9 Puzzles

How to play Sums

Play Sums

Sums is a crossword where the clues are numbers and the answers are digits. Every run of empty cells has to add up to its clue. The arithmetic never goes beyond adding small numbers; the puzzle is in working out which digits can make each sum.

The rules

  • The grid has empty cells and solid blocks. Fill every empty cell with a digit from 1 to 9.
  • A run is a line of empty cells, across or down, from one block to the next or to the edge. Every run has a clue in the block just before it.
  • A block can carry two clues, split by its diagonal: the number above the diagonal is the sum of the run to its right, the number below it the sum of the run underneath.
  • The digits in a run add up to its clue, and no digit appears twice in the same run.

Every puzzle has exactly one solution, and you can always reach it by reasoning, without guessing.

171622171713711317161724231071061116
An easy 7×7 puzzle, before any moves. From Easy puzzle #1 — play it
17162217171371131716172423107106111698797969812865134431371289
Solved. Every run adds up to its clue, and no run uses a digit twice.

Your first moves

Start with the runs that can only be made one way. Two cells that add up to 3 must be 1 and 2; two that add up to 17 must be 8 and 9. You won't know which way round yet, but nothing else can go there.

171622171713711317161724231071061116123456789123456789
Only 8 and 9 add up to 17 in two different digits, so every other candidate in the 17 across in row 2 is crossed out. From Easy puzzle #1 — play it

Then look where runs cross. A cell belongs to one run across and one run down, and its digit has to suit both sums. When the digits the two runs can use share just one, that's the cell's digit.

423167211229457302817415109718931
The 4 across in row 2 can only use 1 or 3; the 7 down in column 5 can only use 1, 2 or 4. The one digit in both is 1, so the cell where they cross, at row 2, column 5, is 1. From Easy puzzle #13 — play it

From there, each digit you place makes its runs smaller and their sums easier to read. The strategy guide covers the rest, from the sums worth knowing by heart to the what-if.

Playing on this site

  • Tap a cell, then a number on the pad. Tap the same number again to clear it.
  • Notes switches the pad to pencil marks. When you place a number, it's removed from the notes in both its runs for you.
  • The two runs of the selected cell are shaded lightly, so you can see what you're adding up.
  • A number repeated in a run is marked straight away, and so is a run whose digits overshoot its clue, or fill it and miss — its clue is struck through. A run that's filled and adds up fades back. Check shows any cell that doesn't match the solution, but only when you ask.
  • Hint names the next deduction and where to look, without giving away the number. One hint per puzzle is free; after that there's a one-minute wait between hints.

Keyboard

  • Arrow keys move; number keys place a number; Shift + number writes a note.
  • N toggles notes, Z undoes, Y redoes, H asks for a hint.

Difficulty and the daily puzzle

  • Easy (7×7, clues included): the sets of digits that make each run, where runs cross, and a digit with one place left in its run.
  • Medium (8×8): those sets worked through what each cell can still hold — one cell is 1 or 2, so its partner in a run of 10 is 9 or 8.
  • Hard (9×9): a short what-if — suppose a digit, follow it a few steps, and watch a run go wrong.

Each day's first puzzle is the same for everyone, and they follow the week from Monday's gentlest to Saturday's toughest, with Sunday the next toughest. After it, Next puzzle gives you as many more as you like.