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, 4 subtraction cages and 13 multiplication cages.

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Where to start

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

9 cages allow a single combination of digits — the safest openings this grid offers: the 20× cage at row 1, column 1 (1 × 4 × 5); the 70× cage at row 1, column 5 (2 × 5 × 7); the 32× cage at row 5, column 4 (1 × 4 × 8); the 112× cage at row 6, column 1 (2 × 7 × 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 20× cage at row 1, column 1 (3 cells) can only be 1 × 4 × 5.
• The 70× cage at row 1, column 5 (3 cells) can only be 2 × 5 × 7.
• The 32× cage at row 5, column 4 (3 cells) can only be 1 × 4 × 8.
• The 112× cage at row 6, column 1 (3 cells) can only be 2 × 7 × 8.
• The 210× cage at row 6, column 3 (3 cells) can only be 5 × 6 × 7.
• The 21× cage at row 6, column 8 (3 cells) can only be 1 × 3 × 7.
• The 35× cage at row 2, column 6 (2 cells) can only be 5 × 7.
• The 3× cage at row 3, column 4 (2 cells) can only be 1 × 3.
• The 40× cage at row 8, column 4 (2 cells) can only be 5 × 8.
• The 22+ cage at row 1, column 7 (3 cells) could be 6 + 8 + 8 or 7 + 7 + 8 — the grid leaves only 6 + 8 + 8.
• The 40× cage at row 6, column 2 (3 cells) could be 1 × 5 × 8 or 2 × 4 × 5 — the grid leaves only 1 × 5 × 8.
• The 6+ cage at row 5, column 7 (2 cells) could be 1 + 5 or 2 + 4 — the grid leaves only 2 + 4.
• The 6+ cage at row 7, column 3 (2 cells) could be 1 + 5 or 2 + 4 — the grid leaves only 2 + 4.
• The 9+ cage at row 1, column 2 (3 cells) could be 1 + 2 + 6, 1 + 3 + 5 or 2 + 3 + 4 — the grid leaves only 2 + 3 + 4.
• The 96× cage at row 3, column 6 (3 cells) could be 2 × 6 × 8, 3 × 4 × 8 or 4 × 4 × 6 — the grid leaves only 3 × 4 × 8.
• The 12× cage at row 3, column 7 (3 cells) could be 1 × 2 × 6, 1 × 3 × 4 or 2 × 2 × 3 — the grid leaves only 1 × 2 × 6.
• The 36× cage at row 4, column 1 (3 cells) could be 1 × 6 × 6, 2 × 3 × 6 or 3 × 3 × 4 — the grid leaves only 2 × 3 × 6.
• The 5− cage at row 7, column 6 (2 cells) could be 6 − 1, 7 − 2 or 8 − 3 — the grid leaves only 6 − 1.
• The 7+ cage at row 8, column 7 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 3 + 4.
• The 10+ cage at row 2, column 2 (3 cells) could be 1 + 2 + 7, 1 + 3 + 6, 1 + 4 + 5 or 2 + 3 + 5 — the grid leaves only 1 + 3 + 6.
• The 2− cage at row 4, column 8 (2 cells) could be 3 − 1, 4 − 2, 5 − 3, 6 − 4, 7 − 5 or 8 − 6 — the grid leaves only 7 − 5.
• The 1− cage at row 3, column 3 (2 cells) allows 7 combinations, from 2 − 1 to 8 − 7 — too many to start from; the cells around it settle it, as 8 − 7.
• The 1− cage at row 6, column 5 (2 cells) allows 7 combinations, from 2 − 1 to 8 − 7 — too many to start from; the cells around it settle it, as 3 − 2.

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 59. Work the 36× cage: with what the board already holds, only 6 can sit in row 4, column 1 and still hit the target. On its own that cage could be 1 × 6 × 6, 2 × 3 × 6 or 3 × 3 × 4; the digits already placed around it leave one.
2. Step 2 of 59. Somewhere in this line the 8 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 59. Work the 96× cage: with what the board already holds, only 8 can sit in row 4, column 5 and still hit the target. On its own that cage could be 2 × 6 × 8, 3 × 4 × 8 or 4 × 4 × 6; the digits already placed around it leave one.

How this grid gives way

Of the 59 cells left to fill after the 5 printed ones, 29 by a cage’s arithmetic leaving only one value that fits; 24 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