A Sudoku grid is a 9 × 9 Latin square further constrained to have nine non-overlapping 3×3 mini-grids each of which contains the values 1–9. In ?-Quasi-Magic Sudoku a further constraint is imposed such that every row, column and diagonal in each mini-grid sums to an integer in the interval [15-?, 15+?]. The problem of proving certain (computationally known) results for ? = 2 concerning mini-grids and bands (rows of mini-grids) was posed at the British Combinatorial Conference in 2007. These proofs are presented and extensions of these provide a full combinatorial enumeration for the total number of completed 2-Quasi-Magic Sudoku grids. It is also shown that there are 40 isomorphism classes of completed 2-Quasi-Magic Sudoku grids.