I'm not sure what your rules are. Are the following definitions and constraints correct?

- The usual Sudoku constraints.
- A cage is a specified group of cells which is connected orthogonally, i.e. each cell in the cage can be accessed from any other via a series of vertical and/or horizontal one-cell moves within the cage.
- No two cages overlap.
- All the cages together comprise the entire puzzle, i.e. every cell is in one cage.
- No digit is repeated within a cage.
- Every cage has an upper left cell. For example r4c5, r5c4, r5c5 would not constitute a legitimate cage, because there is no upper left cell to write the sum in.

Following these rules, I attempted to solve your puzzle, but soon ran into a contradiction. (Either that, or I made a dumb error somewhere.)

This leads me to question my interpretation of the axioms, especially the last one. Perhaps it is OK to have a cage with no upper left cell. In that case, where is the sum written? Is it in the leftmost cell in the top row, or in the topmost cell in the left column? Note that these two are not the same.

Bill Smythe