Daily calcudoku

The puzzle

Fill every row and every column with the digits 1 to 6, each exactly once, so that the digits in every outlined cage combine to its target with its operation. This one has 2 addition cages, 3 subtraction cages and 8 multiplication cages.

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Where to start

4 cells are printed for you: a 1 at row 1, column 3; a 3 at row 1, column 5; a 5 at row 1, column 6; and a 6 at row 4, column 5. Fill those in first, then let each one rule its digit out of its row and column.

One cage allows a single combination of digits, and it is the natural place to begin: the 8× cage at row 1, column 1 (2 × 4).

Cage by cage

What each cage’s arithmetic allows on its own, before the rows and columns have a say — the most constrained first, which is the order worth looking at them in. Digits may repeat inside a cage only where the cells share no row or column.

• The 8× cage at row 1, column 1 (2 cells) can only be 2 × 4.
• The 15+ cage at row 1, column 4 (3 cells) could be 3 + 6 + 6 or 4 + 5 + 6 — the grid leaves only 4 + 5 + 6.
• The 6× cage at row 2, column 2 (3 cells) could be 1 × 1 × 6 or 1 × 2 × 3 — the grid leaves only 1 × 2 × 3.
• The 30× cage at row 4, column 2 (3 cells) could be 1 × 5 × 6 or 2 × 3 × 5 — the grid leaves only 2 × 3 × 5.
• The 20× cage at row 4, column 6 (3 cells) could be 1 × 4 × 5 or 2 × 2 × 5 — the grid leaves only 1 × 4 × 5.
• The 30× cage at row 6, column 1 (3 cells) could be 1 × 5 × 6 or 2 × 3 × 5 — the grid leaves only 1 × 5 × 6.
• The 24× cage at row 6, column 4 (3 cells) could be 1 × 4 × 6 or 2 × 3 × 4 — the grid leaves only 2 × 3 × 4.
• The 6× cage at row 2, column 5 (2 cells) could be 1 × 6 or 2 × 3 — the grid leaves only 2 × 3.
• The 6× cage at row 3, column 5 (2 cells) could be 1 × 6 or 2 × 3 — the grid leaves only 1 × 6.
• The 4− cage at row 5, column 3 (2 cells) could be 5 − 1 or 6 − 2 — the grid leaves only 6 − 2.
• The 3− cage at row 3, column 4 (2 cells) could be 4 − 1, 5 − 2 or 6 − 3 — the grid leaves only 4 − 1.
• The 7+ cage at row 4, column 1 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 3 + 4.
• The 1− cage at row 2, column 1 (2 cells) could be 2 − 1, 3 − 2, 4 − 3, 5 − 4 or 6 − 5 — the grid leaves only 6 − 5.

The moments that matter

Most of a solve is crossing candidates off a line until one is left. These are the steps where this grid actually turns:

1. Step 1 of 32. Work the 20× cage: with what the board already holds, only 5 can sit in row 5, column 5 and still hit the target. On its own that cage could be 1 × 4 × 5 or 2 × 2 × 5; the digits already placed around it leave one.
2. Step 2 of 32. Somewhere in this line the 4 must go — and row 1, column 2 is the only cell that can still take it. Scanning a line for where a digit can still go — rather than what a cell can still hold — is what finds this one.
3. Step 24 of 32. Work the 30× cage: with what the board already holds, only 2 can sit in row 4, column 3 and still hit the target. On its own that cage could be 1 × 5 × 6 or 2 × 3 × 5; the digits already placed around it leave one.

How this grid gives way

Of the 32 cells left to fill after the 4 printed ones, 18 by a cage’s arithmetic leaving only one value that fits; 13 by crossing off what a cell’s row and column already hold until one digit was left; and 1 by finding the one place left in a line where a digit could still go. No step needs a guess: every Cages daily is checked to have exactly one solution, reachable by reasoning alone.

The full solution