Still on the subject of rookeries/templates/

etc. ... perhaps someone should try to prove there's no(?) 13-clue 2Ã—4 sudoku puzzle. It would be a relatively convenient means of testing and refining "rookery" methods.

Let me make up some terminology. This is a

(1,2,2)-scaffold:

- Code: Select all
`. . 2 #|. . . #`

. # . .|. # . #

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

. . . 1|# . 2 .

# . . #|. # . .

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

. # . .|# . # .

# . # .|. . 3 .

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

# # # .|. . . .

. . . .|# 3 . #

In general a (c(1),c(2),..,c(k))-scaffold consists of 8

k cells, c(1) of which contain the digit 1 and so on up to c(k) containing the digit

k, while the remaining 8

k-c(1)-c(2)-...-c(k) cells are blank ('#');

and the given digits are sufficient to 'solve' the whole 8

k cell structure (i.e. there's only one way to place the other 8-c(1) 1's up to 8-c(k)

k's). This is somewhat similar to what Wolfgang was proposing with his 'subpuzzles'.

Now suppose there was 13-clue puzzle. Then up to isomorphism it is a scaffold of one of these types:

- (0,1,1,1,1,1,1,7)
- (0,1,1,1,1,1,2,6)
- (0,1,1,1,1,1,3,5)
- (0,1,1,1,1,1,4,4)
- (0,1,1,1,1,2,2,5)
- (0,1,1,1,1,2,3,4)
- (0,1,1,1,1,3,3,3)
- (0,1,1,1,2,2,2,4)
- (0,1,1,1,2,2,3,3)
- (0,1,1,2,2,2,2,3)
- (0,1,2,2,2,2,2,2)
- (1,1,1,2,2,2,2,2)

So you 'just' have to prove that none of those scaffold-types can exist. Here's one way to do that for, say, the (0,1,1,1,1,1,1,7)-scaffold:

- Write down all (0)-scaffolds up to isomorphism. There's only one.
- Extend that to all (0,1)-scaffolds up to isomorphism as follows. In as many ways as possible { add 8 '#' characters, one per row/col/box, and replace one of those '#' chars with the digit 2; if it's a new shape up to isomorphism and is 'solved' by its given digits, add it to the list of (0,1)-scaffolds }.
- Extend that to a list of all (0,1,1)-scaffolds up to isomorphism in a similar manner.
- Continue extending until we are forced to give up, e.g. by failing to extend from (0,1,1,1).

I've not tried it so have no idea if it's workable. I expect that it's not hard to cross off the first few scaffold-types from the list above; but the last few might require some cunning if they're to complete in reasonable time.