Normal sudoku rules apply. Each of the following clue types has exactly one liar in the puzzle: Digits along an arrow sum to the digit in that arrow’s circle; adjacent digits separated by a black dot have a ratio of 1:2; cells with a grey square contain even digits; adjacent digits along a green line have a difference of at least 5; clues outside the grid indicate the digit which has to be placed in the Nth cell in the corresponding direction, where N is the digit placed in the first cell in that direction; cells with a grey circle contain odd digits; digits in a white circle must appear in at least one of the four surrounding cells (for this clue type to be correct, all digits in the circle must satisfy this condition); cells along a purple line must be a set of consecutive digits; digits along a thermometer must increase from the bulb end; cells separated by a V must contain digits summing to 5; cells connected by a white dot have to contain consecutive digits; and cells separated by an X must contain digits summing to 10.