Calcudoku

Choose a side and a level and a fresh square of cages is cut, checked and put on screen. No limit, no account, and nothing leaves the machine you are reading this on.

A latin square with arithmetic drawn on top of it

The grid rule comes first and never changes: on a side of six, every row and every column holds the digits 1 to 6 exactly once. That much is a latin square and it is thousands of years old. What makes this a puzzle is the second layer — the square is chopped into outlined regions called cages, and each cage carries a small number and a sign in its corner saying what its digits have to produce together.

A cage marked 11+ holds digits that add to eleven. One marked 3− holds two digits three apart, either way round, so a 2 and a 5 satisfy it and so do a 5 and a 2. One marked 2÷ holds two digits where the larger is exactly twice the smaller. A cage with a single cell and no sign is simply that digit, printed for you.

Nothing else is hidden. There are no given digits scattered about waiting to be found, no second grid overlaid on the first, and no region that has to contain each digit once. The cages are the entire clue set and the whole board is the empty square underneath them, which is why a board that looks bare is not necessarily a hard one.

The rule that costs beginners an hour

A digit may appear twice inside one cage. It may not appear twice in a row and it may not appear twice in a column, but a cage is an arithmetic statement rather than a region with a uniqueness rule of its own — so an L-shaped cage whose two ends sit in different rows and different columns is perfectly entitled to hold the same digit at both ends.

This trips up almost everybody who arrives from sudoku, where a box behaves exactly like a row. Read a 48× cage over three cells in a four as needing three different digits and you will conclude it is impossible, because no three different digits from one to four multiply to forty-eight. Allow the repeat and the answer appears immediately: four, four and three, with the two fours where the grid permits them.

The habit worth building is to check the SHAPE of a cage before working out its combinations. Two cells side by side must differ. Two cells in an L may match. A four-cell block spanning two rows and two columns may hold two pairs. Deciding that first turns most cage arithmetic from a long list into a short one.

What size and what level actually change

Side and difficulty are separate choices and they do genuinely different things. A larger side means more cells, longer arithmetic and a longer sitting; it does not by itself mean a harder argument. A 9×9 filled with two-cell cages is a long, gentle board, and a 5×5 with one awkward corner can stop an experienced solver dead.

The four levels name the reasoning a board forces instead. Easy is finished by copying printed digits in and filling the last gap in a row. Medium adds the moment where a digit has only one cell left to go to. Hard makes you add a row up and price what is left. Expert is where a cage and a row have to be held in mind together before anything moves at all.

The grade is measured rather than estimated. Every published board is put through a program that always takes the cheapest argument available to it, and the hardest argument it was ever forced onto is the level printed on the board. Ask for an expert 4×4 and you may well be handed a hard one, because sixteen cells sometimes do not leave room for the argument that expert means.

Common questions

More Calcudoku pages

• Daily calcudokuA new square every day
• Calcudoku rulesThree rules, shown
• 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