How to Solve Sum Grid Puzzles: Runs, Totals and Digit Sets

To solve a sum grid puzzle, look first for runs whose total can be made from only one set of different digits, place those digits where the crossing runs agree, and let each new digit shrink the choices in the runs that cross it. Every white cell holds a digit from 1 to 9, and no digit may appear twice in the same run, so a handful of totals give themselves away at once.

This guide covers how the grid works, the totals worth memorising, a five-step method, one fully worked grid and a second grid to try yourself.

What is a sum grid puzzle?

A sum grid puzzle looks like a small crossword, but every white cell takes a digit instead of a letter. The white cells form runs: a run is an unbroken line of white cells, read either across or down.

Each run has a clue. The clue sits in the shaded cell just before the run, to its left for an across run or above it for a down run. The clue gives the total that the digits in that run must add up to.

Two rules decide everything:

• Each white cell holds one digit from 1 to 9. Zero is never used.
• Within a single run, no digit repeats.

The same digit may appear more than once in a row of the grid, as long as the copies sit in different runs. A well-made grid has exactly one way to fill it.

How do the clues work?

Every white cell belongs to one across run and one down run. That is what makes the puzzle solvable: each digit must satisfy two totals at the same time.

In the grids below, a shaded cell is drawn as a filled square. A shaded cell that carries a clue says, for example, 5 down, meaning the cells below it add up to five, or 14 across, meaning the cells to its right add up to fourteen.

When you read a clue, think of it as two facts at once: a total and a length. A total of 4 across two cells is a very different clue from a total of 4 across three cells. The second one is impossible, because the three smallest different digits already add up to six.

Which totals give themselves away?

Some totals can be built from only one set of different digits. Call these key totals; it pays to learn them by heart. The order of the digits inside the run is still open, but the set is fixed.

The pattern is easy to remember. The smallest total in a run uses the smallest digits, and the next total up swaps the largest of them for the next digit. The same works from the top end.

Totals in the middle of the range are much looser. A total of 10 in two cells has four possible pairs: 1 and 9, 2 and 8, 3 and 7, or 4 and 6. Leave such runs for later, once crossing digits have narrowed them down.

How is a run total like splitting a number into parts?

Mathematicians call a way of writing a number as a sum of whole numbers a partition of that number. A paper published on 23 August 2024 uses the standard definition: the parts are positive whole numbers, they may repeat, and two lists holding the same parts in a different order count as one partition ^[1].

That paper splits the count by size, noting that the total number of partitions of n equals the number with one part, plus the number with two parts, and so on up to n parts ^[1].

Its new results concern partitions in which no even part may appear twice, though odd parts may ^[1]. For two parts, it shows that the count of such partitions of n is n divided by 2 and rounded down, with 1 taken away when n is a multiple of 4 ^[1]. The paper also gives exact formulas for three and four parts ^[1].

The link to sum grids is our own comparison, not something the paper discusses. Run length plays the role of the number of parts: you never ask for all the ways to make a total, only the ways with a fixed number of cells.

A run in a sum grid is a partition with even tighter rules: the number of parts is the run length, every part must be different, and no part may be larger than nine. Because the order of the parts does not matter, the set of digits is fixed first and their order inside the run is settled later by the crossing clues.

For example, six can be split into three parts in three ways, but only one of them, 3 plus 2 plus 1, has no repeated part, which is why 6 in three cells always means 1, 2 and 3.

How is a sum grid related to a Latin square?

A Latin square is a square grid in which no symbol repeats along any row or column. A paper published on 12 May 2006 also uses a looser kind of array, called a partial column-Latin square, in which a cell may be left empty and the no-repeat rule is checked only down each column, not along the rows ^[2].

The next comparison is our own, not the paper's. A sum grid goes further than that array: it applies the no-repeat idea only inside each run, not along a whole row or column of the grid. It also adds the totals, which a Latin square does not have. So the habit of checking a line for repeats carries over, while the counting work comes from the sums.

What is a reliable step-by-step method?

1. Mark the key runs. Find every run whose total and length allow only one digit set, using the table above. Pencil the set lightly beside the run.
2. Look where two key runs cross. The shared cell must hold a digit that belongs to both sets. Sometimes only one digit fits.
3. Work the short runs next. A two-cell run with one cell known gives the other cell by simple subtraction.
4. Use the no-repeat rule as a test. When a run still has two possible fillings, try each one and see whether it forces a repeated digit in a crossing run. A repeat rules that filling out.
5. Check every total at the end. Add each run across and down. It shows up any slip made while placing a digit.

How does the method solve a full grid?

Here is an original grid with ten white cells, labelled a to j so the steps are easy to follow. Shaded cells are drawn as filled squares, and each clue names its direction.

So the runs are: a and b add to 3; c, d and e add to 14; f, g and h add to 23; i and j add to 4. Down, a and c add to 5; b, d and f add to 19; e, g and i add to 9; h and j add to 11. Have a go before you read the steps.

Step 1: the key runs. Three of the across clues are key totals. The pair a and b, adding to 3, must be 1 and 2. The pair i and j, adding to 4, must be 1 and 3. The trio f, g and h, adding to 23, must be 6, 8 and 9.

Step 2: where key runs cross. The down run h and j adds to 11. Cell h comes from 6, 8 and 9, and cell j comes from 1 and 3. The only pair that reaches 11 is 8 and 3. So h is 8 and j is 3, which leaves 1 for i.

Step 3: subtraction. The down run e, g and i adds to 9, and i is 1, so e and g add to 8. Cell g is now 6 or 9, because 8 is taken by h in the same across run. If g were 9, e would have to be minus 1, which is impossible. So g is 6 and e is 2. That leaves 9 for f.

Step 4: the repeat test. The down run b, d and f adds to 19 and f is 9, so b and d add to 10. The across run c, d and e adds to 14 and e is 2, so c and d add to 12. Cell b is 1 or 2, so try both.

• If b is 1, then d is 9, so c is 3 and a is 2. But then the down run b, d and f reads 1, 9, 9, and the digit 9 repeats. That filling is ruled out.
• If b is 2, then d is 8, c is 4 and a is 1. The down run a and c gives 1 plus 4, which is 5. Everything fits.

Step 5: the full check. Here is the finished grid, with the clues kept in their shaded cells and every white cell filled.

Across: 1 plus 2 is 3, 4 plus 8 plus 2 is 14, 9 plus 6 plus 8 is 23, and 1 plus 3 is 4. Down: 1 plus 4 is 5, 2 plus 8 plus 9 is 19, 2 plus 6 plus 1 is 9, and 8 plus 3 is 11. No run repeats a digit. This grid has exactly one solution; this guide's own computer check tried every possible filling.

Which mistakes cost the most time?

• Forgetting the length. A total means little until you know how many cells share it.
• Allowing a repeat. It is easy to write the same digit twice in a long run, especially when its total seems to need it.
• Starting in the middle. Middle totals have many digit sets. Start at the extremes, where the sets are fixed.
• Ignoring the crossing run. Every cell has two clues. A digit that fits one run but breaks the other is wrong.
• Guessing without a test. When you must try two options, follow each one only until it breaks a rule, and note the trial beside the grid rather than in it.

How do you get faster?

Learn the key totals in the table for two-, three- and four-cell runs. Spotting them on sight turns the opening into a few seconds of work.

Next, get used to the subtraction habit. Whenever a run has one empty cell left, its digit is the clue minus the digits already there. After a good start, this one move can carry you a long way.

Finally, keep the no-repeat test in mind for two-way choices. In the worked example, the decisive step was not a sum at all but the discovery that one option forced a repeated 9.

Can you solve this practice grid?

This grid has seven white cells, labelled p to v.

So p and q add to 4; r, s and t add to 24; u and v add to 6. Down, p and r add to 9; q, s and u add to 12; t and v add to 13. Fill it in before reading the answer.

Answer: the rows read 1, 3 in the first row; 8, 7, 9 in the second row; and 2, 4 in the third row, so p is 1, q is 3, r is 8, s is 7, t is 9, u is 2 and v is 4.

Here is the reasoning. The run r, s and t adding to 24 must use 7, 8 and 9, and p and q adding to 4 must be 1 and 3. The down run p and r adds to 9, and the only pair that works is p as 1 with r as 8, since 3 would need a 6. That makes q equal to 3.

The down run q, s and u then needs s and u to add to 9. Cell s is 7 or 9, and 9 would leave nothing for u, so s is 7 and u is 2.

That leaves 9 for t, then 4 for v, and the last down run checks: 9 plus 4 is 13. There is no other filling.

Where can you practise digit sums next?

Sum grids reward the same skill as any good number puzzle: knowing which totals can only be built one way. Try a riddle about a four-digit number: its digits must add to a fixed total, and a few more clues tie the digits to one another.