Normal sudoku rules apply. Yin–Yang: shade some cells in the grid so that all shaded cells are orthogonally connected, all unshaded cells are orthogonally connected, and no 2×2 region is entirely shaded or entirely unshaded. Every shaded cell containing digit x has value 1/x; every unshaded cell has value equal to its digit. Product/Sum cages: each cage has a corner mark in the leftmost cell of its top row identifying its type. If it shows Σ=X, the values in the cage sum to X; if it shows Π=X, the values in the cage multiply to X (X is not necessarily an integer). Digits or values MAY repeat within a cage. The grid is partially covered in fog. Placing correct digits reveals their own cell and may also lift fog elsewhere in the grid. No guessing is required.