## 2 X 4 sudoku

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: 2 X 4 sudoku

solution:
Hidden Text: Show
Code: Select all
`3761842554823167764325812158473615267348837416526237581448156273`

I was able to solve this sudoku in 4 steps basicly with contradiction and finding an assumption which led to an empty cell so it can be eliminated. Like to know how to do it in a better way.
My solvepath was basicly to prove (in 3 steps) that R5C3 can't be 1 so that R5C1 must be 1, and further assuming that R7C6=4 which leads (through a chain) to a contradiction, so R7C6=8.
Any more methodical approach?
Code: Select all
`..618.2............3258..158..3..5..734..3741............8.562..`

Wondering why with so many cluess given this sudoku is so difficult to solve? Or am I missing something?
Last edited by urhegyi on Sat Nov 06, 2021 7:55 am, edited 1 time in total.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2 X 4 sudoku

My conclusion when starting to solve 2X3 and 2X4 sudokus is that through the symmetry they are less difficult to solve as the 3X3 one's.
I found a post on 2X3 where all different grids were calculated and stated that the maximum difficulty was around SE 8.3.
Is there simular work done on 2X4 and what's the hardest puzzle possible?
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2 X 4 sudoku

Code: Select all
`..618.2............3258..158..3..5..734..3741............8.562.. 2Rx4C`

The difficulty level is about right for this grid.

Code: Select all
`Pencilmarks, after basics, a couple naked pairs and a naked triplet:+-------------------------+-------------------------+| 35    47    6     1     | 8     47    2     35    || 358   247   38    27    | 35    1     6     47    |+-------------------------+-------------------------+| 67    67    4     3     | 2     5     8     1     || 2     1     5     8     | 4     67    3     67    |+-------------------------+-------------------------+| 168   5     128   26    | 7     3     4     68    || 68    3     7     4     | 1     68    5     2     |+-------------------------+-------------------------+| 13467 267   123   267   | 35    48    17    3458  || 1347  8     13    5     | 6     2     17    34    |+-------------------------+-------------------------+Either:6.6, Turbot Fish (w/4 nodes): R7C1.3 off: r7c1<>36.8, Bidirectional Y-Cycle (w/12 nodes): R1C6,R4C6,R6C6,R5C8,R5C4,R2C4,R1C2: r5c1<>6, r2c2<>7, r7c4<>27.1, Forcing Chain (w/6 nodes): R3C2.6 on: r3c2<>7, r3c2=61.0, Hidden Single: R3C1: 7 in block: r3c1=71.5, Hidden Single: R8C7: 7 in row: r8c7=71.0, Hidden Single: R7C7: 1 in column: r7c7=14.2, XY-Wing: Cells R7C1,R7C6,R6C1 on value 8: r6c6<>8stteED=7.1/1.2/1.2Or:3.7, Two-String Kite: Cells R1C1,R1C8,R2C5,R7C5: 3: r7c1<>36.8, Bidirectional Y-Cycle (w/12 nodes): R1C6,R4C6,R6C6,R5C8,R5C4,R2C4,R1C2: r5c1<>6, r7c4<>2, r2c2<>74.1, W-Wing: Cells R3C1,R3C2,R7C2,R7C4: 7 and 6: r8c1<>7, r7c1<>71.5, Hidden Single: R3C1: 7 in column: r3c1=71.0, Hidden Single: R3C2: 6 in block: r3c2=61.5, Hidden Single: R8C7: 7 in row: r8c7=71.0, Hidden Single: R7C7: 1 in column: r7c7=14.2, XY-Wing: Cells R7C1,R7C6,R6C1 on value 8: r6c6<>8stteED=6.8/1.2/1.2`

I think very long chains are not possible in 6x6 and 8x8 grids, so the max difficulty is in the ED=8.x region.

Some hard 8x8 grids, from about 1000 randomly generated grids:
Code: Select all
`..........2..456.4...6....3.2.....54..387..8....28.........51.2. 2Rx4C 19 ED=8.4/1.2/1.2.53....7..8.1....8...2..1...5.6...45.......8.4.3.2..8........17. 2Rx4C 19 ED=8.4/1.2/1.2....2.....5..6..1.........8..74....7...3...3..8462.....58..5..7. 2Rx4C 18 ED=8.3/1.2/1.212.74.6....5.......8....2...3..17......4.5.46....7...2...6.2..1. 2Rx4C 20 ED=8.3/1.2/1.2..28.....3.7...2.........6.3.5.4....1...7.....25.27.51.31...4... 2Rx4C 20 ED=8.3/1.2/1.23.7.8......1..76..2..813.....2........4.1......77.8....12.....5. 2Rx4C 19 ED=7.7/1.2/1.26.1......3..6.8.7.2....4...3..5.32..4........5.....23.6.4......2 2Rx4C 19 ED=7.6/1.2/1.2`
1to9only

Posts: 4084
Joined: 04 April 2018

### 2Rx4C-DG-X

Here is a Hard Sudoku 8x8:
Code: Select all
`..........6.47...3....2..8............6..2....7...85.6.......... 2Rx4C-DG-X 12 ED=6.5/1.5/1.5`

And a couple of harder ones:
Code: Select all
`......7.6..8.2..2...7......................3...1..2.4..3.4...... 2Rx4C-DG-X 12 ED=7.1/1.5/1.5..8..3...................1....8..5...7..4.2.1.........2......... 2Rx4C-DG-X 10 ED=8.4/1.2/1.2`
1to9only

Posts: 4084
Joined: 04 April 2018

### Re: 2 X 4 sudoku

After my first attempts with the hard level, I tried four evil ones.
Most of all solve with up to 4 chains.
Code: Select all
`.1.45....5.3...2.8.1...3...5.8....3.6...8...3.7.4...2.1....78.6...3...2...873......8..4.31...6....2...65.6..4......124...3...7...8..6...6.4..7......2..8.6...57..76...4.8..2......8..3.4...1..5.......6...7.83.2...15.2...8....43....5...8.42...1.43.7...6......`
Last edited by urhegyi on Tue Nov 23, 2021 2:28 pm, edited 3 times in total.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2Rx4C-DG-X

1to9only wrote:Here is a Hard Sudoku 8x8:
Code: Select all
`..........6.47...3....2..8............6..2....7...85.6.......... 2Rx4C-DG-X 12 ED=6.5/1.5/1.5`

And a couple of harder ones:
Code: Select all
`......7.6..8.2..2...7......................3...1..2.4..3.4...... 2Rx4C-DG-X 12 ED=7.1/1.5/1.5..8..3...................1....8..5...7..4.2.1.........2......... 2Rx4C-DG-X 10 ED=8.4/1.2/1.2`

For the first I was able to place all the 6's then all the 2's.
Last edited by urhegyi on Tue Nov 09, 2021 5:17 pm, edited 1 time in total.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2Rx4C-DG-X

urhegyi wrote:
1to9only wrote:Here is a Hard Sudoku 8x8:
Code: Select all
`..........6.47...3....2..8............6..2....7...85.6.......... 2Rx4C-DG-X 12 ED=6.5/1.5/1.5`

And a couple of harder ones:
Code: Select all
`......7.6..8.2..2...7......................3...1..2.4..3.4...... 2Rx4C-DG-X 12 ED=7.1/1.5/1.5..8..3...................1....8..5...7..4.2.1.........2......... 2Rx4C-DG-X 10 ED=8.4/1.2/1.2`

For the first I was able to place all the 6's then all the 2's.

Edit:
I'm I correct?
Hidden Text: Show
Code: Select all
`8471523635624781431765285826741317532864624813757185364226348157`
Last edited by urhegyi on Tue Nov 09, 2021 5:21 pm, edited 1 time in total.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2Rx4C-DG-X

1to9only wrote:Here is a Hard Sudoku 8x8:
Code: Select all
`..........6.47...3....2..8............6..2....7...85.6.......... 2Rx4C-DG-X 12 ED=6.5/1.5/1.5`

And a couple of harder ones:
Code: Select all
`......7.6..8.2..2...7......................3...1..2.4..3.4...... 2Rx4C-DG-X 12 ED=7.1/1.5/1.5..8..3...................1....8..5...7..4.2.1.........2......... 2Rx4C-DG-X 10 ED=8.4/1.2/1.2`

Solutions: Show
Code: Select all
`8471523635624781431765285826741317532864624813757185364226348157 2Rx4C-DG-X 12 ED=6.5/1.5/1.55142637867381245281675347354281682613457457386211625478334875162 2Rx4C-DG-X 12 ED=7.1/1.5/1.51287436536452871283671547154628365138742472815365471362883625417 2Rx4C-DG-X 10 ED=8.4/1.2/1.2`
1to9only

Posts: 4084
Joined: 04 April 2018

### Re: 2 X 4 sudoku

Any idea if Disjoint groups(offset sudoku) and diagonals also works with 2X3(6x6) sudoku?
If so, can it be added to the existing Sukaku6Explainer? For now there is no DG support.
urhegyi

Posts: 618
Joined: 13 April 2020

### 6x6-DG-X

I think 6x6-DG generates some valid grids, but they are all singles only, ED=1.2/1.5 [edit: this is from a small sample. on a much larger sample, about 2% are non-singles only!].
6x6-DG-X is probably not possible, as I've failed so far to generate a solution grid!!
This probably means DG will not make it into Sukaku6Explainer.
1to9only

Posts: 4084
Joined: 04 April 2018

### Re: 2 X 4 sudoku

urhegyi wrote:My conclusion when starting to solve 2X3 and 2X4 sudokus is that through the symmetry they are less difficult to solve as the 3X3 one's.
I found a post on 2X3 where all different grids were calculated and stated that the maximum difficulty was around SE 8.3.
Is there simular work done on 2X4 and what's the hardest puzzle possible?

the relevant thread is here. there are nine 6x6 SE 8.4's after singles
Code: Select all
`1...56...1..21...5.35..13.1.6..62.1.....5.4....3...5.1...34.3.2.1..6..3..2......61..2.4...61.....4.6.5.6..3..2.4....6...2.4...61......2..5..123......64...2.23...1..5..2...61....2.41....6....2..3..6.6.5.4....61....2.4..3....5.1....456...5..234....5.1.....34...5.....3..6..1.3.234....5.1...1...5......3...5.1...34...2..556...4`

the first one has 16 clues. the other 8 all have 12 clues and the complements of the clue grid in the solution grid are all isomorphic, but i'm not sure if the 8 puzzles are all isomorphic.

come to think of it i only rated about 70% of the 542257 no-first-single 6x6 puzzles because they have a 9x9 representation and i could plug them into champagne's skfr. after two years, i never got around to doing the other 30% of them with 1to9only's 6x6 explainer! i'll get that done at some point in the hope that another 8.4 or harder puzzle will show up

i highly doubt that an exhaustive search of all 8x8 puzzles would be possible as my guess is that the search space would be too big

1to9only wrote:I think 6x6-DG generates some valid grids, but they are all singles only, ED=1.2/1.5 [edit: this is from a small sample. on a much larger sample, about 2% are non-singles only!].
6x6-DG-X is probably not possible, as I've failed so far to generate a solution grid!!
This probably means DG will not make it into Sukaku6Explainer.

no 6x6-DG-X grid exists. i did an exhaustive pencil-and-paper search this morning because of boredom (don't ask)
boring manual proof of non existence: Show
assume 2Rx3C boxes, and fill in row 1 with 123456
notation: "1/" and "1\" mean a hidden single or locked candidate elimination by looking at the 1s in the "/" and "\" diagonals respectively. "ns" means naked single(s)

split into cases, based on the order of 4,5,6 in r2c123

case r2c123=456:
6\ r34c5-6
6b4 r4c4=6
(*) 14/ r4c2-14
ns r4c2=3
3\ r6c4-3
oddagon 3r2c4,r2c6,r6c6,r6c1,r3c4 => no solution

case r2c123=465:
5\ r4c4=5
same as (*) from above

case r2c123=546:
r4c4=6 as from case 456
5/ r3c3-5
ns r3c3=2, r5c5=3, r5c2=1, no 1 in c3

case r2c123=564:
r3c3=2 as from case 546
4\ r5c5=4
4/ r6c1=4
4b3 r4c2=4
ns r4c4=3, r6c6=5, r3c4=5, no 5 in c2

case r2c1=6:
6\ r5c5=6, no 6 in b4
999_Springs

Posts: 585
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: 2 X 4 sudoku

urhegyi wrote:After my first attempts with the hard level, I tried a few evil ones.
Most of all solve with up to 4 chains.
Code: Select all
`.1.45....5.3...2.8.1...3...5.8....3.6...8...3.7.4...2.1....78.6...3...2...873......8..4.31...6....2...65.6..4......124...3...7...8..6...6.4..7......2..8.6...57..76...4.8..2......8..3.4...1..5....7.3....6....4.7....81..14..6..1..86..85....3.1....7....3.5.........6...7.83.2...15.2...8....43....5...8.42...1.43.7...6......`

The last one needed an AIC, a discontinuous nice loop and two xy-chains.

Conclusion of solving above examples is that after basics the further solvepath highly depends on finding a nice opening set. The question now is how to find out this was the most elegant solvepath or eventually there were other better solutions. So taking another startposition can determine the way a sudoku solves. A computer program usually starts with the simplest step that it finds. But the human solver can( sometimes with a bit of luck) find an elegant way to break open the puzzle. For the last puzzle I like to know which are the first steps SudokuExplainer(2X4) finds after basics? I estimate the rating around SE 7.3 based on following solvepath.

My way to start was whatever the value of R2C2(1 or not 1) R1C5 is always 7. Second step was proving that R5C7<>1 by xy-chain. Third step was starting with R5C4=6 a contradiction chain led to R5C4<>6. Then a locked candidate and a few singles followed by another xy-chain proving that R6C3<>1=6. A naked triple in row 5 finally solved this puzzle. Curious to compare this with a solver based solution.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2 X 4 sudoku

Code: Select all
`5..8...1.37.5...7.....1..2.6.5....3.4.5..5.....3...2.84.8...7..2`
Last edited by urhegyi on Wed Dec 01, 2021 9:05 pm, edited 1 time in total.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2 X 4 sudoku

I could solve it when assuming R5C8=7 leads to R4C1<>1/3/4 is an empty cell and so R4C8 must be 7.
But there must be better ways to solve this.
The solution I found was a lucky shot, but for me it's not the logical way I prefer.
urhegyi

Posts: 618
Joined: 13 April 2020

### Re: 2 X 4 sudoku

When I compare an evil 8x8 sudoku like
Code: Select all
`5..8...1.37.5...7.....1..2.6.5....3.4.5..5.....3...2.84.8...7..2`

with a hard one like
Code: Select all
`..4736.13..6....1......4.4...12..71...4.8......2....2..84.2873..`

I find the hard one much harder then the evil one based on the way I solved them.
Can the rating prove this or am I missing some obvious steps in the hard one that force me to use harder logic then necessary?
urhegyi

Posts: 618
Joined: 13 April 2020

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