The puzzle goes like this - you're given a line of 27 cells, and the goal is to fill each cell with a value from 1 to 9. Each value must appear exactly 3 times, and the special constraint is there must be exactly n cells spaced between each cell holding value n.

To demonstrate this special constraint, let's try a smaller example: with a line of 6 cells, fill in the values 1 to 3, exactly twice for each. Then it seems the following is the only possible solution (barring reflection):

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`2 3 1 2 1 3`

Another smaller example: with a line of 8 cells, fill in the values 1 to 4, exactly twice for each. Then the following should be the only possible solution (again, barring reflection):

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`2 3 4 2 1 3 1 4`

So, I was told the following 3 puzzles each gives a unique solution, and these are essentially all the possible solution lines for the case of 3x9=27 cells (barring reflection):

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`_ _ _:_ _ _:_ _ _|_ _ _:_ _ _:_ _ 4|_ _ _:_ _ _:_ _ _`

_ _ _:_ _ _:_ _ _|_ _ _:_ _ _:_ _ 5|_ _ _:_ _ _:_ _ _

_ _ _:_ _ _:_ _ _|_ _ _:_ _ _:_ _ 6|_ _ _:_ _ _:_ _ _

Some questions:

1. Are these problems only solvable by computer programs? Is there any logical approach to find out the answer, or is T&E the only possible way?

2. If some more clues are given, can these puzzles be made more solvable to a competent human Sudoku player? If so, what is the minimum number of clues to make them "human solvable"?

3. Can this puzzle be generalised to a two-dimensional version? If so what are the new rules?

4. I've also been told that for the case of 3x10=30 cells, there are only 5 possible solution lines (barring reflection). But I don't have any idea on how to work them out. Can any programming guru help me out on this?

Thanks heaps in advance!