afj wrote:... I no longer think that brute force is hopeless, although one has to put in quite a bit of thinking to reduce the size of the search first...since we can reorder the columns in block 2 and in block 3, and can interchange the blocks, this reduces the possibilities by a further factor of 72, to 36288. I'm hopeful of finding sufficiently many more reductions that the possibilities for blocks 1-3 can be reduced to around 1000 or even fewer (i.e., working out how many ways these blocks can be completed to a full grid is enough to solve the problem)....
Some earlier posts on this thread combined with the above got me to thinking. (I suspect afj has already been down this road)
There's probably a relatively small number of PATTERNS to filling in blocks 2 and 3 given block 1.
For the rest of this discussion, assume block1 is filled in as
123
456
789
For example, for blocks 123, consider
123 bbb ccc
456 ccc aaa
789 aaa bbb where a is any one of (123), b is any one of (456) and c is any one of (789) that satisfies the block rule.
If 123 winds up in the same row in either block 2 or 3, then the rest are 'forced'. afj's swapping of middle and right chutes then account for all of the other possibilities.
You guys seem to have a way to 'brute force' the possibilities from here for each permutation of a row grouping. But I'll assert that
123 4
56 789
456 789 123
789 123 456
and
123 4
65 789
456 789 123
789 123 456
should have exactly the same number of valid solutions for blocks 3-9, as would any other permutation of abc.
Then there's
123 bbC ccB
456 ccA aaC
789 aaB bbA
where a capitalized letter shows where the same value must appear. So the value for A (say '1') must be the same in blocks 2 and 3, but the rest of the values for 'a' can be either 2 or 3. Note that A can be any one of (123) in turn. Swapping middle and right vertical chutes cover off the 'other' possibility.
As far as I can tell, this enumerates all of the row possibilities. (123) either appear together somewhere in a row in block 2 or 3, or they do not. Where they do not, then two of them must be together in one of these blocks. The rest fall into place 'automatically' once 123 (or aaA) are placed in block 2 or 3, and chute-swapping makes the decision to place two of (123) into rows 2 or 3 of block 2 equivalent.
So it may just be a matter of counting permutations of these patterns, and brute forcing a sample for each of them.