The New Sudoku Players' Forum

by **ryokousha** » Sun Aug 28, 2022 3:05 pm

Quick explanation where these come from:

On the compatibility graph of the 46656 1-digit templates, I searched for maximal cliques of size <9 (that is collections of compatible 1-digit templates, such that no further compatible template exists). Clearly there are countless maxcliques of size 7, such as

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

with a nice 9-loop for the remaining two digits.

More interestingly, there exists a large number (no idea how many) of maxcliques with size 6, of which I generated 59308:
https://drive.google.com/file/d/1yd_V9YmHOsGk9KrBJ3pD5KL9XbzrnuKh/view?usp=sharing

The patterns of empty cells within these, when minlexed, come down to 40 impossible 3-digit patterns:

Hidden Text: Show

Code: Select all: ......111..1.11...11.1.........11..1..11...1.11....1......111....11...1.11......1 .....1.11..1.1.1..11...1.......1..11..111....11....1.....1...11..11.1...11....1.. .....1.11..1.1.1..11...1.......1..11..111....11....1.....1.11....1....1111.1..... .....1.11..1.1.1..11.1..........1.11..1.1.1..11.1..........11.1..11...1.11..1.... .....1.11..1.1.1..11.1..........1.11..1.1.1..11.1.........1..11..11..1..11...1... .....1.11..1.1.1..11.1..........1.11..1.1.1..11.1.........1..11..11.1...11....1.. .....1.11..1.1.1..11.1..........1.11..111....11....1......111....1....1111.1..... .....1.11..1.1.1..11.1..........1.11..111....11....1.....1...11..1.1.1..11...1... .....1.11..1.1.1..11.1..........1.11..111....11....1.....1...11..1.11...11....1.. .....1.11..1.1.1..11.1..........1.11..111....11....1.....1...11.1...11..1.1.1.... .....1.11..1.1.1..11.1..........1.11.1..1.1..1.11........1...11.11.1....1....11.. .....1.11..1.1.1..11.1..........11.1..11...1.11..1......1..1..1.1..1..1.1..1..1.. .....1.11..1.1.1..11.1.........1..11..11.1...11....1......1..11..11.1...11....1.. .....1.11..1.1.1..11.1.........1..11..11.1...11....1.....1...11..1.1.1..11...1... .....1.11..1.1.1..11.1.........1..11..11.1...11....1.....1.1..1..1.1.1..11.....1. .....1.11..1.1.1..11.1.........1..11..11.1...11....1.....1.11....1....1111..1.... .....1.11..1.1.1..11.1.........1..11..11.1...11....1....1..1..1.1..1..1.1..1..1.. .....1.11..1.1.1..11.1.........1..11..11.1...11....1....1.1.1...1...1..11..1...1. .....1.11..1.1.1..11.1.........1..11..11.1...11....1....11....1.1...11..1...1..1. .....1.11..1.1.1..11.1.........1..11.1.1..1..1.1..1......1.11...1.....111.1.1.... .....1.11..1.1.1..11.1.........1..11.1.1.1...1.1...1......1..11.11...1..1..1.1... .....1.11..1.1.1..11.1.........1..11.1.1.1...1.1...1.....1.11...11.1....1......11 .....1.11..1.1.1..11.1.........1.1.1..11.1...11.....1.....1.11...11.1...11......1 .....1.11..1.1.1..11.1.........11..1..11...1.11....1......111....11...1.11......1 .....1.11..1.1.1..11.1.........11..1..11..1..11.....1.....11.1...11..1..11......1 .....1.11..1.1.1..11.1........1...11..1.11...11....1.....11.1....1....1111...1... .....1.11..1.1.1..11.1........1..1.1.1..1..1.1.1..1.....11...1..1...11..1...1...1 .....1.11..1.1.1..11.1.......1..1..1.1..1..1.1..1..1....11...1..1...11..1...1...1 .....1.11..1.1.1..11.1.......1..11...1..1...11..1...1...11....1.1...1.1.1...1.1.. .....1.11..1.11....11...1....1..11..1..1....11..1...1..1..1...1.1..1..1.1..1..1.. .....1.11..1.11....11...1....11..1..1...1...11...1..1..1.1....1.1.1...1.1....11.. .....1.11..111....11....1.......1.11..111....11....1....1..11...1..1...11..1...1. .....1.11..111....11....1.......1.11.1.11....1.1...1....1.1...1.1.1...1.1....11.. .....1.11..111....11....1......111....1....1111.1.......1.1.1...1...1..11..1...1. .....1.11..111....11....1.....11.1....1....1111...1.....1..11...1..1...11..1...1. .....1.11..111....11....1....1..11...1..1...11..1...1...1..11...1..1...11..1...1. .....1.11..111....11....1....1..11...1..1...11..1...1...1..11...1..1..1.1..1....1 .....1.11..111....11....1....1..11...1..1...11..1...1...1.1...1.1...1.1.1..1..1.. .....1.11..111....11....1....1..11...1..1...11..1...1...1.1..1..1.1..1..1....1..1 ..1..1..1.1..1..1.1..1..1....1..1..1.1..1..1.1..1..1....1..1.1..1..1.1..1..1....1

Note that these, as complements to maximal cliques, are also impossible 1-digit patterns, when interpreted as corner pencilmarks referring to boxes, rows or columns.

From some of the 59308 maxcliques of size 6, all instances of one more digit can be removed, such that the sudoku still has no solutions. This results in a list of 1616 grids with impossible 4-digit patterns for the missing digits.
Extracting those patterns and minlexing resulted in the five unique 4-digit patterns I gave earlier:

Code: Select all: ..1..1.11..1.11..111.1..1....1..111..1.11.1..11..1...1.1..111..1..1...111.11...1. ..1..1.11..1.11..111.1..1....1.1.11..1.11.1..11...1..1.1..111..1..1...111.11...1. ..1..1.11..1.11..111.1..1....1.1.1.1.1.11.1..11...1.1..1..111..1..1...111.11...1. ....11.11.11..1..11.11..1....1.1..11.1..111..11.1..1...1.1...111...111..1.11...1. ....11.11.11..1..11.11..1....1.1..11.1..111..11.1..1....11...111...111..11.1...1.

And from these one can of course generate a number of "minimal impossibles", some of which I've also shown.

by **StrmCkr** » Wed Aug 31, 2022 6:18 am

..1..1.11..1.11..111.1..1....1..111..1.11.1..11..1...1.1..111..1..1...111.11...1.

pretty sure the the empty cells would also be a 5 digit colourable pattern and a 6 digit non colourable.
however the 5 digit full colourable pattern leaves the 4 digit uncolourable pattern. so the 5 digit pattern is "impossible" as it leaves a uncolourable state.

by **ryokousha** » Thu Sep 01, 2022 7:40 am

Yes, they clearly cannot be 4-colored (5 cells in each house) and a 5-coloring implies the impossible 4-coloring of the complement cells. The induced connection graph of these cells (ignoring the rest of the sudoku) however is 5-chromatic, just by the method the pattern was gained.
I am not sure if the pattern is even 6-colorable. The solver is really taking its time on this one.

Edit: It is not 6-colorable, in the sense that the sudoku has no solutions. (for clarity, we're talking about the empty cells in above pattern)

by **marek stefanik** » Thu Sep 01, 2022 9:59 am

The empty cells do have solutions as a 5-digit pattern.
I believed that an exhaustive search has been made here, which only found one invalid 5-rookery, but now I see that it only covered the complements of valid 4-rookeries.

Code: Select all: +---------+---------+---------+ | 5 6 . | 7 8 . | 9 . . | | 7 8 . | 9 . . | 5 6 . | | . . 9 | . 6 5 | . 7 8 | +---------+---------+---------+ | 8 9 . | 5 7 . | . . 6 | | 6 . 7 | . . 9 | . 8 5 | | . . 5 | 6 . 8 | 7 9 . | +---------+---------+---------+ | 9 . 6 | 8 . . | . 5 7 | | . 5 8 | . 9 7 | 6 . . | | . 7 . | . 5 6 | 8 . 9 | +---------+---------+---------+

Added:
If we require the entire grid to have solutions, I can cofirm it is not 6-colorable.
Despite there being several ways to fill 26 of the remaining cells with just three digits, the last digit always breaks.

Marek

by **StrmCkr** » Thu Sep 01, 2022 5:48 pm

the conundrum here is this 5 colour pattern leaves the 4 grid cells left as impossible correct? { no solution}
which means the 5 colour pattern leaves an impossible grid so it is technically also an impossible pattern

or am i wrong in thinking.

these all should balance complementary to each other sizes just like
LS + HS = 9 digits and 9 Cells.

ie
size 2 HS has a size 7 LS

by **ryokousha** » Wed Oct 19, 2022 12:22 pm

Here's a list of 122 essentially different impossible 4-digit (= 5-chromatic) patterns with 4 cells in every house; found with the method described above, just applied a little bit more efficiently (we can assume some fixed template being present in every maximal clique, since all templates can be morphed to each other):

Hidden Text: Show

Code: Select all: .....1111..111...1111..1.....1.11..1.1.1..11.11.1...1...11.1..11...1.11.11..1.1.. .....1111.11..1..11.111........1.1111.11.1...11..1...1.1.1..11..1.1..11.1.1.11... .....1111.11..1..11.111.......11..11.11.1.1..11.1....1.1111....1....111.1....111. .....1111.11..1..11.111.......11.11.1.1..1..111..1...1.1.1..11..1.1..11.1.1.11... .....1111.11..1..11.111......1..111.1..11..1.11..1...1.1.11.1...111..1..1....1.11 .....1111.11..1..11.111.....1..1..11.1..1.1.11.11.1....1.1..11.1..1..11.1.1.11... .....1111.11..1..11.111.....1..1..11.1..1.11.1.11.1....1.1..1.11..1..11.1.1.11... .....1111.11..1..11.111.....1..1..11.1..1.11.1.11.1....1.1..11.1..1..1.11.1.11... .....1111.11..1..11.111.....1..1..111...1.1.11.11.1....1.1..11..1.1..11.1.1.11... .....1111.11..1..11.111.....1.11...1.11.1..1.1..1..1.1.1111....1....111.1....111. .....1111.11.1...1.11.11....11..1..11..1..11.1..11..1..111..1..1....111.1..11...1 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....11.11.11..11..1.11....1..1..1.111..11.1..11..1.1...1.1...11.1.11..1.1.1..11.. ....11.11.11..11..1.11....1..1.1.11..1.11..1.11.1....1.1..1..111..1.11..1.1..11.. ....11.11.11..11..1.11....1..11..11.1...111..11.1....1.1..11.1..1.1...111.1.1.1.. ....11.11.11..11..1.11....1.1..1..111..1.11..1.1..11...1.1...11.1.11..1.1.1.1.1.. ....11.11.11..11..1.11....1.1..11.1..1.1..1.11.1.1.1...1.1...111...111..1.11...1. ....11.11.11..11..1.11..1......111.1.111....111.1...1...11...111...111..11..1..1. ....11.11.11..11..1.11..1......111.1.111....111.1...1..111....11...1.11.1...11.1. ....11.11.11..11..1.11..1....1.1..111..11...111...11....1.111...1.1...1111.1...1. ....11.11.11..11..1.11..1....1.111..1..1...1111.1....1.1..11.1..111..1..1...1..11 ....11.11.11..11..1.11..1....11...11.1.11.1..11..1.1...1111....1....1.111....1.11 ....11.11.11..11..1.11..1....11...111..1.1..111..1.1....11.11...1..1..1111..1..1. ....11.11.11..11..1.11..1....11.11...1..1..1111..1...1.1..111..1..1...111.11...1. ....11.11.11..11..1.11..1....111.1..1....1.1111..1.1...1.1...11.1.1...111.1.11... ....11.11.11..11..1.11..1...1..111..1..1...111.1.1...1.1.1...11.11..11..1..11..1. ....11.11.111..1...111..1...11..1..11...1..111..1.11...11.1..1.1...111..1..1...11 ....11.11.111..1...111..1...11..1..11...1..111..11.1...11.1.1..1....1.111..1.1.1. ....11.11.111..1..1.11..1...1...1.11.1.1.1..11.1.1.1...1..111..1..1...111.1.1..1. ....11.11.111..1..1.11..1...1...1.11.11..1..11..11.1...1..111..1..1...111.1.1..1. ....11.11.111..1..1.11..1...1...1.11.11.1...11..1.1.1..1.1...111...111..1.1.1.1.. ....11.11.111..1..1.11..1...1...1.11.11.1...11..11.1...11..1.1.1...111..1..1...11 ....11.11.111..1..1.11..1...1...11.1.11.1...11..11..1..1.1...111...111..1.1..1.1. ..1..1.11..1.11..111.1..1....1..111..1.11.1..11..1...1.1..111..1..1...111.11...1. ..1..1.11..1.11..111.1..1....1..111..1.11.1..11.1....1.1..11..11..1..11.1.1.1..1. ..1..1.11..1.11..111.1..1....1.1.1.1.1.1.1.1.11.1...1...11..1.11...11.1.11..1.1.. ..1..1.11..1.11..111.1..1....1.1.1.1.1.11.1..11...1.1..1..111..1..1...111.11...1. ..1..1.11..1.11..111.1..1....1.1.11..1.1.11..11..1...1.1...11.11..11..1.1.11...1. ..1..1.11..1.11..111.1..1....1.1.11..1.11.1..11...1..1.1...11.11..11..1.1.11...1. ..1..1.11..1.11..111.1..1....1.1.11..1.11.1..11...1..1.1..111..1..1...111.11...1. ..1..1.11..1.11..111.1..1....11..1.1.1.1.1.1.11..1.1...1..1..111..1.1.1.1.1.1.1.. ..1..1.11..1.11..111.1..1....11..1.1.1.1.11..11..1..1..11.1.1..1...1..111..1.1.1. ..1..1.11..1.11..111.1..1....11..11..1..11..111..1..1..111..1..1...111..1..1...11 ..1..1.11..1.11..111.1..1....11..11..1..11.1.11..1...1.111..1..1...111..1..1...11 ..1..1.11..1.11..111.1..1....11..11..1.11.1..11...1..1.1.1.1..11...1.11.1.1.1..1. ..1..1.11..1.11..111.1..1...1...1.11.1.11..1.1.11..1...111..1..1...1..111...111.. ..1..1.11..1.11..111.1..1...1...11.1.111..1..1..11..1..1..11.1.1...1..111.11..1.. ..1..1.11..1.11..111.1..1...1..1..11.1..111..1.11...1..111..1..1...111..1..1...11 ..1..1.11..1.11..111.1..1...1..1..11.1..111..1.11..1...111..1..1...1..111..1.1.1. ..1..1.11..1.11..111.1..1...1..1..11.1..111..1.11..1...111..1..1...11.1.1..1...11 ..1..1.11..1.111..11.1....1..11.1..1.1..1.11.11.1..1...1..111..1...1..111.11...1.

There are certainly more, but the colab notebook I did the clique search in disconnected after 23h.

And yes, the complements of these must have more than 5 digits in a valid sudoku, but are perfectly 5-colorable in a vacuum. So the reason they don't work is their impossible complement alone.

by **ryokousha** » Thu Oct 20, 2022 12:02 pm

This one is pretty nice, a minimization of one of the 122 patterns with 21 cells:

Code: Select all: ------------------------------ X . . | . . . | . . . X . . | . . . | . . . . X . | . . X | X X X ------------------------------ . . . | . . X | . . . X . . | X . X | X . X . . . | . . X | . . . ------------------------------ X . . | . . X | X X X . X . | . . . | . . . . X . | . . . | . . . ------------------------------

(again, 5 colorable in a vacuum, but the sudoku fails)
This one is not difficult to prove and comprehend. So at least some of the 122 patterns are not so mysterious at all.

by **shye** » Wed Nov 30, 2022 8:35 pm

(at risk of mentioning something already discussed)

here is a chromatic pattern for 3 digits
i found it in a puzzle generated by jovi, and marek came up with a very elegant proof for it

Code: Select all: +-------+-------+-------+ | # . . | . . . | # . . | | . # . | . . . | . # . | | . . . | . . . | . . # | +-------+-------+-------+ | . . . | . . . | . . . | | # # . | . . . | . # . | | . . . | . . . | . . # | +-------+-------+-------+ | . . . | . . . | . . . | | . # . | . . . | . # . | | # . . | . . . | . . # | +-------+-------+-------+

c12, c28, c89 have the same permutation parity, so all four columns have the same parity
at the same time c189 must all have different permutations (alignment in r5, r9, and s3)
r1c1, r2c8 and r3c9 form a remote triple, and break r1c7

this was the puzzle that had the pattern being useful

Code: Select all: .-----------------.----------------.-------------------. |#789 5789 46 | 4789 56 #789 | 1 2 3 | | 12 3789 #789 | 2789 #789 13 | 4 5 6 | | 12 35 46 | 24 56 13 | 7 8 9 | :-----------------+----------------+-------------------: |#789 6789 1 |#789 2 4 | 35689 3679 578 | | 3 4 #789 | 6 #789 5 | 289 179 1278 | | 5 6789 2 | 1 3 #789 | 689 4679 478 | :-----------------+----------------+-------------------: |#789 2 #5-789| 3 #789 6 | 589 14 14 | | 4 789 3 | 5 1 #789 | 2689 679 278 | | 6 1 5789 | 789 4 2 | 3589 379 578 | '-----------------'----------------'-------------------' ......123......456......789..1.24...34.6.5...5.213.....2.3.6...4.351....61..42... SER 10.5/10.5/2.6

from here it solves with a tridagon and RT relabeling logic using that tridagon (solves with just chromatic patterns!

)

Website · by **denis_berthier** » Thu Dec 01, 2022 3:15 am

shye wrote:
Code: Select all
+-------+-------+-------+ | # . . | . . . | # . . | | . # . | . . . | . # . | | . . . | . . . | . . # | +-------+-------+-------+ | . . . | . . . | . . . | | # # . | . . . | . # . | | . . . | . . . | . . # | +-------+-------+-------+ | . . . | . . . | . . . | | . # . | . . . | . # . | | # . . | . . . | . . # | +-------+-------+-------+

This 3-digit pattern can be proven contradictory in T&E(2)

shye wrote:this was the puzzle that had the pattern being useful
Code: Select all
......123......456......789..1.24...34.6.5...5.213.....2.3.6...4.351....61..42... SER 10.5/10.5/2.6

from here it solves with a tridagon and RT relabeling logic using that tridagon (solves with just chromatic patterns! )

Is this puzzle from mith's collection ?

I haven't coded this pattern so I can't say if it's useful, but If you use both tridagon rules and replacement, you can get a very elementary solution:

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 789 5789 46789 ! 4789 56789 789 ! 1 2 3 ! ! 12789 3789 789 ! 2789 789 13789 ! 4 5 6 ! ! 12 35 46 ! 24 56 13 ! 7 8 9 ! +-------------------+-------------------+-------------------+ ! 789 6789 1 ! 789 2 4 ! 35689 3679 578 ! ! 3 4 789 ! 6 789 5 ! 289 179 1278 ! ! 5 6789 2 ! 1 3 789 ! 689 4679 478 ! +-------------------+-------------------+-------------------+ ! 789 2 5789 ! 3 789 6 ! 589 1479 14578 ! ! 4 789 3 ! 5 1 789 ! 2689 679 278 ! ! 6 1 5789 ! 789 4 2 ! 3589 379 578 ! +-------------------+-------------------+-------------------+ 170 candidates. hidden-pairs-in-a-row: r7{n1 n4}{c8 c9} ==> r7c9≠8, r7c9≠7, r7c9≠5, r7c8≠9, r7c8≠7 hidden-pairs-in-a-column: c6{n1 n3}{r2 r3} ==> r2c6≠9, r2c6≠8, r2c6≠7 hidden-pairs-in-a-column: c1{n1 n2}{r2 r3} ==> r2c1≠9, r2c1≠8, r2c1≠7 hidden-pairs-in-a-column: c5{n5 n6}{r1 r3} ==> r1c5≠9, r1c5≠8, r1c5≠7 hidden-pairs-in-a-column: c3{n4 n6}{r1 r3} ==> r1c3≠9, r1c3≠8, r1c3≠7 +-------------------+-------------------+-------------------+ ! 789 5789 46 ! 4789 56 789 ! 1 2 3 ! ! 12 3789 789 ! 2789 789 13 ! 4 5 6 ! ! 12 35 46 ! 24 56 13 ! 7 8 9 ! +-------------------+-------------------+-------------------+ ! 789 6789 1 ! 789 2 4 ! 35689 3679 578 ! ! 3 4 789 ! 6 789 5 ! 289 179 1278 ! ! 5 6789 2 ! 1 3 789 ! 689 4679 478 ! +-------------------+-------------------+-------------------+ ! 789 2 5789 ! 3 789 6 ! 589 14 14 ! ! 4 789 3 ! 5 1 789 ! 2689 679 278 ! ! 6 1 5789 ! 789 4 2 ! 3589 379 578 ! +-------------------+-------------------+-------------------+ OR2-anti-tridagon[12] for digits 7, 8 and 9 in blocks: b4, with cells: r4c1, r5c3, r6c2 b5, with cells: r4c4, r5c5, r6c6 b7, with cells: r7c1, r9c3, r8c2 b8, with cells: r7c5, r9c4, r8c6 with 2 guardians: n6r6c2 n5r9c3

Trid-OR2-whip[4]: c9n5{r4 r9} - OR2{{n5r9c3 | n6r6c2}} - b6n6{r6c7 r4c8} - r4n3{c8 .} ==> r4c7≠5

Code: Select all: hidden-single-in-a-block ==> r4c9=5 +----------------+----------------+----------------+ ! 789 5789 46 ! 4789 56 789 ! 1 2 3 ! ! 12 3789 789 ! 2789 789 13 ! 4 5 6 ! ! 12 35 46 ! 24 56 13 ! 7 8 9 ! +----------------+----------------+----------------+ ! 789 6789 1 ! 789 2 4 ! 3689 3679 5 ! ! 3 4 789 ! 6 789 5 ! 289 179 1278 ! ! 5 6789 2 ! 1 3 789 ! 689 4679 478 ! +----------------+----------------+----------------+ ! 789 2 5789 ! 3 789 6 ! 589 14 14 ! ! 4 789 3 ! 5 1 789 ! 2689 679 278 ! ! 6 1 5789 ! 789 4 2 ! 3589 379 78 ! +----------------+----------------+----------------+

***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE *****
RELEVANT DIGIT REPLACEMENTS WILL BE NECESSARY AT THE END, based on the original givens.

Code: Select all: Trying in block 8 +-------------------+-------------------+-------------------+ ! 789 5789 46 ! 4789 56 789 ! 1 2 3 ! ! 12 3789 789 ! 2789 789 13 ! 4 5 6 ! ! 12 35 46 ! 24 56 13 ! 789 789 789 ! +-------------------+-------------------+-------------------+ ! 789 6789 1 ! 789 2 4 ! 36789 36789 5 ! ! 3 4 789 ! 6 789 5 ! 2789 1789 12789 ! ! 5 6789 2 ! 1 3 789 ! 6789 46789 4789 ! +-------------------+-------------------+-------------------+ ! 789 2 5789 ! 3 9 6 ! 5789 14 14 ! ! 4 789 3 ! 5 1 8 ! 26789 6789 2789 ! ! 6 1 5789 ! 7 4 2 ! 35789 3789 789 ! +-------------------+-------------------+-------------------+ finned-x-wing-in-columns: n9{c6 c1}{r1 r6} ==> r6c2≠9 finned-x-wing-in-columns: n9{c1 c4}{r4 r1} ==> r1c6≠9 singles ==> r1c6=7, r2c5=8, r5c5=7, r6c6=9, r4c4=8

Trid-OR2-whip[2]: OR2{{n5r9c3 | n6r6c2}} - b4n8{r6c2 .} ==> r9c3≠8

Code: Select all: whip[1]: r9n8{c9 .} ==> r7c7≠8 biv-chain[3]: c1n7{r4 r7} - r7n8{c1 c3} - r5c3{n8 n9} ==> r4c1≠9 stte

by **shye** » Thu Dec 01, 2022 6:21 am

denis_berthier wrote:Is this puzzle from mith's collection ?

no it was from a search jovi is doing on tridagon puzzles with 4 empty boxes (ongoing search i believe)

thanks for looking into it, the replacement method is very powerful here! can solve with just replacement and a handful of very easy deductions
12 hidden pair in c1
9r1c1 = 9r4c1 - 9r4c4 = 9r123c4 => -9r1c6
(9=8)r1c1 - 8r13c2 = 8r6c2 - (8=9)r5c3 => -9r123c3, -9r4c1
stte

this puzzle wasnt designed around the aforementioned pattern though, its just where i happened to find it. so naturally, there will be other arguably easier methods

(especially when it was designed to have a near-tridagon lol)

by **shye** » Thu Dec 01, 2022 7:09 am

shye wrote:this puzzle wasnt designed around the aforementioned pattern though, its just where i happened to find it

here is a quickly thrown together 9.0 that should show it being a bit more useful hopefully

Code: Select all: +-------+-------+-------+ | . 6 . | . . . | . 1 2 | | 1 . 5 | . . . | 3 . 4 | | 2 3 . | 8 . . | 5 6 . | +-------+-------+-------+ | 4 1 . | . . 9 | . 3 5 | | . . 3 | 5 . . | . . 1 | | 5 2 . | . . 3 | . 4 . | +-------+-------+-------+ | 3 . . | . 7 . | . 2 6 | | 6 . . | . . . | . . 3 | | . 4 . | . . 6 | . 5 . | +-------+-------+-------+ .6.....121.5...3.423.8..56.41...9.35..35....152...3.4.3...7..266.......3.4...6.5.

solves pretty easily with:
789 naked triple in r5
chromatic pattern on 789 => +5r8c2
singles
avoidable rectangle (58r78c26) => -8r7c2
dual empty rectangle (8r2c8 = r5c8 - b4p45 = b4p39 - r7c3 = r7c7) => -8r1c7
stte

Website · by **denis_berthier** » Thu Dec 01, 2022 8:28 am

.
As I said in my previous post, I haven't coded his pattern. However, I can simulate its action.
This will eliminate the other 3 candidates in r8c2 before any other rule is applied, and therefore r8c2=5 will be asserted at the start:
(bind ?*simulated-eliminations* (create$ 782 882 982))

Code: Select all: (solve ".6.....121.5...3.423.8..56.41...9.35..35....152...3.4.3...7..266.......3.4...6.5.") Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 789 6 4789 ! 3 5 47 ! 789 1 2 ! ! 1 789 5 ! 2679 269 27 ! 3 789 4 ! ! 2 3 479 ! 8 149 147 ! 5 6 79 ! +-------------------+-------------------+-------------------+ ! 4 1 678 ! 267 268 9 ! 2678 3 5 ! ! 789 789 3 ! 5 2468 2478 ! 26789 789 1 ! ! 5 2 6789 ! 167 168 3 ! 6789 4 789 ! +-------------------+-------------------+-------------------+ ! 3 89 189 ! 149 7 5 ! 1489 2 6 ! ! 6 5 1279 ! 1249 1289 128 ! 1479 79 3 ! ! 789 4 12789 ! 129 3 6 ! 1789 5 789 ! +-------------------+-------------------+-------------------+ 141 candidates. finned-x-wing-in-columns: n7{c2 c6}{r5 r2} ==> r2c4≠7 whip[1]: c4n7{r6 .} ==> r5c6≠7 naked-triplets-in-a-row: r5{c1 c2 c8}{n9 n8 n7} ==> r5c7≠9, r5c7≠8, r5c7≠7, r5c6≠8, r5c5≠8 singles ==> r8c6=8, r3c6=1 finned-x-wing-in-rows: n7{r3 r8}{c3 c9} ==> r9c9≠7 biv-chain[3]: r1c6{n7 n4} - b1n4{r1c3 r3c3} - r3n7{c3 c9} ==> r1c7≠7 biv-chain[3]: r1c7{n9 n8} - r2n8{c8 c2} - r7c2{n8 n9} ==> r7c7≠9 z-chain[3]: b9n8{r9c7 r9c9} - c1n8{r9 r5} - c8n8{r5 .} ==> r1c7≠8 singles ==> r1c7=9, r3c9=7, r2c8=8 whip[1]: r5n8{c2 .} ==> r4c3≠8, r6c3≠8 finned-x-wing-in-columns: n9{c1 c8}{r5 r9} ==> r9c9≠9 stte

That's a solution in Z3. Without the pattern, the solution is in W7 (unless you use uniqueness, which flattens everything).
Is this an example of a useful case of the patttern? It's hard to say until one analyses the complexity of the pattern itself. Sure, a puzzle in T&E(2) would be more conclusive.

Website · by **denis_berthier** » Thu Jan 19, 2023 8:06 am

eleven wrote:I have generated now 680 non equivalent 3-digit patterns in 6 boxes (10 cells: 31, 12: 38, 13: 290, 14: 159, 15: 102, 16: 10). This should be all, if i made no mistake.
I didn't see one more, which i would suspect to be as hard as the TH or the mentioned 15 cell pattern.
Added: i have copied a zip file to my old google drive: here (list). As patterns here (wait a bit, until it is shown). [Edit: updated link to patterns (10 cells were missing)]

I had completely missed this post.
What's interesting is, instead of one pattern potentially in T&E(3) - i.e. that can't be proven contradictory in the simplified version of T&E(2), restricted to cells in the pattern - we now have 10:

Code: Select all: 000001001000010010000100100000001001000010100000100010000000000000000000000000000 #38 12-cells 000000000001001001001010010000001010000010001001100100000000000000000000000000000 #171 13-cells 000000000001001001001010010001000000010001010010010001000000000000000000000000000 #180 13-cells 000000000001001001001010010001000000010001010100010001000000000000000000000000000 #181 13-cells 000000001000011010001100100001000001010100001100100010000000000000000000000000000 #56 15-cells 000000001001001010010010100001001000010100001100010001000000000000000000000000000 #66 15-cells 000001001001000010010010100000010001001001100010100010000000000000000000000000000 #97 15-cells (ex #37) 000001001001010010001010010001000001010001100010001100000000000000000000000000000 #5 16-cells 000001001001010010001010010001000001010001100100001100000000000000000000000000000 #6 16-cells 000001001001010010001010100001001000010100001100100001000000000000000000000000000 #9 16-cells

For a more visual representation:
(I've tried to find morphs with as many anti-diag cells as possible.)

Code: Select all: +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . . ! . 1 . ! . 1 . ! ! . . . ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . . ! . 1 . ! 1 . . ! ! . . . ! 1 . . ! . 1 . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000001001000010010000100100000001001000010100000100010000000000000000000000000000 #38 12-cells +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . 1 ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! +-------+-------+-------+ ! . . . ! . . 1 ! . 1 . ! ! . . . ! . 1 . ! . . 1 ! ! . . 1 ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000000000001001001001010010000001010000010001001100100000000000000000000000000000 #171 13-cells isomorphic to: +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . 1 ! . . 1 ! . 1 . ! ! . . 1 ! . 1 . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . . ! . 1 . ! . 1 . ! ! . . 1 ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . 1 ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . . ! ! . 1 . ! . . 1 ! . 1 . ! ! . 1 . ! . 1 . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000000000001001001001010010001000000010001010010010001000000000000000000000000000 #180 13-cells isomorphic to: +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . 1 ! . 1 . ! . 1 . ! ! . . 1 ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . . ! ! . 1 . ! . 1 . ! 1 . . ! ! . 1 . ! 1 . . ! . 1 . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . 1 ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . . ! ! . 1 . ! . . 1 ! . 1 . ! ! 1 . . ! . 1 . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000000000001001001001010010001000000010001010100010001000000000000000000000000000 #181 13-cells +-------+-------+-------+ ! . . . ! . . . ! . . 1 ! ! . . . ! . 1 1 ! . 1 . ! ! . . 1 ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . 1 ! ! . 1 . ! 1 . . ! . . 1 ! ! 1 . . ! 1 . . ! . 1 . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000000001000011010001100100001000001010100001100100010000000000000000000000000000 #56 15-cells +-------+-------+-------+ ! . . . ! . . . ! . . 1 ! ! . . 1 ! . . 1 ! . 1 . ! ! . 1 . ! . 1 . ! 1 . . ! +-------+-------+-------+ ! . . 1 ! . . 1 ! . . . ! ! . 1 . ! 1 . . ! . . 1 ! ! 1 . . ! . 1 . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000000001001001010010010100001001000010100001100010001000000000000000000000000000 #66 15-cells isomorphic to: +-------+-------+-------+ ! . . . ! . . . ! . . 1 ! ! . . 1 ! . . 1 ! . 1 . ! ! . 1 . ! 1 . . ! 1 . . ! +-------+-------+-------+ ! . . 1 ! . . 1 ! . . . ! ! . 1 . ! . 1 . ! . . 1 ! ! 1 . . ! 1 . . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . 1 ! . . . ! . 1 . ! ! . 1 . ! . 1 . ! 1 . . ! +-------+-------+-------+ ! . . . ! . 1 . ! . . 1 ! ! . . 1 ! . . 1 ! 1 . . ! ! . 1 . ! 1 . . ! . 1 . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000001001001000010010010100000010001001001100010100010000000000000000000000000000 #97 15-cells (ex #37) isomorphic to: +-------+-------+-------+ ! . . . ! X . . ! . . X ! ! . . X ! . . . ! . X . ! ! . X . ! . . X ! X . . ! +-------+-------+-------+ ! . . . ! . . X ! . . X ! ! . X . ! . X . ! . X . ! ! . . X ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ also iosmorphic to a saucepan with lid +-------+-------+-------+ ! . . . ! . . X ! . . X ! ! . . . ! . X . ! . X . ! ! . . . ! X . . ! X . . ! +-------+-------+-------+ ! . . . ! . . X ! X . . ! ! . . . ! . X . ! . . . ! ! . . . ! X . . ! . . X ! +-------+-------+-------+ ! . . . ! X . . ! . X . ! ! . . . ! . X . ! X . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! ! . . 1 ! . 1 . ! . 1 . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . 1 ! ! . 1 . ! . . 1 ! 1 . . ! ! . 1 . ! . . 1 ! 1 . . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000001001001010010001010010001000001010001100010001100000000000000000000000000000 #5 16-cells +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! ! . . 1 ! . 1 . ! . 1 . ! +-------+-------+-------+ ! . . 1 ! . . . ! . . 1 ! ! . 1 . ! . . 1 ! 1 . . ! ! 1 . . ! . . 1 ! 1 . . ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000001001001010010001010010001000001010001100100001100000000000000000000000000000 #6 16-cells +-------+-------+-------+ ! . . . ! . . 1 ! . . 1 ! ! . . 1 ! . 1 . ! . 1 . ! ! . . 1 ! . 1 . ! 1 . . ! +-------+-------+-------+ ! . . 1 ! . . 1 ! . . . ! ! . 1 . ! 1 . . ! . . 1 ! ! 1 . . ! 1 . . ! . . 1 ! +-------+-------+-------+ ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! ! . . . ! . . . ! . . . ! +-------+-------+-------+ 000001001001010010001010100001001000010100001100100001000000000000000000000000000 #9 16-cells

I don't know if any of these patterns (except ex-37) has a real puzzle with a reasonable number of guardians.
[Edit]:added s the saucepan representation of ex-#37

by **totuan** » Thu Jan 19, 2023 2:12 pm

denis_berthier wrote:I don't know if any of these patterns (except ex-37) has a real puzzle with a reasonable number of guardians.

Perhaps, this one:

Code: Select all: ...4.6...4.7.892.6.8.27....2.46.7.9...8.92.4.9..84.......72..6.......5........1.3;504318;29;3;11.1;10.9;10.5;DCFC+MFC

After some moves, come to below: 15 cells impossible pattern (don't know on eleven’s list or not):

Code: Select all: *-----------------------------------------------------------------------------* |*135 12359 12359 | 4 *135 6 | 379 8 1579 | | 4 *135 7 |*135 8 9 | 2 *135 6 | | 6 8 1359 | 2 7 *135 | 349 135 1459 | |-------------------------+-------------------------+-------------------------| | 2 *135 4 | 6 *135 7 | 38 9 158 | |*135 13567 8 |*135 9 2 | 367 4 157 | | 9 *1357 1356 | 8 4 *135 | 367 *135 2 | |-------------------------+-------------------------+-------------------------| |*1358 13459 1359 | 7 2 *13458 | 489 6 489 | | 1378 123469 12369 | 139 136 1348 | 5 27 489 | | 578 24569 2569 | 59 56 458 | 1 27 3 | *-----------------------------------------------------------------------------*

Impossible pattern(135) * marked cells => (7)r6c2=(48)r7c16

Code: Select all: *-----------------------------------------------------------* | 135 . . | . 135 . | . . . | | . 135 . | 135 . . | . 135 . | | . . . | . . 135 | . . . | |-------------------+-------------------+-------------------| | . 135 . | . 135 . | . . . | | 135 . . | 135 . . | . . . | | . 135 . | . . 135 | . 135 . | |-------------------+-------------------+-------------------| | 135 . . | . . A135 | . . . | | . . . | . . . | . . . | | . . . | . . . | . . . | *-----------------------------------------------------------* A=(1|3|5) => impossible

Move: (7)r6c2==(489)r7c1679-(49=135)r247c2-(135=7)r6c2 => r6c2=7
I solved this one and have to use 13, 14, 15, 17 cells impossible pattern.

totuan

Website · by **denis_berthier** » Thu Jan 19, 2023 4:48 pm

totuan wrote:
denis_berthier wrote:I don't know if any of these patterns (except ex-37) has a real puzzle with a reasonable number of guardians.

Perhaps, this one:
Code: Select all
...4.6...4.7.892.6.8.27....2.46.7.9...8.92.4.9..84.......72..6.......5........1.3;504318;29;3;11.1;10.9;10.5;DCFC+MFC

After some moves, come to below: 15 cells impossible pattern (don't know on eleven’s list or not):

Hi totuan;
It's always interesting to have a real puzzle for a pattern. 2 remarks:
- it can't be on eleven's list: all his patterns span only 2 bands;
- it's contradictory in T&E(2).

To check the latter, select only T&E(2) in SudoRules configuration file and type:
(solve-k-digit-pattern-string 3 "X...X.....X.X...X......X....X..X....X..X......X...X.X.X....X.....................")
After 2s or so, you will get :
"PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RC-CELL r2c8"

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