The New Sudoku Players' Forum

by **eleven** » Tue Apr 12, 2022 7:59 pm

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

So i can find the pattern quickly

by **jco** » Wed Apr 13, 2022 4:50 pm

Probably there is a much shorter and simpler solution, but my findings are the following.

0. After basics (to 47 known numbers) and applying eleven's replacement approach

Step 1: r3c1 = x, r5c2 = y and r6c3 = z.

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | 129 xyz59 8 | 7 235 135 | | 5 xz 9 | 6 xyz 7 | 13 4 8 | | 8 7 xy | 4 xyz5 xyz | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | 129 z49 z4 | yz 7 6 |<--- r4c7=y* | 7 y* 4 | 8 6 5 | xz 9 13 | | 6 9 z* | 7 xy xy | 5 8 4 | |----------------------+----------------------+----------------------| | yz4 6 xy7 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | yz4 xz xy7 | 5 8 124 | 9 6 37 | '--------------------------------------------------------------------'

=> r4c7 = y, due to r4c2 = y and (xy)r6c56.

Step 1.1

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | 129 xyz59 8 | 7 235 135 | | 5 xz 9 | 6 xyz 7 | 13 4 8 |<---r2c5=y* | 8 7 xy | 4 xyz5 xyz | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | 129 z49 z4 | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 13 | | 6 9 z* | 7 xy xy | 5 8 4 | |----------------------+----------------------+----------------------| | yz4 6 xy7 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | yz4 xz xy7 | 5 8 124 | 9 6 37 | '--------------------------------------------------------------------'

=> r2c5 = y, since it is the only place for y in row 2.

Step 1.2

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | 129 z59 8 | 7 235 135 | | 5 xz 9 | 6 y* 7 | 13 4 8 |<--- | 8 7 xy | 4 z5 xz | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z49 z4 | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 13 |<--- | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------------+----------------------| | yz4 6 xy7 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | yz4 xz xy7 | 5 8 124 | 9 6 37 | '--------------------------------------------------------------------'

due to (y*)r4c7, replace (13)r5c9 and r2c7 by xz
-----
Step 2

Code: Select all: .--------------------------------------------------------------------. |(yz) 4 6 | 129 59-z 8 | 7 235 135 | | 5 xz 9 | 6 y* 7 | xz 4 8 | | 8 7 (xy) | 4 z5 (xz) | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z49 z4 | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 xz | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------------+----------------------| | yz4 6 xy7 | 12 z4 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | yz4 xz xy7 | 5 8 124 | 9 6 37 | '--------------------------------------------------------------------'

Y-Wing (z=y)r1c1 - (y=x)r3c3 - (x=z)r3c6 => -z r1c45
-----
Step 3

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | x9 59 8 | 7 235 135 | | 5 xz 9 | 6 y* 7 | xz 4 8 | | 8 7 xy | 4 (z)5 x(z) | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z9-4 (z4) | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 xz | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+-----------------|----+----------------------| | yz4 6 xy7 | 12 (z4) 9 | | 8 35 357 | | 9 5 8 | 3 7 6 v | 4 1 2 | | yz4 xz xy7 | 5 8 12-4 | 9 6 37 | '--------------------------------------------------------------------'

Almost W-wing

(4=z)r4c6 - (z=x)r3c6 - (x=y)r3c3 - (y=z)r1c1 - (z)r2c2 = (z)r2c7 - (z*)r3c8 = [(4=z)r7c5 - (z)r3c5 =* (z)r3c6 - (z=4)r4c6]

=> -4 r4c5, -9 r9c6

Step 3.1

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | x9 59 8 | 7 235 135 | | 5 xz 9 | 6 y* 7 | xz 4 8 | | 8 7 xy | 4 z5 xz | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z9 4* | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 xz | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------------+----------------------| | yz 6 xy7 | xz 4* 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4* xz xy7 | 5 8 xz | 9 6 37 | '--------------------------------------------------------------------'

=> x,z=1,2 and y=3 (due to block 8)

Step 4

Code: Select all: .--------------------------------------------------------------------. |(3)z 4 6 | x9 59 8 | 7 235 135 | | 5 xz 9 | 6 3* 7 | xz 4 8 | | 8 7 x(3) | 4 z5 xz | 6 2(3)-5 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z9 4* | 3* 7 6 | | 7 3* 4 | 8 6 5 | xz 9 xz | | 6 9 z* | 7 x* 3* | 5 8 4 | |----------------------+----------------------+----------------------| |(3)z 6 x37 | xz 4* 9 | 8 (35) 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4* xz x37 | 5 8 xz | 9 6 37 | '--------------------------------------------------------------------'

(5=3)r7c8 - (3)r7c1 = (3)r1c1 - (3)r3c3 = (3)r3c8 => -5 r3c8 [since x and z are not 3]

[r3c5 = 5, r1c5 = 9, r4c4 = 9 and more singles]

Step 4.1

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

Noticing block 1, we infer that z=1, so x = 2, solving the puzzle.

Hidden Text: Show

Thanks for the puzzle!

Edit: I am very thankful to Marek for two observations:
. the w-wing stands without need of that additional chain due to (z) at block 2.
. in my step 3.1, I should have (xyz) at r7c4 instead of just (xz)r7c4.
In this case, placing 4s, using the naked pair (xz)r9c26, there is a SS that solves the puzzle (details hidden).

Hidden Text: Show

Code: Select all: .--------------------------------------------------------------------. | yz 4 6 | x9 59 8 | 7 235 135 | | 5 #xz 9 | 6 y* 7 | xz 4 8 | | 8 7 y-x | 4 z5 #xz | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | z9 z9 4* | y* 7 6 | | 7 y* 4 | 8 6 5 | xz 9 xz | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------- ----+----------------------| | yz 6 xy7 | xyz 4* 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4* #xz y7 | 5 8 #xz | 9 6 37 | '--------------------------------------------------------------------'

SS: (x) r2c2 = r9c2 - r9c6 = r3c6 => -x r3c3[/code]
Then,

Code: Select all: .--------------------------------------------------------------------. | z* 4 6 | x* 9* 8 | 7 235 135 | | 5 x* 9 | 6 y* 7 | z* 4 8 | | 8 7 y* | 4 z5 z* | 6 235 9 | |----------------------+----------------------+----------------------| | x* 8 5 | 9* z* 4* | y* 7 6 | | 7 y* 4 | 8 6 5 | x* 9 xz | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------------+----------------------| | y* 6 x* | z* 4* 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4* z* 7* | 5 8 x* | 9 6 37 | '--------------------------------------------------------------------'

y=3 (block 8)

Code: Select all: .--------------------------------------------------------------------. | z* 4 6 | x* 9* 8 | 7 3* 5* | | 5 x* 9 | 6 y* 7 | z* 4 8 | | 8 7 3* | 4 z5 z* | 6 2* 9 | |----------------------+----------------------+----------------------| | x* 8 5 | 9* z* 4* | 3* 7 6 | | 7 3* 4 | 8 6 5 | x* 9 z* | | 6 9 z* | 7 x* y* | 5 8 4 | |----------------------+----------------------+----------------------| | 3* 6 x* | z* 4* 9 | 8 5* 7* | | 9 5 8 | 3 7 6 | 4 1 2 | | 4* z* 7* | 5 8 x* | 9 6 3* | '--------------------------------------------------------------------'

=> z = 1 and x = 2, solving the puzzle.

Website · by **denis_berthier** » Thu Apr 14, 2022 5:07 am

.
SER = 9.0

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 123 4 6 ! 129 12359 8 ! 7 235 135 ! ! 5 123 9 ! 6 123 7 ! 13 4 8 ! ! 8 7 123 ! 4 1235 123 ! 6 235 9 ! +-------------------+-------------------+-------------------+ ! 123 8 5 ! 129 12349 1234 ! 123 7 6 ! ! 7 123 4 ! 8 6 5 ! 123 9 13 ! ! 6 9 123 ! 7 123 123 ! 5 8 4 ! +-------------------+-------------------+-------------------+ ! 1234 6 1237 ! 12 124 9 ! 8 35 357 ! ! 9 5 8 ! 3 7 6 ! 4 1 2 ! ! 1234 123 1237 ! 5 8 124 ! 9 6 37 ! +-------------------+-------------------+-------------------+ 107 candidates

Like jco, I'll apply eleven's replacement method, but in my own way, with no variables
In order to better show the difference, I'll choose the same block b4. (In the present case, b1 would work as well.)
In block b4, digits 1, 2, 3 appear in only 3 places and each of them must therefore occupy one of these places.
A priori, we can't arbitrarily choose which place each of them occupies, because some of these choices wouldn't be compatible with the givens (i.e. with r8c8 and r8c9).
But suppose we replace every occurrence of 1 2 3 (including the givens) in the above resolution state, by the 3 digits:

Code: Select all: +-------------------+-------------------+-------------------+ ! 123 4 6 ! 129 12359 8 ! 7 1235 1235 ! ! 5 123 9 ! 6 123 7 ! 123 4 8 ! ! 8 7 123 ! 4 1235 123 ! 6 1235 9 ! +-------------------+-------------------+-------------------+ ! 1 8 5 ! 1239 12349 1234 ! 123 7 6 ! ! 7 2 4 ! 8 6 5 ! 123 9 123 ! ! 6 9 3 ! 7 123 123 ! 5 8 4 ! +-------------------+-------------------+-------------------+ ! 1234 6 1237 ! 123 1234 9 ! 8 1235 12357 ! ! 9 5 8 ! 123 7 6 ! 4 123 123 ! ! 1234 123 1237 ! 5 8 1234 ! 9 6 1237 ! +-------------------+-------------------+-------------------+

Now, we can choose any arrangement of values among {1, 2, 3} for r4c1, r5c2, r6c3. If we get a solution, it will be a solution of the original puzzle modulo some renaming of 1, 2, 3 to be found based on the givens:

Code: Select all: +-------------------+-------------------+-------------------+ ! 123 4 6 ! 129 12359 8 ! 7 1235 1235 ! ! 5 123 9 ! 6 123 7 ! 123 4 8 ! ! 8 7 123 ! 4 1235 123 ! 6 1235 9 ! +-------------------+-------------------+-------------------+ ! 1 8 5 ! 1239 12349 1234 ! 123 7 6 ! ! 7 2 4 ! 8 6 5 ! 123 9 123 ! ! 6 9 3 ! 7 123 123 ! 5 8 4 ! +-------------------+-------------------+-------------------+ ! 1234 6 1237 ! 123 1234 9 ! 8 1235 12357 ! ! 9 5 8 ! 123 7 6 ! 4 123 123 ! ! 1234 123 1237 ! 5 8 1234 ! 9 6 1237 ! +-------------------+-------------------+-------------------+

No x, y, z here. The advantage of proceeding this way is, you can use your usual solving methods or any solver.
Using SudoRules simplest-first strategy, one gets a 2-)step solution in BC3:

Code: Select all: (solve-sukaku-grid +-------------------+-------------------+-------------------+ ! 123 4 6 ! 129 12359 8 ! 7 1235 1235 ! ! 5 123 9 ! 6 123 7 ! 123 4 8 ! ! 8 7 123 ! 4 1235 123 ! 6 1235 9 ! +-------------------+-------------------+-------------------+ ! 1 8 5 ! 1239 12349 1234 ! 123 7 6 ! ! 7 2 4 ! 8 6 5 ! 123 9 123 ! ! 6 9 3 ! 7 123 123 ! 5 8 4 ! +-------------------+-------------------+-------------------+ ! 1234 6 1237 ! 123 1234 9 ! 8 1235 12357 ! ! 9 5 8 ! 123 7 6 ! 4 123 123 ! ! 1234 123 1237 ! 5 8 1234 ! 9 6 1237 ! +-------------------+-------------------+-------------------+ ) singles ==> r4c7=2, r2c5=2, r6c5=1, r6c6=2 finned-x-wing-in-columns: n1{c6 c2}{r9 r3} ==> r3c3≠1 singles ==> r3c3=2, r1c1=3, r2c2=1, r2c7=3, r5c7=1, r5c9=3, r9c2=3 biv-chain[3]: r1c4{n1 n9} - r4c4{n9 n3} - c6n3{r4 r3} ==> r3c6≠1 stte 346198725 519627348 872453619 185934276 724865193 693712584 261349857 958276431 437581962

By observing cells r8c8 and r8c9, one can see that this is not a solution to the original puzzle; but replace 3 by 1 and 1 by 2 (and therefore 2 by 3) and you'll get the solution.

The same method can be applied to find a 1-step solution, but it will be in Z5 instead of BC3:

[For the user of SudoRules], the simplest way of doing this is:

Code: Select all: (defglobal ?*RS* = (create$ (clean-grid-list +-------------------+-------------------+-------------------+ ! 123 4 6 ! 129 12359 8 ! 7 1235 1235 ! ! 5 123 9 ! 6 123 7 ! 123 4 8 ! ! 8 7 123 ! 4 1235 123 ! 6 1235 9 ! +-------------------+-------------------+-------------------+ ! 1 8 5 ! 1239 12349 1234 ! 123 7 6 ! ! 7 2 4 ! 8 6 5 ! 123 9 123 ! ! 6 9 3 ! 7 123 123 ! 5 8 4 ! +-------------------+-------------------+-------------------+ ! 1234 6 1237 ! 123 1234 9 ! 8 1235 12357 ! ! 9 5 8 ! 123 7 6 ! 4 123 123 ! ! 1234 123 1237 ! 5 8 1234 ! 9 6 1237 ! +-------------------+-------------------+-------------------+ ))) (find-sukaku-1-steppers-wrt-W1 ?*RS*)

[/For the user of SudoRules]

Code: Select all: z-chain[5]: b1n1{r3c3 r2c2} - r9c2{n1 n3} - c6n3{r9 r4} - r4c4{n3 n9} - r1c4{n9 .} ==> r3c6≠1 stte

(with the same isomorphism at the end.)

Remark:
Notice that the original puzzle was in W7.
The method has allowed an apparent reduction of complexity from W7 to BC3. See my analyses here http://forum.enjoysudoku.com/eleven-s-variable-replacement-method-and-its-complexity-t39277.html for an explanation of this reduction.

by **marek stefanik** » Thu Apr 14, 2022 6:59 pm

Similar beginning to JCO's:

Code: Select all: .-----------------.------------------.---------------. | 123 4 6 | 129 12359 8 | 7 235 135 | | 5 123 9 | 6 y123 7 | 13 4 8 | | 8 7 123 | 4 1235 123 | 6 235 9 | :-----------------+------------------+---------------: |x123 8 5 | 129 12349 1234 |y123 7 6 | | 7 y123 4 | 8 6 5 | 123 9 13 | | 6 9 z123 | 7 x123 y123 | 5 8 4 | :-----------------+------------------+---------------: | 1234 6 1237 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 1234 123 1237 | 5 8 124 | 9 6 37 | '-----------------'------------------'---------------'

relabeling xyzb4p159, HS yb6, yr2, yb5, xb5

Personally, I like this proof of –4r4c5 better, though the y-wing this skips is still helpful later.

Code: Select all: .-----------------.------------------.---------------. |yz123 4 6 | 129 59–123 8 | 7 235 135 | | 5 123 9 | 6 y 7 | 13 4 8 | | 8 7 xy123 | 4 1235 xz123 | 6 235 9 | :-----------------+------------------+---------------: | x 8 5 | 129 1239-4 1234 | y 7 6 | | 7 y 4 | 8 6 5 | 123 9 13 | | 6 9 z | 7 x y | 5 8 4 | :-----------------+------------------+---------------: | 1234 6 1237 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 1234 123 1237 | 5 8 124 | 9 6 37 | '-----------------'------------------'---------------'

9r1c4 – 9r4c4 = 9r4c5
||
[123b124p159 => the permutations in b25 are the same => zr3c6] – (z=4)r4c6
–4r4c5;
(z=y)r1c1 – (y=x)r3c3 – (x=z)r3c6 => –zr1c45, –123r1c5

Code: Select all: .----------------.-----------------.---------------. | 123 4 6 | 129 59 8 | 7 235 135 | | 5 12–3 9 | 6 y 7 |w13 4 8 | | 8 7 123 | 4 1235 123 | 6 235 9 | :----------------+-----------------+---------------: | x 8 5 | 129 1239 4 | y 7 6 | | 7 y 4 | 8 6 5 | 12–3 9 w13 | | 6 9 z | 7 x y | 5 8 4 | :----------------+-----------------+---------------: | 123 6 1237 | 12 4 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4 w13–2 1237 | 5 8 12 | 9 6 37 | '----------------'-----------------'---------------'

Let w be the digit in r2c7. HS wb6, wc2
kraken y:
|1y – (1=3)w
|2y – (2=1)r9c6 – (1=3)w
|3y
–3r2c2, -3r5c7

Code: Select all: .----------------.-----------------.---------------. | 123 4 6 | 129 59 8 | 7 235 135 | | 5 12 9 | 6 y 7 | 13 4 8 | | 8 7 123 | 4 1235 123 | 6 235 9 | :----------------+-----------------+---------------: | x 8 5 | 129 1239 4 | y 7 6 | | 7 y 4 | 8 6 5 | 12 9 13 | | 6 9 z | 7 x y | 5 8 4 | :----------------+-----------------+---------------: | 123 6 1237 | 12 4 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4 13 1237 | 5 8 12 | 9 6 37 | '----------------'-----------------'---------------' Kraken r7c1: (u in {12})r7c1 – ur7c4 = ur9c6 – uy \\ \ux ================= uz 3r7c1 – 3r9c2 = 3y -3z

Code: Select all: .----------------.----------------.---------------. |#123 4 6 | 129 59 8 | 7 #235 #135 | | 5 12 9 | 6 123 7 | 1–3 4 8 | | 8 7 123 | 4 1235 123 | 6 235 9 | :----------------+----------------+---------------: |#123 8 5 | 129 129 4 |#123 7 6 | | 7 123 4 | 8 6 5 | 12 9 13 | | 6 9 12 | 7 123 123 | 5 8 4 | :----------------+----------------+---------------: | 123 6 1237 | 12 4 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 4 13 1237 | 5 8 12 | 9 6 37 | '----------------'----------------'---------------'

3r14\c17b3 => –3r2c7, stte

Marek

by **eleven** » Thu Apr 14, 2022 11:48 pm

Thanks, i just had found another example of my patterns posted here (@mith 6 boxes probably are better than 9 to find puzzles, but only 4 seem to allow really high ratings). My solution was:

Code: Select all: +----------------------+-----------------------+-----------------------+ | #123 4 6 | #12+9 12359 8 | 7 235 135 | | 5 #123 9 | 6 #123 7 | #13 4 8 | | 8 7 #123 | 4 1235 #123 | 6 235 9 | +----------------------+-----------------------+-----------------------+ | #123 8 5 | a129 b12349 #123+4 | #123 7 6 | | 7 #123 4 | 8 6 5 | #123 9 13 | | 6 9 #123 | 7 #123 #123 | 5 8 4 | +----------------------+-----------------------+-----------------------+ | 1234 6 1237 | 12 124 9 | 8 35 357 | | 9 5 8 | 3 7 6 | 4 1 2 | | 1234 123 1237 | 5 8 124 | 9 6 37 | +----------------------+-----------------------+-----------------------+

Impossible pattern 123 in marked cells: 4r4c6 == 9r1c4 - r4c4 = (9-4)r4c5 = 4r4c6 => 4r4c6,r7c5,r9c1

Code: Select all: +-----------------------+----------------------+-----------------------+ | z123 4 6 | 129 12359 8 | 7 235 135 | | 5 x123 9 | 6 y123 7 | z13 4 8 | | 8 7 y123 | 4 1235 z123 | 6 235 9 | +-----------------------+----------------------+-----------------------+ | 123 8 5 | 129 1239 4 | y123 7 6 | | 7 y123 4 | 8 6 5 | 123 9 z13 | | 6 9 123 | 7 123 y123 | 5 8 4 | +-----------------------+----------------------+-----------------------+ | y123 6 1237 | z12 4 9 | 8 35 357 | | 9 5 8 | y3 7 6 | 4 1 2 | | 4 z123 1237 | 5 8 x12 | 9 6 37 | +-----------------------+----------------------+-----------------------+

Setting r259c2 to x,y,z with singles we get zr2c7 and yr8c4 => y=3, z=1, x=2
(singles yr4c7, zr2c7, yr2c5, yr6c6, xr9c6, zr3c6, zr1c1, yr3c3, yr7c1, zr7c4, yr8c4)

The New Sudoku Players' Forum

123pattern

123pattern

Re: 123pattern

Re: 123pattern

Re: 123pattern

Re: 123pattern