  A Sudoku puzzle often has a regular pattern in the arrangement of initial
digits and it is typically made solvable with known solving techniques, called
strategies. In this paper, we consider the problem of generating such Sudoku
instances. We introduce a rigorous framework to discuss solvability for Sudoku
instances with respect to strategies. This allows us to handle not only known
strategies but also general strategies under a few reasonable assumptions. We
propose an exact method for determining Sudoku clues for a given set of clue
positions that is solvable with a given set of strategies. This is the first
exact method except for a trivial brute-force search. Besides the clue
generation, we present an application of our method to the problem of
determining the minimum number of strategy-solvable Sudoku clues. We conduct
experiments to evaluate our method, varying the position and the number of
clues at random. Our method terminates within $1$ minutes for many grids.
However, as the number of clues gets closer to $20$, the running time rapidly
increases and exceeds the time limit set to $600$ seconds. We also evaluate our
method for several instances with $17$ clue positions taken from known minimum
Sudokus to see the efficiency for deciding unsolvability.
