carls.haven wrote: .... Pat could you please elaborate on your reply to Smythe Dakota? ....
I think Pat just means that, within a 4-cell cage, if all four cells are in the same row or column, then (to avoid duplicate digits in the row or column) the minimum 4-digit sum is 1+2+3+4 which is 10, and the maximum is 9+8+7+6 which is 30.
The same would be true even if the four cells are not all in the same line, as long as you add a no-repeat-within-cage rule.
Without such a rule, the minimum would be 1+1+1+1 which is 4, and the maximum would be 9+9+9+9 which is 36.
However, if you invoke a rule that a cage must be rook-connected, then you'll find that it is not possible to have a 4-cell cage whose cells are all in different rows (or different columns). Therefore (try it!), even without a no-repeat-within-cage rule, the minimum is 1+2+1+2 which is 6, and the maximum is 9+8+9+8 which is 34, as long as you still have the no-repeat-within-row and no-repeat-within column rules.
By "rook-connected" I mean that it must be possible to travel from any cell in the cage to any other cell in the cage, staying within the cage, with a sequence of 1-cell horizontal and vertical movements. This is a standard constraint that almost everybody invokes in all these kinds of variations. No disconnected cages (like the Hawaiian Islands) allowed!
I think the three (or more) variations put forth in this thread are all based on the same general idea (cages), but have different constraints. Still another such variation is the jigsaw Sudoku, where instead of having nine 3x3 boxes you have nine 9-cell cages (irregularly shaped but each cage is rook-connected), with no-repeat-within rules applied to rows, columns, and cages, rather than to rows, columns, and boxes. I even saw a 7-by-7 version once (digits 1 through 7), where the cages could not be boxes because 7 is a prime number.