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 5 addition cages, 6 subtraction cages, 6 multiplication cages and 1 division cage.

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Where to start

5 cells are printed for you: a 3 at row 2, column 1; a 7 at row 2, column 4; a 5 at row 2, column 5; a 1 at row 2, column 6; and a 1 at row 6, column 4. Fill those in first, then let each one rule its digit out of its row and column.

5 cages allow a single combination of digits — the safest openings this grid offers: the 40× cage at row 1, column 6 (2 × 4 × 5); the 56× cage at row 4, column 1 (2 × 4 × 7); the 15× cage at row 4, column 2 (1 × 3 × 5); the 4÷ cage at row 3, column 1 (4 ÷ 1), among others.

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 40× cage at row 1, column 6 (3 cells) can only be 2 × 4 × 5.
• The 56× cage at row 4, column 1 (3 cells) can only be 2 × 4 × 7.
• The 15× cage at row 4, column 2 (3 cells) can only be 1 × 3 × 5.
• The 4÷ cage at row 3, column 1 (2 cells) can only be 4 ÷ 1.
• The 30× cage at row 7, column 1 (2 cells) can only be 5 × 6.
• The 72× cage at row 3, column 3 (3 cells) could be 2 × 6 × 6 or 3 × 4 × 6 — the grid leaves only 3 × 4 × 6.
• The 30× cage at row 4, column 4 (3 cells) could be 1 × 5 × 6 or 2 × 3 × 5 — the grid leaves only 1 × 5 × 6.
• The 5− cage at row 3, column 5 (2 cells) could be 6 − 1 or 7 − 2 — the grid leaves only 7 − 2.
• The 5− cage at row 6, column 7 (2 cells) could be 6 − 1 or 7 − 2 — the grid leaves only 6 − 1.
• The 5− cage at row 7, column 3 (2 cells) could be 6 − 1 or 7 − 2 — the grid leaves only 7 − 2.
• The 15+ cage at row 3, column 7 (3 cells) could be 2 + 6 + 7, 3 + 5 + 7 or 4 + 5 + 6 — the grid leaves only 3 + 5 + 7.
• The 8+ cage at row 4, column 6 (2 cells) could be 1 + 7, 2 + 6 or 3 + 5 — the grid leaves only 2 + 6.
• The 7+ cage at row 7, column 5 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 3 + 4.
• The 3− cage at row 6, column 5 (2 cells) could be 4 − 1, 5 − 2, 6 − 3 or 7 − 4 — the grid leaves only 7 − 4.
• The 9+ cage at row 1, column 3 (3 cells) could be 1 + 1 + 7, 1 + 2 + 6, 1 + 3 + 5, 1 + 4 + 4, 2 + 2 + 5 or 2 + 3 + 4 — the grid leaves only 1 + 2 + 6.
• The 1− cage at row 1, column 1 (2 cells) could be 2 − 1, 3 − 2, 4 − 3, 5 − 4, 6 − 5 or 7 − 6 — the grid leaves only 7 − 6.
• The 1− cage at row 1, column 4 (2 cells) could be 2 − 1, 3 − 2, 4 − 3, 5 − 4, 6 − 5 or 7 − 6 — the grid leaves only 4 − 3.
• The 10+ cage at row 5, column 3 (3 cells) allows 7 combinations, from 1 + 2 + 7 to 3 + 3 + 4 — too many to start from; the cells around it settle it, as 2 + 3 + 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 44. Work the 30× cage: with what the board already holds, only 5 can sit in row 4, column 4 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.
2. Step 2 of 44. Somewhere in this line the 7 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 31 of 44. Work the 15+ cage: with what the board already holds, only 7 can sit in row 5, column 7 and still hit the target. On its own that cage could be 2 + 6 + 7, 3 + 5 + 7 or 4 + 5 + 6; the digits already placed around it leave one.

How this grid gives way

Of the 44 cells left to fill after the 5 printed ones, 22 by a cage’s arithmetic leaving only one value that fits; 18 by crossing off what a cell’s row and column already hold until one digit was left; and 4 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