In this note, we will study two different methods for the mathematical evaluation of difficulty level of Sudoku puzzles in terms of boolean Groebner bases. One is b- and s-rank, and the other is the Inoue invariant. We first study b-rank and several ranks related to or derived from s-rank in detail, and we then define and study the improved version of computable Inoue invariants. We finally exhibit the hardest Sudoku puzzle in regard to these two indicators of difficulty level.

boolean Groebner bases, Sudoku puzzles, s-rank, Inoue invariant.