Calcudoku,
arithmetic in a Latin square.

Logic with a little arithmetic

Calcudoku, also called Mathdoku, fills a square grid so that each row and column holds every number from 1 up to the grid's size exactly once. On top of that, the grid is split into outlined cages, and each cage's numbers must reach a target using the operation shown. The cages supply almost all of the information, so the skill is turning each target into a short list of possible numbers.

Reading a cage

Before placing anything, ask which sets of numbers could make each cage label.

• Single-cell cages. A label with no operation is simply the number that goes there. Fill these first.
• Subtraction and division. These cages always cover exactly two cells, and order does not matter. On a 4×4 grid, a 3− cage can only be 1 and 4. A 2÷ cage can be 1 and 2 or 2 and 4.
• Large products. Multiplication targets are often the most revealing. On the 6×6 board, a two-cell 30× can only be 5 and 6. A 20× is 4 and 5. Factor the target and see which pairs fit the range.
• Small sums. A two-cell 3+ is always 1 and 2. A two-cell 4+ is always 1 and 3, because 2 and 2 would repeat a number in one line.

Note that a cage that bends around a corner can repeat a number, as long as the two copies are not in the same row or column. An L-shaped 3-cell 12× could be 2, 3 and 2 if the two 2s sit in different rows and columns.

Combining cages with rows and columns

Here is a worked example on the 4×4 Easy board. Say the top row contains a 3− cage across its first two cells, so those cells are 1 and 4 in some order. The other two cells of the row must then be 2 and 3. Say the third cell starts a two-cell 6× cage that runs downward. On a 4×4 grid the only way to make 6 from two numbers is 2 and 3, so the top cell is 2 or 3 and the cell below it is the other. Now look at the second row. If a 2 already sits elsewhere in that row, the lower cell cannot be 2. It must be 3, which makes the top cell 2, and the last cell of the top row 3.

A few habits help on every size:

• Use the row total. Every row adds up to the same amount: 10 on 4×4, 15 on 5×5 and 21 on 6×6. If a row is covered by known cages plus one unknown cell, subtract to find it.
• Pairs lock out numbers. If two cells in a column must be 1 and 4, no other cell in that column can be 1 or 4.
• Watch the cage tag. A tag turns green when its numbers reach the target, and red when a full cage misses it.

Pitfalls worth knowing

• Forgetting division can include 1. A 3÷ cage is 1 and 3 just as easily as 2 and 6.
• Assuming cages never repeat. Only cages in a straight line are sure to hold different numbers. Bent cages can repeat a number across different rows and columns.
• Guessing to move forward. The board has no candidate notes, so keep options in your head or on paper. Each puzzle has a single solution, so a guess is never needed.
• Missing a red repeat. A number repeated in a row or column turns red. Fix it before continuing, because it will mislead every cage nearby.

Progressing through the grid sizes

Easy is 4×4, where most cages have only one or two possible sets and the row total of 10 finishes nearly anything. Medium is 5×5, with bigger cages and more ways to make each target. Hard is 6×6, where products grow large and factoring becomes the main skill.

A good drill is to list the possible sets for every cage before placing a single number. On Easy this can solve the board on its own. On Hard it shows you where to start. The hint fills one correct cell and locks it. Sudoku at /games/sudoku shares the one-of-each rule, and Kakuro at /games/kakuro works the same sum-reading skill with longer runs.