The Rules of Calcudoku

Three rules, and only the third takes any learning. Here is each of them shown on a small board rather than merely stated.

Rule one: each digit once along every line

On a board of side n the digits run from 1 to n, and each of them appears exactly once in every row and exactly once in every column. That is the entire grid rule. There is no third kind of line — no box, no diagonal, no shape that has to contain a full set — and nothing carries over from one board to the next.

It is worth being precise about what "exactly once" buys you, because it is more than it looks. A row is not merely free of repeats; it is a complete set, so the digits in it always add to the same total and always multiply to the same product whatever else is going on. On a six that total is twenty-one on every row and every column of every board ever printed, and it is free information nobody wrote on the puzzle.

The figure below shows a finished four alongside a grid where every cage is satisfied and the board is still wrong, because two cells in a row hold the same digit. Getting the arithmetic right is not the same as getting the puzzle right, and the two can come apart in exactly this way.

Rule two: the cage says what its digits produce

Each cage carries a number and a sign. Add, and its digits total the number. Multiply, and they multiply to it. Subtract or divide, and the cage has exactly two cells: the number is the difference between them or the result of dividing the larger by the smaller, and neither cares which cell holds which digit. A cage of one cell carries only a number, and that number is the digit.

The two-cell restriction on subtraction and division is not a house style, it is arithmetic. Subtraction and division are not associative, so "these four digits subtract to two" does not name a value — it names four different values depending on the order you take them in. Addition and multiplication are associative, which is exactly why they are the two operations allowed to span a cage of any size.

The figure here shows a three with a 3÷ cage filled both ways round, which is legal, and the same cage holding a pair that does not divide at all, which is not. A quotient cage needs one digit to go into the other a whole number of times; two and three are three apart in one sense and no use whatsoever in this one.

Rule three: a cage is not a region

This is the one people get wrong, and the one worth reading twice. A cage constrains arithmetic and nothing else. Two of its cells may hold the same digit, provided they are not in the same row and not in the same column — because the only thing forbidding a repeat is rule one, and rule one talks about lines rather than about cages.

The consequence shows up immediately in the combinations. On a four, a three-cell cage marked 48× has exactly one set of digits behind it, and that set is four, four and three. Read the cage as needing three different digits and there is no set at all, the board looks broken, and the usual reaction is to assume the puzzle is faulty. It is not; the reading was.

The figure below shows that cage twice with the same three digits in it. One arrangement is legal, with the two fours in different rows and different columns. The other has both fours side by side in the top row, which breaks rule one while satisfying the cage perfectly — the clearest possible demonstration that the two rules are checking different things.

Four things people assume the rules say

They do not say a cage contains each digit at most once. That is rule three above, and it is the single most expensive misreading available on this puzzle.

They do not say cages are rectangles. A cage is any connected group of cells — an L, a T, a staircase, a single square — and its shape is what decides whether a repeat inside it is available at all.

They do not say the cage sign tells you the cage size. A plus or a times cage may hold one cell, two, or six; only minus and divide are pinned to exactly two. A large number does not imply a large cage either: 40× fits in two cells on a nine.

And they do not permit guessing. Every board published here was checked to be finishable by argument alone before it was shown to anybody, so a board that will not move is one where something has been missed rather than one that requires a coin. That is a promise about the generator, and it is why the generator stops merging cages exactly when it does.

Common questions

More Calcudoku pages

• CalcudokuUnlimited boards
• Daily calcudokuA new square every day
• Solving a squareA 6×6, worked
• Calcudoku techniquesThe whole ladder
• Free cellsTechnique — easy
• Cage combinationsTechnique — easy
• SinglesTechnique — medium
• Row totalsTechnique — hard
• Cage interactionTechnique — expert
• Guessing at cagesTechnique — refused here too
• Printable calcudokuFor paper
• Calcudoku archiveEvery past square