The New Sudoku Players' Forum

[JPF]

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

[marek stefanik]

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

[JPF]

Hi Marek,

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

Good point!
your remark leads to a more general challenge:

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

[coloin]

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


[JPF]

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

JPF

[JPF]

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

[coloin]

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|
+---+---+---+ 


[JPF]

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