evert wrote:r.e.s. wrote:- Code: Select all
` x x x x x x`

x x x x

x x x x x x x x x x

x x x x x x x

x x x x

x x x x x x x x x x

x x x x x

x x x

x x x x x x x x

x x x x x x

where rows and columns can have "gaps" in them. I.e., in the picture, row#1 has length 6 (so contains 1-6), and col#1 has length 3 (so contains 1-3, etc.)

Should be possible with boxes also.

I'd like to see a puzzle like this with box constraint added.

Here's a simple way to generate special cases -- which I'll call

sub-sudokus -- of this type of puzzle with box constraints, by producing them from a starting sudoku puzzle. I

think these always have a unique solution (assuming we start with a unique-solution sudoku). I've verified solution-uniqueness for all three of the following examples ...

Example 1In three steps ...

(1) Start with an ordinary sudoku puzzle ...

- Code: Select all
`6 2 . | 3 . . | . . 9`

. . 7 | . 6 8 | . . .

. 4 3 | . 9 . | . . .

------+-------+------

2 . . | . 3 6 | . . .

. . 9 | 8 . 5 | 6 . .

. . . | 7 4 . | . . 5

------+-------+------

. . . | . 1 . | 8 4 .

. . . | 6 8 . | 3 . .

1 . . | . . 3 | . 9 6

(2) solve it ...

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

5 9 7 | 2 6 8 | 4 1 3

8 4 3 | 1 9 7 | 5 6 2

------+-------+------

2 5 4 | 9 3 6 | 1 7 8

7 1 9 | 8 2 5 | 6 3 4

3 6 8 | 7 4 1 | 9 2 5

------+-------+------

9 3 6 | 5 1 2 | 8 4 7

4 7 2 | 6 8 9 | 3 5 1

1 8 5 | 4 7 3 | 2 9 6

(3) remove all 7,8,9 digits, then replace with '.' all remaining digits that were not originally clues ...

- Code: Select all
`6 2 . | 3 . . | `

. | . 6 | . . .

4 3 | . | . . .

------+-------+------

2 . . | 3 6 | .

. | . 5 | 6 . .

. . | 4 . | . 5

------+-------+------

. . | . 1 . | 4

. . | 6 | 3 . .

1 . | . 3 | . 6

This is now the first example puzzle, which has only one rule:

Replace each '.' with a digit such that each row, column, and box contains all six digits 1,...,6.

(I've put '.'s instead of 'x's so printing with a large fixed-width font will give a bit more suitable grid for pencil-and-paper solving.)

Example 2Using the same method, here's the result of reducing a 16-digit sudoku (with 4x4 boxes) to a 9-digit puzzle ...

- Code: Select all
` 1 | . . | . . | 5 2 7 .`

. 2 . . | 8 | . | . . .

. . . | . 7 . 3 | 5 . |

. | . . | . . . 6 | . .

--------+---------+---------+--------

9 . . | . . | . . | 1 .

. . | . . | 6 7 | 9 3 .

2 4 . | 7 . | . . 3 | 6

. | . 3 4 | . 1 | 2 . .

--------+---------+---------+--------

. | . 5 . | 6 . | . 3 .

. . . 4 | 3 2 | . | 6 8

. . | . . | 4 . . . | .

8 . | . . | . . | . 9 2

--------+---------+---------+--------

4 . | . 8 | . 7 . | . .

. . 5 | . 1 . | . 9 | 7

. 8 | . . . | . | 4 . .

3 . | . | . . . | . 5 9

Replace the '.'s to make each row, column, and box contain all nine digits 1,...,9.

Example 3And finally this one by reducing a 12-digit sudoku (with 3x4 boxes) to a 10-digit puzzle ...

- Code: Select all
`. . 4 | 1 2 . | 6 3 . .`

. . 2 . | 3 . 9 4 | . .

. 6 . | . . 5 | 9 4 . .

--------+---------+--------

5 3 . | . . . . | . 2 8

. . . 0 | 4 . 8 | . 9 .

. . 4 | . . . | . . . .

--------+---------+--------

. . 1 | . . 7 | . . 6 .

3 0 . . | 6 . . . | 4 .

. . . | 2 . . | . . . 7

--------+---------+--------

1 . . . | . . 6 . | . 7

. . | 1 0 4 . | 2 . . .

. 7 5 . | 8 . | 0 . . .

Replace the '.'s to make each row, column, and box contain all ten digits 1,...,9,0 (abbreviating 10 as 0 for convenience).

The same method can reduce any n-digit sudoku to such an m-digit

sub-sudoku (1 < m < n), with the only rule being to replace the '.'s with digits to make each row, column, and box contain all the digits 1,...,m. (I suppose sub-sudokus can be extracted from other types of sudoku variant, e.g. jigsaws, etc., as well as from themselves.)

As expected, it seems that the sub-sudokus of a given n-digit starting sudoku increase in difficulty as m increases from 1 to n. Also, from a given n-digit starting sudoku, various "disjoint sub-sudokus" can be produced in an obvious way by reducing to disjoint sets of digits.