Diagonal Puzzles

Everything about Sudoku that doesn't fit in one of the other sections

Diagonal Puzzles

Diagonal Puzzles

Find the minimum and maximum number of clues of valid puzzles such that:
• r9c1=r1c9=0
and for any 1≤i≤9 and 1≤j≤9 :

• if ricj=0 then r(i+1)c(j+1)=0
• if 0<ricj<8 then r(i+1)c(j+1)={(ricj)+1} or 0
• if ricj=9 then r(i+1)c(j+1)= 1 or 0

Here is an example 25c:
Code: Select all
`+-------+-------+-------+| 1 . . | 2 . . | . 4 . || . 2 . | . 3 . | . . 5 || . . 3 | . . 4 | . . . |+-------+-------+-------+| 7 . . | 4 . . | 5 . . || . 8 . | . 5 . | . 6 . || . . 9 | . . 6 | . . 7 |+-------+-------+-------+| 4 . . | 1 . . | 7 . . || . 5 . | . 2 . | . 8 . || . . 6 | . . . | . . 9 |+-------+-------+-------+`

JPF
JPF
2017 Supporter

Posts: 6117
Joined: 06 December 2005
Location: Paris, France

Re: Diagonal Puzzles

Hi JPF,

It seems like an interesting challenge, but without minimality as a requirement it was simple to find the maximum:
Code: Select all
`+-------+-------+-------+| 1 4 9 | 3 8 7 | 2 6 . || 6 2 5 | 1 4 9 | 8 3 7 || 8 7 3 | 6 2 5 | 1 9 4 |+-------+-------+-------+| 5 9 8 | 4 7 3 | 6 2 1 || 2 6 1 | 9 5 8 | 4 7 3 || 4 3 7 | 2 1 6 | 9 5 8 |+-------+-------+-------+| 9 5 4 | 8 3 2 | 7 1 6 || 7 1 6 | 5 9 4 | 3 8 2 || . 8 2 | 7 6 1 | 5 4 9 |+-------+-------+-------+`

Here is a 25-clue minimal from the same grid:
Code: Select all
`+-------+-------+-------+| 1 . . | . 8 . | . 6 . || 6 2 . | . . 9 | . . 7 || . 7 3 | . . . | . . . |+-------+-------+-------+| . . 8 | 4 . . | . . . || . . . | 9 5 . | . . . || . . . | . 1 6 | . . . |+-------+-------+-------+| 9 . . | . . 2 | 7 . . || 7 1 . | . . . | 3 8 . || . 8 . | . . . | . 4 9 |+-------+-------+-------+`

Marek
marek stefanik

Posts: 353
Joined: 05 May 2021

Re: Diagonal Puzzles

Hi Marek,
you wrote:It seems like an interesting challenge, but without minimality as a requirement it was simple to find the maximum

Good point!

1) Consider this variant sudoku: Diagonal (increasing) sudoku, say DI sudoku having theses properties:

Edited to remove a typo. Thanks to dobrichev.
For any 1≤i≤9 and 1≤j≤9 except for r1c9 and r9c1:

• if ricj=0 then r(i+1)c(j+1)=0
• if 0<ricj<9 then r(i+1)c(j+1)={(ricj)+1} or 0
• if ricj=9 then r(i+1)c(j+1)= 1 or 0

You gave a DI solution grid.
The question may be: how many essentially different DI are there?

2) What is the minimum and maximum number of clues for minimal DI puzzles?

Note that I removed the first constraint which is not really useful.

Here is a 24 clue minimal puzzle:
Code: Select all
`+-------+-------+-------+| 1 . . | 2 . . | . 4 . || . 2 . | . 3 . | . . 5 || . . 3 | . . 4 | . . . |+-------+-------+-------+| 7 . . | 4 . . | 5 . . || . 8 . | . 5 . | . 6 . || . . 9 | . . 6 | . . . |+-------+-------+-------+| 4 . . | 1 . . | 7 . . || . 5 . | . 2 . | . 8 . || . . 6 | . . . | . . 9 |+-------+-------+-------+`

JPF
Last edited by JPF on Fri Mar 25, 2022 7:24 pm, edited 1 time in total.
JPF
2017 Supporter

Posts: 6117
Joined: 06 December 2005
Location: Paris, France

Re: Diagonal Puzzles

Not sure that I completely understand .... this might be valid 21C
Code: Select all
`+---+---+---+|1..|...|4..||.2.|.8.|.5.||..3|..9|..6|+---+---+---+|...|4..|...||7..|.5.|1..||.8.|..6|.2.|+---+---+---+|...|.1.|7..||.6.|..2|.8.||...|...|..9|+---+---+---+  21C`
coloin

Posts: 2362
Joined: 05 May 2005
Location: Devon

Re: Diagonal Puzzles

That puzzle is not valid because in a DI puzzle: r7c1=0 ==> r8c2=0

JPF
JPF
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Posts: 6117
Joined: 06 December 2005
Location: Paris, France

Re: Diagonal Puzzles

note : due to a typo pointed out by dobrichev, I edited the definition of DI puzzles given here

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

JPF
JPF
2017 Supporter

Posts: 6117
Joined: 06 December 2005
Location: Paris, France

Re: Diagonal Puzzles

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

Posts: 2362
Joined: 05 May 2005
Location: Devon

Re: Diagonal Puzzles

coloin wrote:20 clues
Code: Select all
`+---+---+---+|1..|.3.|4..||.2.|...|.5.||..3|..9|..6|+---+---+---+|...|4..|...||7..|.5.|1..||5..|..6|...|+---+---+---+|...|.1.|7..||...|.7.|.8.||..6|...|..9|+---+---+---+  `

The puzzle doesn't satisfy the first rule : r8c2=0 ==> r9c3=0

JPF
JPF
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Posts: 6117
Joined: 06 December 2005
Location: Paris, France