Daily calcudoku

The puzzle

Fill every row and every column with the digits 1 to 7, each exactly once, so that the digits in every outlined cage combine to its target with its operation. This one has 10 addition cages, 2 subtraction cages, 5 multiplication cages and 2 division cages.

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Where to start

4 cells are printed for you: a 5 at row 2, column 5; a 5 at row 4, column 6; a 7 at row 7, column 1; and a 6 at row 7, column 2. Fill those in first, then let each one rule its digit out of its row and column.

3 cages allow a single combination of digits — the safest openings this grid offers: the 35× cage at row 1, column 4 (1 × 5 × 7); the 13+ cage at row 1, column 6 (6 + 7); the 7÷ cage at row 6, column 2 (7 ÷ 1).

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 35× cage at row 1, column 4 (3 cells) can only be 1 × 5 × 7.
• The 13+ cage at row 1, column 6 (2 cells) can only be 6 + 7.
• The 7÷ cage at row 6, column 2 (2 cells) can only be 7 ÷ 1.
• The 60× cage at row 2, column 2 (3 cells) could be 2 × 5 × 6 or 3 × 4 × 5 — the grid leaves only 3 × 4 × 5.
• The 48× cage at row 3, column 4 (3 cells) could be 2 × 4 × 6 or 3 × 4 × 4 — the grid leaves only 2 × 4 × 6.
• The 16+ cage at row 3, column 5 (3 cells) could be 3 + 6 + 7 or 4 + 5 + 7 — the grid leaves only 3 + 6 + 7.
• The 12× cage at row 1, column 3 (2 cells) could be 2 × 6 or 3 × 4 — the grid leaves only 3 × 4.
• The 12× cage at row 7, column 5 (2 cells) could be 2 × 6 or 3 × 4 — the grid leaves only 3 × 4.
• The 8+ cage at row 2, column 6 (2 cells) could be 1 + 7, 2 + 6 or 3 + 5 — the grid leaves only 2 + 6.
• The 9+ cage at row 3, column 1 (2 cells) could be 2 + 7, 3 + 6 or 4 + 5 — the grid leaves only 3 + 6.
• The 8+ cage at row 3, column 6 (2 cells) could be 1 + 7, 2 + 6 or 3 + 5 — the grid leaves only 1 + 7.
• The 2÷ cage at row 4, column 7 (2 cells) could be 2 ÷ 1, 4 ÷ 2 or 6 ÷ 3 — the grid leaves only 4 ÷ 2.
• The 7+ cage at row 5, column 1 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 2 + 5.
• The 9+ cage at row 5, column 4 (2 cells) could be 2 + 7, 3 + 6 or 4 + 5 — the grid leaves only 3 + 6.
• The 7+ cage at row 1, column 1 (3 cells) could be 1 + 1 + 5, 1 + 2 + 4, 1 + 3 + 3 or 2 + 2 + 3 — the grid leaves only 1 + 2 + 4.
• The 7+ cage at row 5, column 6 (3 cells) could be 1 + 1 + 5, 1 + 2 + 4, 1 + 3 + 3 or 2 + 2 + 3 — the grid leaves only 1 + 2 + 4.
• The 2− cage at row 6, column 7 (2 cells) could be 3 − 1, 4 − 2, 5 − 3, 6 − 4 or 7 − 5 — the grid leaves only 5 − 3.
• The 1− cage at row 7, column 3 (2 cells) could be 2 − 1, 3 − 2, 4 − 3, 5 − 4, 6 − 5 or 7 − 6 — the grid leaves only 2 − 1.
• The 13+ cage at row 4, column 2 (3 cells) allows 8 combinations, from 1 + 5 + 7 to 4 + 4 + 5 — too many to start from; the cells around it settle it, as 1 + 5 + 7.

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 45. Work the 35× cage: with what the board already holds, only 5 can sit in row 1, column 4 and still hit the target. On its own that cage allows only 1 × 5 × 7.
2. Step 2 of 45. Somewhere in this line the 2 must go — and row 6, column 5 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 34 of 45. Work the 9+ cage: with what the board already holds, only 6 can sit in row 6, column 4 and still hit the target. On its own that cage could be 2 + 7, 3 + 6 or 4 + 5; the digits already placed around it leave one.

How this grid gives way

Of the 45 cells left to fill after the 4 printed ones, 20 by a cage’s arithmetic leaving only one value that fits; 19 by crossing off what a cell’s row and column already hold until one digit was left; and 6 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