## Chromatic Patterns

Advanced methods and approaches for solving Sudoku puzzles

### Re: Chromatic Patterns

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:

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.
ryokousha

Posts: 37
Joined: 30 April 2022

### Re: Chromatic Patterns

..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.
Last edited by StrmCkr on Thu Sep 01, 2022 5:50 pm, edited 2 times in total.
Some do, some teach, the rest look it up.
stormdoku

StrmCkr

Posts: 1336
Joined: 05 September 2006

### Re: Chromatic Patterns

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)
Last edited by ryokousha on Thu Sep 01, 2022 6:45 pm, edited 2 times in total.
ryokousha

Posts: 37
Joined: 30 April 2022

### Re: Chromatic Patterns

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 |   +---------+---------+---------+`

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
marek stefanik

Posts: 316
Joined: 05 May 2021

### Re: Chromatic Patterns

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
Some do, some teach, the rest look it up.
stormdoku

StrmCkr

Posts: 1336
Joined: 05 September 2006

### Re: Chromatic Patterns

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.....1111.11.1...1.11.11....111...1.1....11.11...11..1.1111....1..1..11.1..1..11......1111.11.1...11.11.1.......1.1111.11....111..11....1.1..11..1.1..11.1.1.11........1111.11.1...11.11.1....1....1111...11..11.111.....1.1..11..1.1..11.1.1.11........1111.11.1...11.11.1....1...111.1..11...11.1.1...1.1.1..11..1.1..11.1.1.11........1111.11.1...11.11.1....1..11..11...1.11.1.11....1.1.1..11..1.1..11.1.1.11........1111.11.1...11.11.1....1..11.1.1...1.1.11.11....1.1.1..11..1.1..11.1.1.11........1111.11.1...11.11.1....1..11.1.1...1.11.1.11....1.1.1..1.1.1.1..11.1.1.11........1111.11.1...11.11.1....1.1..11..1.1..11.1.1.11....1.1..11.1...11..11.1.1...1.....1111.11.1...11.111........1111.1.11....111..1..1..1.1..11..1.1.11..1.1..1..1.....1111.11.1...11.111.......1.111.1.1..1..111.1...1..1..1.11..1..111..1.11....1.....1111.11.1...11.111.......11.11..11..1..11.1..1..1.1.11..1.1..1..11.11...11.......1111.11.1...11.111.......11.11..11..1..11.1..1..1.1.11..1.1..1.11..11....11......1111.11.1...11.111.......11.11..11..1..111...1..1..111..1.1..1..11.11...11.......1111.11.1...11.111.......11.11..11..1..111...1..1..111..1.1..1.11..11....11......1111.11.1...11.111.......11.11..11..1..111...1..1.1111....1....111.1..1..11......1111.11.1...11.111......11.1..11...1.11.11...1..1.1.1..11..1.1..11.1.1.11........1111.11.1...11.111.....1..1.11..1..11.1.1.11....1.1.1.11..1..1..11.1.1..1..1.....1111.11.1...11.111.....1.11..1..111...1.1....11.1.11..11..1....1.111..11.1......11.11.11...1.11.11.1.......111.11.1.1..1.11.1...1..1.1..1.1.111..1..1...11.1.....11.11.11...1.11.11.1.......111.11.11..1..11.1...1..1..11.1..11.1..1.1..1..1.1....11.11.11...1.11.11.1.......111.111.1...1.11.1...1...1.11.1...11..1.111..1.1......11.11.11...1.11.11.1.......1111.1.111....11....1.1..1..11.1.1.11..1.11.1...1.....11.11.11...1.11.11.1......1.111..1111....11.....11..1..1.111..11.1..11..1.1......11.11.11..1..11.1.1..1..1..1..111..1.11..1.11..1...1.1..1.1.111...1.1...111......11.11.11..1..11.1.1.1...1..11..11..1..11.1.11....1.1.1..11..1.1..11.1.1.11.......11.11.11..1..11.1.1.1...1..111...111...1.1..1...11.1.1..1.11...11.1.1.11..1......11.11.11..1..11.1.1.1...1..111..1..1..11.1.11....1.1.1...11.1.1..11.1.1.11.......11.11.11..1..11.11..1......11.11.11.1..1.11.1..1...111..1..1....111.1..11...1....11.11.11..1..11.11..1......11.11.11.1..1.11.1..1...111..1..1...111..1..1...11....11.11.11..1..11.11..1......111.1.111...1.11.1...1..111....11...1.11.1...111......11.11.11..1..11.11..1......111.1.111...1.11.1..1...1.1..1.11...11.1.1.1.1..1.....11.11.11..1..11.11..1....1..111.1..11..1.11.1....1.1.11.1...11.1.1..1....1.11....11.11.11..1..11.11..1....1.1..11.1..11.1.11.1..1...111..1..1...111..1..1...11....11.11.11..1..11.11..1....1.1..11.1..111..11.1..1....11...111...111..11.1...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...111...1.1...111..1..1...11....11.11.11..1..11.11..1....1.111...1.1..11.11.1...1..111..1..1...1..111...11..1....11.11.11..1..11.11..1....11..1.11...111..11.1...1..1..11..1.11.1..1.1..1..11.....11.11.11..1..11.11..1....11..11.1...111..11.1....1.1..11.1..11.1...11..1..11.....11.11.11..1..11.11..1...1...1.11.11.1..1.1..11.1...111..1..1....111.1..11...1....11.11.11..1..11.11..1...1...1.11.11.1..1.1..11.1...111..1..1...1.1.11..1.1.1.....11.11.11..1..11.11..1...1...1.11.11.1..1.1..11.1...111..1..1...111..1..1...11....11.11.11..1..11.11..1...1..1..11.11..1.1.1..11.1...111..1..1...1.1.11..1.1.1.....11.11.11..1..11.11..1...1..1..11.11..1.1.1..11.1...111..1..1...111..1..1...11....11.11.11..1..11.11..1...1..1..11.11.1..1.1..1.11...111..1..1...111..1..1...11....11.11.11..1..11.11..1...1..1..11.11.11...1..1..11..111..1..1...111..1..1...11....11.11.11..1..11.11..1...1..1.1.1.111...1.1..1.1.1..1.11..1.1...1.1.11.1..11......11.11.11..1..11.11..1...1..1.11..1.1...111.11.1....1..111..1...1..111.11..1......11.11.11..1..11.11..1...1..1.11.1..1.1.1.1.11....1.1..111...1.1..11.1.1.1...1....11.11.11..1..11.11..1...1..111...1.1..11.1.1.1...1.1..111..1..1...111.11...1.....11.11.11..1..11.11..1...1..111...1.1..11.1.1.1...1.1.1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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.
ryokousha

Posts: 37
Joined: 30 April 2022

### Re: Chromatic Patterns

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.
ryokousha

Posts: 37
Joined: 30 April 2022

### Re: Chromatic Patterns

(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! )

shye

Posts: 245
Joined: 12 June 2021

### Re: Chromatic Patterns

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≠7hidden-pairs-in-a-column: c6{n1 n3}{r2 r3} ==> r2c6≠9, r2c6≠8, r2c6≠7hidden-pairs-in-a-column: c1{n1 n2}{r2 r3} ==> r2c1≠9, r2c1≠8, r2c1≠7hidden-pairs-in-a-column: c5{n5 n6}{r1 r3} ==> r1c5≠9, r1c5≠8, r1c5≠7hidden-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, r8c6with 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≠9finned-x-wing-in-columns: n9{c1 c4}{r4 r1} ==> r1c6≠9singles ==> r1c6=7, r2c5=8, r5c5=7, r6c6=9, r4c4=8`

Trid-OR2-whip[2]: OR2{{n5r9c3 | n6r6c2}} - b4n8{r6c2 .} ==> r9c3≠8
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`whip[1]: r9n8{c9 .} ==> r7c7≠8biv-chain[3]: c1n7{r4 r7} - r7n8{c1 c3} - r5c3{n8 n9} ==> r4c1≠9stte`
denis_berthier
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### Re: Chromatic Patterns

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)

shye

Posts: 245
Joined: 12 June 2021

### Re: Chromatic Patterns

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

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

shye

Posts: 245
Joined: 12 June 2021

### Re: Chromatic Patterns

.
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))
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`(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≠7whip[1]: c4n7{r6 .} ==> r5c6≠7naked-triplets-in-a-row: r5{c1 c2 c8}{n9 n8 n7} ==> r5c7≠9, r5c7≠8, r5c7≠7, r5c6≠8, r5c5≠8singles ==> r8c6=8, r3c6=1finned-x-wing-in-rows: n7{r3 r8}{c3 c9} ==> r9c9≠7biv-chain[3]: r1c6{n7 n4} - b1n4{r1c3 r3c3} - r3n7{c3 c9} ==> r1c7≠7biv-chain[3]: r1c7{n9 n8} - r2n8{c8 c2} - r7c2{n8 n9} ==> r7c7≠9z-chain[3]: b9n8{r9c7 r9c9} - c1n8{r9 r5} - c8n8{r5 .} ==> r1c7≠8singles ==> r1c7=9, r3c9=7, r2c8=8whip[1]: r5n8{c2 .} ==> r4c3≠8, r6c3≠8finned-x-wing-in-columns: n9{c1 c8}{r5 r9} ==> r9c9≠9stte`

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.
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

### Re: Chromatic Patterns

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:
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`000001001000010010000100100000001001000010100000100010000000000000000000000000000 #38 12-cells000000000001001001001010010000001010000010001001100100000000000000000000000000000 #171 13-cells000000000001001001001010010001000000010001010010010001000000000000000000000000000 #180 13-cells000000000001001001001010010001000000010001010100010001000000000000000000000000000 #181 13-cells000000001000011010001100100001000001010100001100100010000000000000000000000000000 #56 15-cells000000001001001010010010100001001000010100001100010001000000000000000000000000000 #66 15-cells000001001001000010010010100000010001001001100010100010000000000000000000000000000 #97 15-cells (ex #37)000001001001010010001010010001000001010001100010001100000000000000000000000000000 #5 16-cells000001001001010010001010010001000001010001100100001100000000000000000000000000000 #6 16-cells000001001001010010001010100001001000010100001100100001000000000000000000000000000 #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-cellsisomorphic 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-cellsisomorphic 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-cellsisomorphic 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.
:added s the saucepan representation of ex-#37
Last edited by denis_berthier on Fri Jan 20, 2023 8:07 am, edited 4 times in total.
denis_berthier
2010 Supporter

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Location: Paris

### Re: Chromatic Patterns

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
totuan

Posts: 190
Joined: 25 May 2010
Location: vietnam

### Re: Chromatic Patterns

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;

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"
denis_berthier
2010 Supporter

Posts: 3676
Joined: 19 June 2007
Location: Paris

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