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 10 addition cages, 5 subtraction cages, 7 multiplication cages and 2 division cages.

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Where to start

6 cells are printed for you: a 6 at row 3, column 5; a 3 at row 5, column 5; a 1 at row 5, column 6; a 3 at row 6, column 7; a 8 at row 8, column 4; and a 6 at row 8, column 8. 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 14× cage at row 1, column 8 (1 × 2 × 7); the 16× cage at row 1, column 5 (2 × 8); the 32× cage at row 1, column 7 (4 × 8); the 14× cage at row 5, column 4 (2 × 7), 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 14× cage at row 1, column 8 (3 cells) can only be 1 × 2 × 7.
• The 16× cage at row 1, column 5 (2 cells) can only be 2 × 8.
• The 32× cage at row 1, column 7 (2 cells) can only be 4 × 8.
• The 14× cage at row 5, column 4 (2 cells) can only be 2 × 7.
• The 20× cage at row 7, column 5 (2 cells) can only be 4 × 5.
• The 56× cage at row 4, column 4 (3 cells) could be 1 × 7 × 8 or 2 × 4 × 7 — the grid leaves only 1 × 7 × 8.
• The 19+ cage at row 5, column 1 (3 cells) could be 4 + 7 + 8 or 5 + 6 + 8 — the grid leaves only 5 + 6 + 8.
• The 3÷ cage at row 1, column 1 (2 cells) could be 3 ÷ 1 or 6 ÷ 2 — the grid leaves only 3 ÷ 1.
• The 24× cage at row 5, column 2 (2 cells) could be 3 × 8 or 4 × 6 — the grid leaves only 4 × 6.
• The 6− cage at row 6, column 5 (2 cells) could be 7 − 1 or 8 − 2 — the grid leaves only 7 − 1.
• The 9+ cage at row 3, column 7 (3 cells) could be 1 + 2 + 6, 1 + 3 + 5 or 2 + 3 + 4 — the grid leaves only 1 + 2 + 6.
• The 7+ cage at row 7, column 6 (2 cells) could be 1 + 6, 2 + 5 or 3 + 4 — the grid leaves only 2 + 5.
• The 7+ cage at row 7, column 3 (3 cells) could be 1 + 1 + 5, 1 + 2 + 4, 1 + 3 + 3 or 2 + 2 + 3 — the grid leaves only 1 + 3 + 3.
• The 9+ cage at row 3, column 3 (2 cells) could be 1 + 8, 2 + 7, 3 + 6 or 4 + 5 — the grid leaves only 4 + 5.
• The 2÷ cage at row 6, column 8 (2 cells) could be 2 ÷ 1, 4 ÷ 2, 6 ÷ 3 or 8 ÷ 4 — the grid leaves only 8 ÷ 4.
• The 16+ cage at row 1, column 2 (3 cells) could be 1 + 7 + 8, 2 + 6 + 8, 3 + 5 + 8, 3 + 6 + 7 or 4 + 5 + 7 — the grid leaves only 3 + 5 + 8.
• The 3− cage at row 8, column 6 (2 cells) could be 4 − 1, 5 − 2, 6 − 3, 7 − 4 or 8 − 5 — the grid leaves only 7 − 4.
• The 14+ cage at row 1, column 6 (3 cells) could be 1 + 5 + 8, 1 + 6 + 7, 2 + 4 + 8, 2 + 5 + 7, 3 + 4 + 7 or 3 + 5 + 6 — the grid leaves only 3 + 5 + 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 5 − 3.
• The 2− cage at row 5, column 3 (2 cells) could be 3 − 1, 4 − 2, 5 − 3, 6 − 4, 7 − 5 or 8 − 6 — the grid leaves only 8 − 6.
• The 1− cage at row 2, column 4 (2 cells) allows 7 combinations, from 2 − 1 to 8 − 7 — too many to start from; the cells around it settle it, as 5 − 4.
• The 10+ cage at row 7, column 2 (3 cells) allows 8 combinations, from 1 + 1 + 8 to 3 + 3 + 4 — too many to start from; the cells around it settle it, as 1 + 2 + 7.
• The 15+ cage at row 1, column 3 (3 cells) allows 9 combinations, from 1 + 6 + 8 to 4 + 5 + 6 — too many to start from; the cells around it settle it, as 2 + 6 + 7.
• The 13+ cage at row 3, column 1 (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 + 4 + 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 58. Somewhere in this line the 7 must go — and row 4, 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.
2. Step 5 of 58. Work the 6− cage: with what the board already holds, only 7 can sit in row 6, column 6 and still hit the target. On its own that cage could be 7 − 1 or 8 − 2; the digits already placed around it leave one.
3. Step 44 of 58. Work the 15+ cage: with what the board already holds, only 7 can sit in row 2, column 3 and still hit the target. On its own that cage could be 1 + 6 + 8, 1 + 7 + 7, 2 + 5 + 8, 2 + 6 + 7, 3 + 4 + 8 or 3 + 5 + 7; the digits already placed around it leave one.

How this grid gives way

Of the 58 cells left to fill after the 6 printed ones, 26 by crossing off what a cell’s row and column already hold until one digit was left; 25 by a cage’s arithmetic leaving only one value that fits; and 7 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