How to play Calcudoku

Calcudoku is a number puzzle that mixes the logic of Sudoku with a little arithmetic. The square grid is divided into outlined groups of cells called cages. Each cage shows a target number and an operation - addition, subtraction, multiplication or division - and the digits inside it must produce that target. At the same time every row and every column must contain each digit exactly once.

Every puzzle here is generated fresh by the game when you start it, has exactly one solution, and can be solved by reasoning alone. You never need to guess. The four levels grow from a 4x4 grid on Easy to a 7x7 grid on Crazy, and each level also asks for deeper reasoning than the one before.

How to play on skill.games

When you open the game, the settings popup appears. Choose a level under DIFFICULTY, choose how mistakes are treated under MISTAKES, and press PLAY. A new puzzle is generated in a moment.

• The board takes up most of the screen. Thick lines mark the cages; each cage's target and sign sit in its top-left corner, for example 12+, 3−, 60× or 2÷. A cage of a single cell simply shows the digit that belongs there.
• Choosing a cell: tap any cell. Its row and column are shaded, and every cell holding the same digit is highlighted.
• Entering a digit: with a cell chosen, tap a number on the number pad (under the board, or beside it when your screen is wider than it is tall). The small number under each key shows how many of that digit are still to place; a key fades out once every row has it.
• NOTES: tap NOTES to switch notes on. Numbers you tap are now pencilled into the cell as small candidates, below the cage label, and tapping one again removes it. Tap NOTES again to go back to entering digits. When you enter a digit, it is removed automatically from the notes of every cell in its row and column.
• ERASE clears the chosen cell (its digit or its notes).
• UNDO takes back your last action - a digit, a note or an erase - all the way back to the start. Undo is free and unlimited.
• Highlighting a digit: with no cell chosen, tap a number on the pad to highlight every cell and note holding that digit. Tap it again to clear the highlight.

On a computer you can also use the keyboard: the arrow keys (or W, A, S, D) move between cells, the number keys enter a digit, Shift and a number pencil a note, Backspace, Delete or 0 erase, N switches notes on and off, and Z undoes.

The top line shows the clock, the level, the grid size and your mistakes. The clock runs only while you play: it stops when the menu or a popup is open. The menu button in the top-left corner pauses the game, starts a new puzzle or switches the sound on or off. The game is saved after every move, so if you close the page you can press CONTINUE later and carry on with the same grid, notes, clock and undo history.

When the last cell is filled correctly, the board ripples and the results screen shows your time, your best time at that level (a star marks a new record), your mistakes, the grid size and the puzzle's code, together with your wins, losses, win rate and streaks.

Rules

The grid. On a grid of size N (4 to 7 here) every row and every column must end up holding each digit from 1 to N exactly once.

Cages. Every cell belongs to exactly one cage. The digits of a cage must make its target with its operation:

• + the digits add up to the target;
• × the digits multiply to the target;
• − a two-cell cage: the larger digit minus the smaller one gives the target;
• ÷ a two-cell cage: the larger digit divided by the smaller one gives the target.

Subtraction and division cages always have exactly two cells, and the order of the two cells doesn't matter. A digit may appear twice in one cage, as long as the two cells are not in the same row or column. A one-cell cage gives its digit away.

Mistakes - Unlimited (the default). If you enter a digit that is already in the same row or column, it turns red and the mistakes counter goes up by one. If you fill the last cell of a cage and its digits don't make the target, the cage's label turns red inside a red box and that also counts as a mistake. Nothing else is checked, so nothing about the solution is revealed - just as on paper. Because the solution is unique, a full grid with no red marks is always the right one.

Mistakes - 3 lives. Every digit you enter is checked against the solution at once. A wrong digit is marked in red and costs a life; the third mistake ends the game and counts as a loss. Undo can take a wrong digit back, but not the life it cost.

Abandoned puzzles. Starting a new puzzle before finishing the current one counts as a loss in your statistics for that level.

Solving techniques

These are the methods the game's own solver uses to grade every puzzle, from the simplest up.

• One-cell cages: write them in first - they are free digits.
• Cage arithmetic: a two-cell cage rarely allows many digits. On a 6x6 grid, 11+ can only be 5 and 6; 3÷ can only be 1 and 3 or 2 and 6. When all but one cell of a cage are known, the last digit follows from the target.
• Singles: a digit with only one possible place in a row or column, or a cell with only one possible digit.
• Cage combinations: list the sets of digits a bigger cage can hold and keep only those that still fit the cells' candidates. A 3-cell 6+ cage in one row must be 1, 2 and 3.
• A digit locked in a line: if every combination of a cage needs a certain digit, and that digit can only go in cells of the cage that lie in one row, the digit can't appear anywhere else in that row (the same for columns).
• Pairs and triples: two cells of a row that can only hold the same two digits claim them; the rest of the row loses those digits.
• X-Wing: a digit confined to the same two columns in two rows can be removed from the rest of those columns.
• Row and column totals: every row adds up to the same total (1 + 2 + ... + N) and multiplies to the same product. Subtract the addition cages that lie entirely inside a row, and the few cells left over must make up the difference - a new cage you can work with.
• Advanced: XY-Wing, Swordfish, quads and simple colouring - longer "if this, then that" reasoning, still without guessing.

Strategy tips

• Start with the freebies. One-cell cages and two-cell cages with few combinations give the first digits.
• Look for extremes. Large products need large digits, small sums need small ones. On a 5x5 grid, 3+ over two cells is always 1 and 2.
• Combine cages with lines. A cage that lies in one row tells you which digits that row has used up.
• Use notes once the easy cells run out. Pencil in the possible digits of a cage, then cross off the ones its row and column rule out.
• Count totals. When most of a row is covered by addition cages, the total of the row tells you what the remaining cells add up to.
• Don't guess. Every puzzle here can be solved by logic. If you feel stuck, there is a cage or a line you haven't fully used yet.

Difficulty levels

The game generates every puzzle on the spot and grades it with a solver that works the way a person does, always using the easiest step available. A puzzle is given to you only if its hardest required step matches the level you chose.

• Easy: a 4x4 grid; cage arithmetic on cages with one or two open cells and simple singles are all you need.
• Normal: a 5x5 grid; at least one step needs the combinations of a bigger cage, a digit locked in a line, or a pair.
• Hard: a 6x6 grid with cages of up to four cells; at least one step needs a triple or an X-Wing.
• Crazy: a 7x7 grid; at least one step needs row and column totals, an XY-Wing, a Swordfish, a quad or colouring.

Statistics and best times are kept separately for each level. If Calcudoku is new to you, start on Easy: the 4x4 puzzles teach the cage arithmetic every bigger grid is built on.