Daily calcudoku

The puzzle

Fill every row and every column with the digits 1 to 8, each exactly once, so that the digits in every outlined cage combine to its target with its operation. This one has 6 addition cages, 2 subtraction cages, 11 multiplication cages and 1 division cage.

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Where to start

9 cells are printed for you: a 7 at row 1, column 4; a 1 at row 1, column 6; a 8 at row 3, column 2; a 2 at row 4, column 8; a 4 at row 5, column 8; a 1 at row 6, column 2; a 7 at row 6, column 7; a 6 at row 6, column 8; and a 5 at row 7, column 7. Fill those in first, then let each one rule its digit out of its row and column.

6 cages allow a single combination of digits — the safest openings this grid offers: the 280× cage at row 2, column 1 (5 × 7 × 8); the 90× cage at row 3, column 4 (3 × 5 × 6); the 126× cage at row 4, column 2 (3 × 6 × 7); the 64× cage at row 6, column 3 (2 × 4 × 8), 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 280× cage at row 2, column 1 (3 cells) can only be 5 × 7 × 8.
• The 90× cage at row 3, column 4 (3 cells) can only be 3 × 5 × 6.
• The 126× cage at row 4, column 2 (3 cells) can only be 3 × 6 × 7.
• The 64× cage at row 6, column 3 (3 cells) can only be 2 × 4 × 8.
• The 280× cage at row 8, column 4 (3 cells) can only be 5 × 7 × 8.
• The 5÷ cage at row 2, column 8 (2 cells) can only be 5 ÷ 1.
• The 120× cage at row 1, column 1 (3 cells) could be 3 × 5 × 8 or 4 × 5 × 6 — the grid leaves only 4 × 5 × 6.
• The 120× cage at row 5, column 6 (3 cells) could be 3 × 5 × 8 or 4 × 5 × 6 — the grid leaves only 3 × 5 × 8.
• The 84× cage at row 7, column 8 (3 cells) could be 2 × 6 × 7 or 3 × 4 × 7 — the grid leaves only 3 × 4 × 7.
• The 12× cage at row 8, column 1 (3 cells) could be 1 × 2 × 6 or 1 × 3 × 4 — the grid leaves only 1 × 2 × 6.
• The 6− cage at row 7, column 5 (2 cells) could be 7 − 1 or 8 − 2 — the grid leaves only 8 − 2.
• The 72× cage at row 1, column 5 (3 cells) could be 2 × 6 × 6, 3 × 3 × 8 or 3 × 4 × 6 — the grid leaves only 3 × 4 × 6.
• The 7+ cage at row 3, column 7 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 1 + 6.
• The 8+ cage at row 5, column 4 (2 cells) could be 1 + 7, 2 + 6 or 3 + 5 — the grid leaves only 1 + 7.
• The 24× cage at row 4, column 1 (3 cells) could be 1 × 3 × 8, 1 × 4 × 6, 2 × 2 × 6 or 2 × 3 × 4 — the grid leaves only 2 × 3 × 4.
• The 4− cage at row 6, column 1 (2 cells) could be 5 − 1, 6 − 2, 7 − 3 or 8 − 4 — the grid leaves only 5 − 1.
• The 13+ cage at row 7, column 2 (3 cells) could be 1 + 4 + 8, 1 + 5 + 7, 2 + 3 + 8, 2 + 4 + 7, 2 + 5 + 6 or 3 + 4 + 6 — the grid leaves only 3 + 4 + 6.
• The 11+ cage at row 2, column 3 (3 cells) allows 9 combinations, from 1 + 2 + 8 to 3 + 4 + 4 — too many to start from; the cells around it settle it, as 2 + 2 + 7.
• The 13+ cage at row 1, column 7 (3 cells) allows 10 combinations, from 1 + 4 + 8 to 4 + 4 + 5 — too many to start from; the cells around it settle it, as 2 + 3 + 8.
• The 13+ cage at row 3, column 6 (3 cells) allows 10 combinations, from 1 + 4 + 8 to 4 + 4 + 5 — too many to start from; the cells around it settle it, as 1 + 4 + 8.

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 55. Work the 280× cage: with what the board already holds, only 8 can sit in row 2, column 1 and still hit the target. On its own that cage allows only 5 × 7 × 8.
2. Step 4 of 55. Somewhere in this line the 2 must go — and row 1, column 7 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 41 of 55. Work the 120× cage: with what the board already holds, only 4 can sit in row 1, column 2 and still hit the target. On its own that cage could be 3 × 5 × 8 or 4 × 5 × 6; the digits already placed around it leave one.

How this grid gives way

Of the 55 cells left to fill after the 9 printed ones, 26 by crossing off what a cell’s row and column already hold until one digit was left; 23 by a cage’s arithmetic leaving only one value that fits; 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