Play Sumplete Online
Delete numbers until what is left in every row and column adds up to the target at its end. Seven board sizes, three difficulty levels — and every board is solved by deduction before you see it, so each has exactly one answer.
Created by Brian Hamilton
Click a number to cross it out
How to play Sumplete
Every row has a target at its right-hand end and every column has one underneath. Delete numbers until the numbers still standing in each row add up to that row’s target, and the numbers still standing in each column add up to that column’s target.
- A number is either kept or deleted, and the choice counts for its row and its column at once.
- Numbers can repeat. Two 4s in a row are different cells with the same value.
- Every line must hit its target exactly — not more, not less.
- The small number under each target is what the line currently adds up to. It turns to a tick when it matches.
- Each board has exactly one answer, and it can be reached by deduction alone.
Controls: click or tap a number once to cross it out, again to ring it as kept, and a third time to clear it. Right-click (or long-press) rings a number straight away. Arrow keys move between cells; X crosses out and K rings. Check marks any crossed or ringed cell that disagrees with the answer.
Every line has two targets, and the puzzle only prints one
The number at the end of a row is what must be left when you have finished. Add up the whole row and take the target away, and you have the second target: what must go. The two are equally binding, and at the start of any board the unprinted one is doing most of the work.
Take the second row of the puzzle worked below: 7, 4, 1, 1, 6, with a target of 6. The row adds up to 19, so exactly 6 must stay and exactly 13 must be deleted. The 7 cannot stay, because it is bigger than 6 by itself. That is the whole of the outsize rule, and it runs in both directions: a number bigger than what its line still needs to keep must go, and a number bigger than what its line still needs to delete must stay. The moment a line’s remaining delete budget drops below one of its numbers, that number is safe.
The rule bites harder than it looks, because every decision is made in two lines at once. Deleting the 7 spends 7 of its row’s delete budget and 7 of its column’s. Whichever budget that leaves smaller decides the next number, and the next number does the same for two more lines.
| Board | Cells decided on sight | Boards with no outsize move at all |
|---|---|---|
| 4×4 | 52% | 2 of 200 |
| 5×5 | 36% | 1 of 200 |
| 6×6 | 27% | 4 of 200 |
| 7×7 | 18% | 11 of 200 |
| 9×9 | 8% | 27 of 200 |
So on a small board the outsize rule is not an opening, it is most of the solve. On a 9×9 it is a way in and little more, and you will need the other two ideas below.
A worked opening
Here is a 5×5 Medium board, exactly as the generator produced it. Eleven of its twenty-five cells can be decided without listing a single subset.
- Row 2 (7, 4, 1, 1, 6 → 6) is the one above. The 7 goes.
- Column 1 (7, 7, 7, 8, 6 → 21) adds up to 35, so 14 must go. The 7 from row 2 has just gone, leaving 7 to delete — and the 8 is bigger than 7, so it stays.
- Column 4 (7, 1, 1, 9, 2 → 8): the 9 goes on sight. The column adds up to 20, so 12 must go, and with the 9 gone only 3 remains to delete. The 7 is bigger than 3, so it stays; that uses 7 of the 8, and the 2 is bigger than the 1 left to keep, so it goes too.
- Row 1 (7, 5, 1, 7, 3 → 11): the 7 in column 4 is staying, so the rest of the row must keep 4. The 7 and the 5 both go. They add up to 12, which is the row’s whole delete budget, so the 1 and the 3 stay.
- Column 1 again: its delete budget was 14, and the two 7s that have gone spend all of it. The remaining 7 and the 6 stay.
Eleven cells, no arithmetic beyond adding up a line, and every step forced. Then it stops. Look at any line on the board now and nothing is bigger than either of its budgets. On a 5×5 Medium board this is the normal shape of things: the outsize rule decides 7.6 cells on average and then runs dry.
When the outsize rule stops, list the subsets
Row 3 reads 7, 6, 9, 1, 9 with a target of 17. The 7 is already staying, so the other four must supply exactly 10 between them. Write out every way to make 10 from 6, 9, 1 and 9 and there is only one: a 9 and the 1. So the 6 goes, the 1 stays — and the row has nothing to say about which 9 it is. Two equal numbers in a line are interchangeable as far as that line is concerned; only the columns can tell them apart.
That is the complete second technique. When no number in a line is outsize, list the subsets of its undecided cells that make what it still needs. A cell in every subset stays; a cell in no subset goes; a cell in some but not all is still open. It is slower than the outsize rule, which is why you save it for when the rule has run out.
Here one subset step is enough. Row 3 is not the only line offering one at this point — row 4 and columns 2 and 3 do too — but it is the one that unlocks the rest. With the 6 gone and the 1 kept, the outsize rule takes over again and decides the remaining twelve cells: column 4 is now complete, so its last 1 goes; row 2 then has 5 left to delete and the 6 is bigger than that, so it stays; column 5 now needs only 3 more, which puts both its 9s out; and so on to the finish.
How typical is that? Across 160 generated 5×5 Medium boards, 101 needed exactly one subset step, 43 needed two, and the rest three to six. None needed anything more, which is what Medium means on this page: the outsize rule will not finish the board, but one line at a time will.
Repeated numbers are what make it hard
The intuition is that bigger numbers make a harder puzzle. Measured, it is the reverse. What defeats line-at-a-time reasoning is a line with several subsets that make the same total, and that happens when numbers repeat, not when they are large. A line of 8, 3, 8, 3 can make 11 two ways; a line of 2, 5, 13, 17 can make almost every total only one way.
| Board | Numbers | Boards with one answer | Outsize alone | Line at a time | Needs a what-if |
|---|---|---|---|---|---|
| 3×3 | 1–9 | 100% | 100% | 0% | 0% |
| 5×5 | 1–9 | 98% | 85% | 15% | 0% |
| 5×5 | 1–3 | 34% | 62% | 28% | 10% |
| 6×6 | 1–9 | 94% | 55% | 45% | 0% |
| 6×6 | 1–4 | 35% | 30% | 62% | 9% |
| 7×7 | 1–9 | 91% | 17% | 78% | 5% |
| 7×7 | 1–5 | 32% | 16% | 46% | 38% |
| 8×8 | 1–9 | 74% | 2% | 60% | 38% |
| 8×8 | 1–19 | 99% | 8% | 92% | 0% |
| 9×9 | 1–9 | 35% | 0% | 21% | 67% |
| 9×9 | 1–19 | 98% | 0% | 85% | 15% |
Read the shaded rows against their neighbours. Narrow the numbers on a 6×6 from 1–9 to 1–4 and boards that need a what-if go from none in 300 to about one in eleven. Widen an 8×8 from 1–9 to 1–19 and they vanish: 38% to 0%. The price of narrow numbers is paid elsewhere — only a third of random 6×6 boards of 1–4 have a single answer at all, against 94% with digits — and that is why the generator here checks every board rather than trusting the odds. One row does not add up to 100%: on a 9×9 of digits, the missing 12% needed a what-if inside a what-if, and those are the boards the generator throws away.
The difficulty buttons are set from this table, and they are defined by what it takes to finish the board rather than by how the board looks. Easy boards can be finished by the outsize rule alone, from the first cell to the last. Medium boards cannot, but one line at a time will finish them. Hard boards need at least one what-if. Each size uses the range of numbers that makes its level reachable:
| Board | Easy | Medium | Hard |
|---|---|---|---|
| 3×3 | 1–9 | 1–4 | — |
| 4×4 | 1–9 | 1–5 | — |
| 5×5 | 1–9 | 1–9 | 1–4 |
| 6×6 | 1–9 | 1–9 | 1–4 |
| 7×7 | 1–9 | 1–9 | 1–5 |
| 8×8 | 1–19 | 1–19 | 1–9 |
| 9×9 | 1–29 | 1–19 | 1–9 |
Hard is not offered on 3×3 or 4×4 because no board that small was ever seen to need a what-if: not one of 1,737 random 3×3 boards with a single answer, across six ranges of numbers, and not one of 1,628 4×4s. Easy on a 9×9 needs numbers up to 29 before the outsize rule alone can finish a board at all, which is why the biggest Easy boards look nothing like the small ones.
A what-if, worked
A what-if is not a guess. You suppose one cell is deleted, follow the forced consequences, and if they end in a line that cannot be completed, the supposition was false and the cell stays. Because every board here has exactly one answer, the opposite of a contradiction is a fact. Here is a 5×5 Hard board after the outsize rule and the subset rule have both run dry, with nine cells still open.
Suppose the 1 in row 3 is deleted. Row 3 (2, 1, 2, 3, 2 → 3) has both its 2s in columns 1 and 3 already gone, so with the 1 gone as well only 2 remains in its delete budget, and the 3 is bigger than that: the 3 stays. Column 2 (1, 4, 1, 1, 4 → 10) has both 4s kept and now has nothing left to delete, so its other two 1s stay. Column 4 (2, 2, 3, 1, 4 → 5) now has the 2 and the 3 it needs, so its remaining 2 and 1 go. That leaves row 1 (4, 1, 4, 2, 3 → 3): the 1 is kept, so 2 is left to keep, and 1 is left to delete. The 3 is bigger than both. It can neither stay nor go, so the supposition fails and the 1 in row 3 is kept. From there, one line at a time finishes the board.
Five forced moves to a contradiction is short. On 5×5 Hard boards the two line rules leave 17.7 cells open on average before the first what-if is needed, and on 6×6 Hard it is 24.2, so expect to reach for this earlier than you would like. No board here needs a what-if inside a what-if: the generator throws those away.
Where solvers get stuck
Deleting past the target. The small number under each target is what the line currently adds up to. It only ever goes down, so the moment it drops below the target the line is lost and nothing you delete elsewhere will bring it back. The target turns red with a ! when that happens; the only way out is Undo.
Treating a line that adds up as finished. When a row’s remaining numbers hit the target, every one of them is now a keep — but the board does not know that until you ring them. Ten moves later a column argument will tempt you to delete one of them, and the row will quietly break. Ring the survivors the moment a line completes.
Asking a line to tell equal numbers apart. It cannot. If a row has two 9s and needs one of them, no amount of staring at the row will say which. Turn ninety degrees and ask the columns, which see each 9 next to different neighbours.
Trusting the tidy subset. When three subsets make the target, a cell that is in two of them is not decided, however natural the pairing looks. Only cells in all of the subsets or none of them are settled. Everything else waits.
Three things you can check about these boards
No line is a giveaway. A target that equals the line’s whole total (keep everything) or zero (delete everything) never appears; the generator rejects any board with one. Check the smallest target on any board here against the line it belongs to.
On Easy, there is always a number to strike out on sight. That is the definition, not a tendency: an Easy board is one the outsize rule finishes on its own, so at every stage of the solve, right to the last cell, some line has a number bigger than one of its two budgets. If you ever find yourself listing subsets on Easy, look again.
About half the numbers go. Every board keeps between 38% and 62% of its numbers; a random keep-set outside that band is thrown away before its targets are even worked out. On a 5×5 that means 10 to 15 of the 25 survive.
The step that matters most is invisible. Every board is handed to the same three rules the article describes, run to completion, before it is shown. A board that a sound deduction can finish has exactly one answer, because a second answer would leave some cell that no sound step could ever decide. A separate brute-force count, run offline on 2,280 generated boards across every size and difficulty, has agreed with that every time.
Sumplete, Rullo and Summer
Sumplete went live at sumplete.com on 3 March 2023. Its creator, Daniel Tait, built it by prompting ChatGPT, which proposed the rules, wrote the first playable version and supplied the name; the story of an AI “inventing” a puzzle is what made it spread. The rules were not new. Rullo, an app on the App Store since 2019, and Summer, on Google Play since 2020, play identically: a grid of numbers, a target per row and column, delete until the sums match.
It is sometimes filed next to Kakuro, and the comparison is worth making precisely. In Kakuro the cells start empty, you write digits 1–9 into runs, and a digit may not repeat within a run. In Sumplete every cell is already filled, you only ever remove, and repeats are not just allowed but are the main source of difficulty. The closer relative is Killer Sudoku’s cage arithmetic, where the question “which subsets make this total?” is asked the same way — without the no-repeat rule that makes Killer’s answer so much shorter.
The same board works with letters. In Word Sieve every row and column hides a word instead of a sum, the clue is the word’s length, and what you delete is the filler around it. A sum does not care what order its numbers come in; a word does, which changes which deductions are cheap.
More number puzzles
If adding up subsets is the part you enjoy, these ask the same question in different grids: