Normal sudoku rules apply. Exactly nine cells in the grid are "rooters". The value of a rooter equals the square root of its digit (in all other cells, the value equals the digit). No two rooters share a row, column, box, or digit. A black dot connects two cells where one value is double the other (not all such dots are necessarily given). Values on thermometers strictly increase from bulb to tip. The sum of the values along an arrow equals the value in the attached circle. Along green lines, neighbouring values differ by at least 5. On each purple line, the set of values consists of consecutive integers in any order. All cages have the same product of values (digits or values may repeat within a cage if allowed by other rules). The grid is partially covered in fog. Entering correct digits may clear some fog (not necessarily beneath or around the entered digit). No guessing is required.