Calcudoku strategy: cages, lines and the what-if

Calcudoku mixes two kinds of thinking: arithmetic inside each cage, and Sudoku-style logic across rows and columns. The trick is to let each one feed the other.

How the difficulty works

Each tier is set by the hardest step you'll need:

• Easy (4×4) needs only cage arithmetic and singles.
• Medium (6×6) needs at least one cage that forces a number into a row or column, or a pair.
• Hard (8×8) needs a triple or a short "what if": try a number, follow it a step or two, and watch it fail.

Nothing needs guessing beyond that one short what-if.

Cage arithmetic

Every cage starts with a question: which numbers can make it? Small cages are the most useful, because they have the fewest answers.

• Two-cell differences: in a 6×6, a 5− can only be 1 and 6. A 1− has the most options of all — avoid starting there.
• Two-cell quotients: 3÷ in a 6×6 is 1 and 3, or 2 and 6. Divisions rarely have more than two answers.
• Products are often the most telling: a 7 in a product means a 7 in the cage, and 5× in two cells can only be 1 and 5.
• Sums at the extremes: the smallest and largest sums can only be made one way. In a 4×4, 3+ in two cells is 1 and 2; 7+ is 3 and 4.

Remember the repeat rule: in an L-shaped cage, the two ends don't share a row or column, so they can hold the same number. A 3-cell 4+ is impossible in a straight line but works in an L as 1, 2, 1.

Singles, as in Sudoku

Once cages have narrowed things down, the Latin-square rule does the rest. A number with only one place left in a row or column goes there, and a cell with only one number left takes it.

Medium: a cage that claims a number for its line

Sometimes you can't place anything in a cage yet, but you can tell that every way of making it uses a certain number in a certain row or column. Whichever way it's made, that line gets its number from the cage — so the rest of the line can't have it.

Two-cell cages in a line are the classic case: a 5− in a row of a 6×6 must be 1 and 6, so that row's 1 and 6 are both inside the cage, and nowhere else.

Pairs

Two cells in a row or column that can only hold the same two numbers will use both of them between them, so every other cell in that line can drop those numbers.

Hard: a short what-if

On an 8×8 board you'll sometimes run out of direct deductions. Then pick a cell with only two or three candidates, ideally in a small cage, and ask what happens if it held one of them. Fill in only what follows directly — the cage it's in, the singles it creates — for a step or two. If that leaves a cage that can't be made, or a row with no place for a number, the candidate was wrong.

Choose the try that looks most likely to fail quickly, and stop after a few steps. A what-if that doesn't break fast isn't proving anything; undo it and look elsewhere. Our puzzles never need one longer than that.

Good habits

• Pencil in small cages first. Two-cell cages and products with a 5 or 7 have the fewest options.
• Use the line totals. Every row adds up to the same number (10 on a 4×4, 21 on a 6×6, 36 on an 8×8). If every cell of a row but one lies in addition cages wholly inside that row, the last cell is the total minus their targets.
• Watch for repeats in cages. They only work where the cells don't share a line, which is often what makes an L-shaped cage solvable.
• After each number, revisit its cage. One placed number often settles the rest of a small cage at once.