Fill in the grid with digits from 1 to the grid size, so that every digit appears exactly once in each row and column.
Thick lines divide the grid into cages, each with a clue. A clue’s number is what the cage’s digits make by its sign: added (+) or multiplied (×), or, in a cage of two cells, the larger less the smaller (−) or divided by the smaller (÷). A digit may appear twice in a cage, as long as its row and column allow it.
On a touch screen, a tap does what a click does, and a long press what a right-click does (more ways to right-click). A key the puzzle’s controls show as a button can be pressed there instead of on a keyboard.
To place a number, click in a square to select it, then type the number on the keyboard. To erase a number, click to select a square and then press Backspace.
Right-click in a square and then type a number to add or remove the number as a pencil mark, indicating numbers that you think might go in that square. Pencil mode is sticky: once a right-click switches it on, it stays on wherever you click, until you right-click again. The Marks key is lit while it is on, and pressing the key also switches it. To have it last only until your next click, turn off Right-click toggles a sticky pencil mode (stays on until right-clicked again) in the preferences.
You can also use the arrow keys to move the selected square around. Press Enter to toggle between entering numbers and entering pencil marks, and Backspace or Space to clear a square.
Press Fill marks among the puzzle’s controls to write in every pencil mark wherever nothing is placed and no mark is written yet. The M key does the same, on any board where M is not something you type. Marks you have narrowed are left alone, and pressing again never starts over: see Filling in all the pencil marks. Once every empty square has marks, the press crosses out of each the numbers already placed in its row and column.
Hint explains the next step rather than simply making it. It starts by filling in every cell’s pencil marks (the same as pressing Fill marks) and crossing out the numbers already standing in each row and column, then works from your own marks, so it carries on from wherever you are. If a number you’ve placed is wrong, or a cell’s pencil marks have crossed out its answer, it asks you to fix the highlighted mistakes first. If you would rather it left a cell unmarked until a step needs it, set Hints pencil in to Only as needed in the preferences: a cell with no marks then counts as holding every number not already in its row and column.
A cage’s clue is read out as what its cells must do: “sum to 9”, “differ by 3”, “multiply to 72”, “have a ratio of 2”. The cage steps work cell by cell: a number goes when no way of filling the cage puts it in this cell, and when every way of filling a cage puts, say, a 4 in one row, no other cell of that row can be 4.
On the hardest boards that need no guessing, a step may be a wing. Take a cell whose pencil marks are down to two values, say a or b, and two cells it can’t repeat a value with, both outlined: one marked a and c, the other b and c. Whatever goes in the middle cell, one of the outlined cells must have c, so a cell that can’t repeat a value with either of them can’t be c. If the middle cell could be c as well, the same holds for a cell that can’t repeat a value with the middle cell either.
If you’ve turned on the preference that removes a number from pencil marks in its row and column when you place it, the hint leaves that tidying to the placement rather than spelling it out as a step of its own. On an Unreasonable board, deduction can run out before the grid is full; the hint says so rather than guessing.
These parameters are available from the ‘Custom type…’ option on the ‘Type’ menu.