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The following Python program runs in 30ms and finds the solution, although it is not a generic engine for solving similar problems. It uses a simple strategy to fill out the unknown cells, and it turns out that by considering the possibilities for the cells surrounding the top-left cell this strategy is sufficient.

It would be more satisfactory to adapt my Sudoku solving code to work on this puzzle, but as the code below does its job I didn’t do that.

Solution:Cells 1 and 7 contain honey.