How to Solve Calcudoku Puzzles

The four signs

Minus and divide only ever appear in cages of exactly two squares, and the order does not matter: the bigger digit can be in either square. A cage of a single square simply shows its digit, and the page fills those in for you. Plus and times cages can be any size, usually two to four squares.

Sizes and difficulty

Bigger grids have more digits to choose from, bigger cages and more multiplying and dividing, so each step takes a little more thought. Every daily grid is checked by a solver when it is made, and only grids with exactly one answer are used, so you never need to guess.

Start with the cages that have one answer

Some cages can only be filled one way, and they are the place to begin. Look for:

• Extreme sums. In a 6×6 grid, 3+ over two squares must be 1 and 2, and 11+ over two squares must be 5 and 6.
• Big differences. 5− in a 6×6 grid can only be 1 and 6, because no other pair is five apart.
• Unusual products. 30× over two squares must be 5 and 6. A product with a 5 in it tells you a 5 is in the cage, since no other digit up to 6 has 5 as a factor.
• Divisions with few pairs. 4÷ needs 1 and 4; in a 6×6 grid 3÷ could be 1 and 3 or 2 and 6, which is still only two choices.

Once you know which two digits a cage holds, you often do not yet know which goes where. That is fine: write both as pencil notes in both squares. You still know a lot, because those two digits are now used up in any row or column that contains the whole cage.

A pair that fixes a row

In a 4×4 grid, a 3− cage sits across the first two squares of the top row. It must hold 1 and 4. The other two squares of that row must then hold 2 and 3. If the last square of the row is also a one-square cage showing 3, the third square is 2, and you have filled half a row before doing any real arithmetic.

Use the row and column totals

Every row and every column holds each digit once, so each one adds up to the same total: 10 in a 4×4 grid, 15 in a 5×5 grid and 21 in a 6×6 grid. This is the most useful trick in the puzzle.

When a row is covered entirely by addition cages and one leftover square, add up the cage targets and subtract them from the row total: the difference is the leftover digit. The same works for a column, and for two rows together (twice the total) when cages stick out of one row into the next. Cages that are not addition cages still help when you have narrowed them down to a single digit set, because then you know their sum too.

Notes and elimination

On medium and hard days most of the solving happens in the notes. A good routine:

1. List the digit sets for each cage. For a two-square cage this is quick. For a three-square sum, list the combinations that reach the target with no repeat in the same row or column.
2. Note the candidates. Switch on Notes and add the possible digits to each square of the cage.
3. Cross out what the rows and columns already have. Every digit you place removes itself from the notes in its row and column (the page can do this for you: leave Notes clear themselves switched on).
4. Look for a lonely candidate. A square with one note left is solved, and a digit that can only go in one square of a row or column belongs there.

Repeat until the grid is full. When it stalls, return to the row totals or look at a cage whose digit sets you have not written down yet.

Bent cages and repeated digits

A cage that turns a corner, such as an L of three squares, can hold the same digit twice, as long as the two copies are in different rows and different columns. This surprises many players: 1+1+2 = 4+ is a perfectly good L-shaped cage, and so is 2×2×3 = 12×. Straight cages along a row or column can never repeat a digit. Keep this in mind when you list digit sets, or you will rule out the right answer.

Speed habits

• Fill one-answer cages first, then the row totals. These two steps alone often solve an easy grid.
• Prefer pairs over guesses. Two digits locked in a cage are almost as useful as a placed digit; note them and move on.
• Watch the red marks. With Show mistakes on, a digit repeated in a row or column is underlined in red, and a full cage that misses its target is tinted, so a slip never costs you the whole grid.
• Use the keyboard on a computer. Arrow keys move, digits fill, N switches notes, Backspace erases and Ctrl+Z undoes.

Scoring and the leaderboard

Calcudoku is ranked by time, fastest first; fewer entries break ties. The clock starts with your first move, the grid is checked the moment the last square is filled correctly, and only a member's first solve of the day counts for the leaderboard. Practice puzzles from Play more are never ranked, so they are a good place to try the techniques above at your own pace.