2.....5.4.98..1..7.4..8.`

Can be solved in BC3

Code: Select all
`skfr 8.3.45.........1...7.8...23......9.71........3...8.4.6.2...3.....5.7.8....6......9..`

Can be solved in Z4

shye wrote:a firework always accounts for two line truths, one per row and column

It's always 3. Unless you change the definition you have given in a previous post, you must always add the block.
There's no way of proving the or3 relation without the hypothesis that there's no candidate outside the block (except in one place), i.e. that all the candidates for CSP-Variable rn (except one) are also candidates for CSP-Variable bn and same for column c.
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

### Re: Fireworks

denis_berthier wrote:
shye wrote:a firework always accounts for two line truths, one per row and column

It's always 3. Unless you change the definition you have given in a previous post, you must always add the block.
There's no way of proving the or3 relation without the hypothesis that there's no candidate outside the block (except in one place), i.e. that all the candidates for CSP-Variable rn (except one) are also candidates for CSP-Variable bn and same for column c.

so the method i was using to make that count was T&L, and i think truths must not be inherently the same as CSP-variables. i would like to know a brief definition of what a CSP-variable is

truths and links can be defined briefly:
a truth is a selection of candidates of which at least one must be true (ie: all the positions for x in a house, or all the candidates within a cell)
a link says that at most one of a selection of candidates can be true (ie: at most one of x can be in a house, at most one candidate can be in a cell)

with this definition a firework is made up of two truths (one per line) and one link (the box) all on a single digit
since the candidate is counted twice within the intersecting cell, one from each the row and column, the box link only removes one of those two from that cell
and so the remaining positions, the intersecting cell and the positions outside the intersecting box, form a new truth because there is one more total truths than links. much like forming a strong link at the two ends of a reversible chain

hopefully that makes sense. i can assure you that my counting is correct with this method

if it matters, here is the count of total truths and links in each of the SDC vs FMSHS examples i posted above (counting locked candidates and cannibal elims for both, to be fair and account for every elimination):

...4....9....8.2..8....2154..69183.......7948.....4....34..18.66......3.7.2......
SDC = 15 (7 truths ; 8 links)
FMSHS = 14 (7 truths ; 7 links)

..3.5.6..4..82.3.5.5...3.7.782.3.1549451.2...316...92..3.2.....5.4.98..1..7.4..8.
SDC = 17 (8 truths ; 9 links)
FMSHS = 12 (6 truths ; 6 links)

denis_berthier wrote:None of these 3 examples is very convincing

the point was to show how it is more efficient to use this inverse than the equivalent SDC/ALS technique

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

shye wrote:i would like to know a brief definition of what a CSP-variable is

An informal one would be: any of the 324 2D-cells introduced in my extended Sudoku board (published in May 2007).

shye wrote:truths and links can be defined briefly:
a truth is a selection of candidates of which at least one must be true (ie: all the positions for x in a house, or all the candidates within a cell)
a link says that at most one of a selection of candidates can be true (ie: at most one of x can be in a house, at most one candidate can be in a cell)

"Truth" is Allan Barker's renaming of my CSP-Variables (without ever referring to them). It is a ridiculous name. In his approach (exact covers), there's no reason not to call them "Base Set".
"Link" with this definition is also a ridiculous name. In his approach, it should remain "Cover Set".
At least, he explictly writes what the standard names are: "General logic starts with the basic idea of base sets that are truths, and cover sets that are links."

shye wrote:with this definition a firework is made up of two truths (one per line) and one link (the box) all on a single digit
since the candidate is counted twice within the intersecting cell, one from each the row and column, the box link only removes one of those two from that cell
and so the remaining positions, the intersecting cell and the positions outside the intersecting box, form a new truth because there is one more total truths than links. much like forming a strong link at the two ends of a reversible chain
hopefully that makes sense. i can assure you that my counting is correct with this method

No, it doesn't make sense to add "Truths" to a pattern while trying to prove it. (And, in the present case, I can see no proof at all.)
Instead of trying to prove the or3 conclusion, try to define the pattern without mentioning the block (which is equivalent to not using CSP-Variable bn, as n is fixed).

Allan has never defined any concept of complexity for his patterns. As a result, Xsudo provides no guarantee that patterns with fewer "Truths" are found before some with more "Truths". Indeed, several counter-examples were provided at that time.
Worse, Xsudo doesn't find all the patterns with n "Truths". Said otherwise, Xsudo is not complete (and Allan unambiguously admitted this).
denis_berthier
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Location: Paris

### Re: Fireworks

ok, well naming schemes aside can we agree that for the examples i presented, the firework approach proves more efficient than its inverse (SDC/ALS)? i fear youre veering the conversation further away from the actual point with every remark

at least i think i have an idea of what a CSP-variable is now. might i ask a question then, how much CSP-variables make up a two-string kite? example below
because if it is 2, which i would expect, then i do not see why a firework must be 3. im not that bothered if you use it being 3 for your complexity scaling, but i do not understand the reasoning for it

Code: Select all
`+-------+-------+-------+| 8 . . | 1 2 3 | 4 5 x || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || x . . | . . . | . . -9|+-------+-------+-------+`

9r1c9 = 9r1c23 - 9r23c1 = 9r9c1

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

shye wrote:ok, well naming schemes aside can we agree that for the examples i presented, the firework approach proves more efficient than its inverse (SDC/ALS)? i fear youre veering the conversation further away from the actual point with every remark

There's nothing to agree or disagree upon as long as you haven't defined a measure of complexity. As I said, and this is right on the point, Xsudo doesn't propose one. What's yours? Are you adopting mine (nb of CSP-Variables)?

shye wrote:at least i think i have an idea of what a CSP-variable is now. might i ask a question then, how much CSP-variables make up a two-string kite? example below
because if it is 2, which i would expect, then i do not see why a firework must be 3. im not that bothered if you use it being 3 for your complexity scaling, but i do not understand the reasoning for it
Code: Select all
`+-------+-------+-------+| 8 . . | 1 2 3 | 4 5 x || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| 6 . . | . . . | . . . || 7 . . | . . . | . . . || 1 . . | . . . | . . . |+-------+-------+-------+| 2 . . | . . . | . . . || 3 . . | . . . | . . . || x . . | . . . | . . -9|+-------+-------+-------+`

9r1c9 = 9r1c23 - 9r23c1 = 9r9c1

I need no kite; the elimination is done by a generic g-whip[2] (using the same cells as the kite):
g-whip[2]: r1n9{c9 c123} - c1n9{r2 .} ==> r9c9≠9

As for your pattern, my complexity scale was not designed to measure patterns that don't produce any elimination. But your kite example has convinced me that this can be defined as 2; the block is used only for providing contradiction links.
denis_berthier
2010 Supporter

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Location: Paris

### Re: Fireworks

denis_berthier wrote:
shye wrote:ok, well naming schemes aside can we agree that for the examples i presented, the firework approach proves more efficient than its inverse (SDC/ALS)? i fear youre veering the conversation further away from the actual point with every remark

There's nothing to agree or disagree upon as long as you haven't defined a measure of complexity. As I said, and this is right on the point, Xsudo doesn't propose one. What's yours? Are you adopting mine (nb of CSP-Variables)?

i may have mentioned it earlier in the thread, but i do not have a rigorous system for measuring the complexity of a pattern, it isnt something i really concern myself with. but for the purposes of comparing two techniques as i did above, i had to use some metric; T&L is what i know well enough, and counting the truths in each approach seemed (at least for this case) a nice way to draw that comparison. but i can tell you have your reservations towards Xsudo and its approach, so i wont continue that discussion

I need no kite; the elimination is done by a generic g-whip[2] (using the same cells as the kite):
g-whip[2]: r1n9{c9 c123} - c1n9{r2 .} ==> r9c9≠9

As for your pattern, my complexity scale was not designed to measure patterns that don't produce any elimination. But your kite example has convinced me that this can be defined as 2; the block is used only for providing contradiction links.

ok, so as far as im concerned these are equivalent (and do correct me if this is wrong, ive avoided using your terminology prior only because of lack of understanding)

as a result of each technique, r1c9 & r9c1 contain at least one 9

if we can agree on that full statement, and that it has two CSP-variables, then i would expect the same to be valid in a firework

ill try a slight rewording. i can run the same logic on both of these example grids and get the same result:

Code: Select all
`+-------+-------+-------+     +-------+-------+-------+| 8 . . | 1 2 3 | 4 5 x |     | x . . | 1 2 3 | 4 5 x || . . . | . . . | . . . |     | . . . | . . . | . . . || . . . | . . . | . . . |     | . . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+| 6 . . | . . . | . . . |     | 6 . . | . . . | . . . || 7 . . | . . . | . . . |     | 7 . . | . . . | . . . || 1 . . | . . . | . . . |     | 1 . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+| 2 . . | . . . | . . . |     | 2 . . | . . . | . . . || 3 . . | . . . | . . . |     | 3 . . | . . . | . . . || x . . | . . . | . . . |     | x . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+`

in each grid, we are considering positions for 9 in row 1 and 9 in column 1, making a total of two 9s to consider
at most one 9 can be placed within b1p2347 (one of the two 9s)
so at least one of the remaining positions, marked with x's, must hold that second 9 (or potentially both of them, this is not important here)

i didnt use a third CSP-variable that i was aware of in that definition, so im unsure why it must be 3 like you say it should

im getting very tired now, ill respond to anything newly posted much later. hopefully this is clear or helpful

shye

Posts: 245
Joined: 12 June 2021

### Re: Fireworks

shye wrote:
Code: Select all
`+-------+-------+-------+     +-------+-------+-------+| 8 . . | 1 2 3 | 4 5 x |     | x . . | 1 2 3 | 4 5 x || . . . | . . . | . . . |     | . . . | . . . | . . . || . . . | . . . | . . . |     | . . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+| 6 . . | . . . | . . . |     | 6 . . | . . . | . . . || 7 . . | . . . | . . . |     | 7 . . | . . . | . . . || 1 . . | . . . | . . . |     | 1 . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+| 2 . . | . . . | . . . |     | 2 . . | . . . | . . . || 3 . . | . . . | . . . |     | 3 . . | . . . | . . . || x . . | . . . | . . . |     | x . . | . . . | . . . |+-------+-------+-------+     +-------+-------+-------+`

From a fish’s point of view, the PM grid on the left is different from the PM grid on the right. In the PM grid on the right, xr1c1 is an endo fin.
yzfwsf

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