Sums strategy: one-way runs, crossings and the what-if

Play Sums

Sums is about sets of digits. Every clue says which digits a run could be made from; every cell sits in two runs and has to suit both. Most of the puzzle is narrowing those sets until only one digit fits.

How the difficulty works

Each tier is set by the hardest step you'll need:

Easy needs only the sets of digits that make each run, where runs cross, and a digit with one place left in its run.
Medium needs those sets worked through what each cell can still hold.
Hard needs a short what-if: suppose a digit, follow it a few steps, and watch a run go wrong.

Nothing ever needs a guess.

Runs that can only be made one way

Most sums can be made several ways, but the smallest and largest sums for a run's length can't. These are worth knowing by heart, because each one tells you exactly which digits the run holds:

Cells	Sum	Digits
2	3	1 2
2	4	1 3
2	16	7 9
2	17	8 9
3	6	1 2 3
3	7	1 2 4
3	23	6 8 9
3	24	7 8 9
4	10	1 2 3 4
4	11	1 2 3 5
4	29	5 7 8 9
4	30	6 7 8 9
5	15	1 2 3 4 5
5	16	1 2 3 4 6
5	34	4 6 7 8 9
5	35	5 6 7 8 9

Only 8 and 9 make the 17 across in row 2, so 14 candidates go. From Easy puzzle #1 — play it

Runs that can be made a few ways still narrow things down. Two cells adding up to 10 can't hold a 5, since that would need a second 5, and three cells adding up to 8 always include a 1.

Where runs cross

Every empty cell sits in one run across and one run down. Write down what each run's sum allows, and the cell can only hold a digit both lists share.

The across run through row 2, column 5 can only use 1 or 3; the down run, 1, 2 or 4. The one digit in both is 1. From Easy puzzle #13 — play it

A digit with one place left

Sometimes every set that makes a run needs the same digit. If only one cell of the run can still hold it, it goes there.

Every way to make the 16 down in column 2 uses a 7, and only the cell at row 3, column 2 can still hold one. From Easy puzzle #1 — play it

Medium: work the sets through the candidates

Once some cells are narrowed down, look at a run again with them in mind. A set only works if each of its digits has a cell that can take it, so candidates in one cell decide what's left for the others.

In the 13 across in row 4, the cell at row 4, column 2 can only be 8 or 9, so the cell at row 4, column 3 can only be 5 or 4: 2 candidates go. From Medium puzzle #1 — play it

Longer runs work the same way, with more to keep in mind: a set whose digits can't all find a cell is out, and so is every candidate only it used.

Hard: the what-if

When nothing else works, pick a cell with two candidates and suppose it holds one of them. Follow the easier steps from there: the runs through it get smaller, their sums narrow, and their other cells follow. If a run can no longer add up — or would need a digit twice — the digit you supposed is wrong.

A 5 at row 3, column 2 looks possible, but follow it a few steps and a run can't be made. So it's crossed out. From Hard puzzle #1 — play it

Good cells to try sit where short runs cross, because a single digit there settles a lot. If a trial doesn't go wrong within a few steps, undo it — our puzzles never need a long one.

Good habits

Start with the one-way runs. A 3 or a 17 in two cells, a 6 or a 24 in three, tells you its digits outright.
Check both runs. Before you write a digit, make sure it suits the run across and the run down.
Subtract as you go. Once a run has a digit in it, what's left is a smaller run with a smaller sum, and often a one-way one.
Pencil in pairs. Knowing a two-cell run is "1 and 2" is often enough to settle the runs that cross it.