Sudoku Made and maintained by Brian Hamilton and Karan Hamilton.
Brian builds the playable puzzle tools, while Karan helps test clarity, difficulty feel, and the player experience.
Playable puzzles and controls are tested in the browser for rule accuracy, usability, and a clear solving experience. See how we make and test puzzles, our editorial standards, or report a problem.
What is Sandwich Sudoku?
Sandwich Sudoku is ordinary sudoku with a number outside the end of each row and column. Fill the grid so that:
- every row, column and 3×3 box holds 1 to 9 exactly once;
- and each number outside the grid is the sum of the digits between the 1 and the 9 of its row or column.
The 1 and the 9 are the bread; only the filling between them is added, and it does not matter which of the two comes first. A row reading 4, 1, 3, 5, 7, 9, 2, 6, 8 has a sum of 3 + 5 + 7 = 15. A 0 means the 1 and the 9 sit side by side; 35 means they sit at the two ends with everything else between them. Two sums can never appear: 1, because the smallest filling is a single 2, and 34, because leaving one digit out of 2 + 3 + … + 8 takes away at least 2.
A Killer Sudoku cage tells you which cells add up. A sandwich sum does not: the cells it covers depend on where the 1 and the 9 go, and that is the unknown. So every sum answers two questions at once — how far apart the 1 and the 9 stand, and which digits fill the gap — and the digits it leaves out have to go outside the bread. A sum of 20 across four cells is as much a statement about the other three cells of the line as about the four.
What does a sum tell you before you place anything?
Start with the gap. A gap of n cells holds n different digits from 2 to 8, so the sum limits how wide the gap can be, and a line has only so many places for a gap of that width. Write the candidates in as pencil marks and check each way of placing the bread against them:
| Gap size | Sums it can make | Sums only this gap can make | Sums with one filling | Places |
|---|---|---|---|---|
| 0 | 0 | 0 | nothing | 16 |
| 1 | 2–8 | 2, 3, 4 | the sum itself | 14 |
| 2 | 5–15 | — | 5, 6, 14, 15 | 12 |
| 3 | 9–21 | — | 9, 10, 20, 21 | 10 |
| 4 | 14–26 | — | 14, 15, 25, 26 | 8 |
| 5 | 20–30 | — | 20, 21, 29, 30 | 6 |
| 6 | 27–33 | 31, 32, 33 | 27, 28, 29, 30, 31, 32, 33 | 4 |
| 7 | 35 | 35 | 35 | 2 |
The gap is the number of cells between the 1 and the 9, filled with different digits from 2 to 8. “Only this gap” means the sum fixes how far apart the 1 and the 9 are; “one filling” means that, at that width, the digits between are fixed too. “Places” is how many ways a gap of that size can sit in a line, counting both orders of the 1 and the 9. Sums from 5 to 30 can be made more than one way, and that is where the reading is.
Read every sum like that before placing a digit. Across 40 boards per level, the first read strikes a median of 70 pencil marks on Easy and 114 on Medium, and settles 24 and 27 cells outright — about half the empty grid on Easy. An empty cell starts with about three candidates on Easy and ends the read with one and a half. On Expert, with eight sums of eighteen, the read still strikes 93 marks and settles nine cells. The commonest sum by far is 0, more than one line in five: the 1 and the 9 sit next to each other more often than any other arrangement.
Do the sums actually do any work?
Every board this page deals is built so that the answer depends on them: the generator removes digits until the puzzle has one answer under the sums it shows, then checks the same givens without the sums and keeps digging until they no longer have one. On the board above:
We checked 160 fresh boards, 40 per level: every one has a single answer with its sums and more than one without them. Without the sums, Easy givens have a median of six answers and up to 98; Medium, Hard and Expert givens have more than 200 on every board we tried. A solver that ignores the sums stalls on every board at every level.
That was not true until this rewrite. The old generator never solved anything: it blanked random cells down to a count and showed a random choice of sums. Measured the same way, half its boards had more than one answer — 6 of 40 on Easy, 13 on Medium, 26 on Hard and 35 of 40 on Expert — and so did 27 of the next 60 daily sandwich puzzles. On those boards a digit that broke no rule could still be marked a mistake, because the page checked every digit against the one answer it happened to have stored.
How do the four levels differ?
We ran 40 fresh boards per level through a solver that knows one technique at a time, then took techniques away to see which boards stopped:
| Level | Given digits | Sums shown | Marks the first read strikes | Cells it settles | Reading and singles finish it | Needs locked candidates | Beyond our solver |
|---|---|---|---|---|---|---|---|
| Easy | 30–34 | 18 of 18 | 70 | 24 | 100% | 0% | 0% |
| Medium | 27 | 18 of 18 | 114 | 27 | 100% | 0% | 0% |
| Hard | 21 | 12 of 18 | 131 | 19 | 95% | 3% | 3% |
| Expert | 17 | 8 of 18 | 93 | 9 | 65% | 18% | 18% |
40 boards per level from this page’s generator, fixed seed; medians, and shares rounded. The solver reads every shown sum to a fixed point, then knows naked and hidden singles, locked candidates, naked pairs and triples, and the X-Wing; “beyond our solver” boards still have exactly one answer. Easy comes in a few givens under 34 when that is what it takes to make the sums load-bearing. Which sums are hidden on Hard and Expert is chosen at random.
Easy and Medium differ in how much you are given; Hard and Expert take sums away. Sum reading plus naked and hidden singles finish every Easy and Medium board, and all but two of the Hard ones. Expert is the real step: with only eight sums, the lines without one have to be solved as ordinary sudoku, and about one board in six needs locked candidates and one in six goes past everything our solver knows. The levels used to be 38, 30, 25 and 22 givens. That was far more than the sums need — with all eighteen shown, boards still had one answer when we dug them down to twelve givens — and the old Easy was an ordinary sudoku with numbers round the edge 15 times in 40.
The generator fills a grid, works out its eighteen sums, picks the ones the level shows, and removes digits one at a time, keeping a removal only while the puzzle still has exactly one answer under those sums; a search it cannot finish answers “not proven” and the digit stays. We re-solved 160 fresh boards and 60 daily ones with a separate counter written for the purpose, and a second of our own where the first ran out of steps: every one had exactly one answer. An Expert board gives 17 digits, the fewest a plain sudoku can have and still have one answer; here eight sums do the rest. A board takes 3 to 4 milliseconds to build, 34 at worst.
Which level should you play?
Start on Medium. Twenty-seven givens and all eighteen sums: the reading does real work on every board — the same givens have more than 200 answers without it — and it never needs more than singles once the sums are read. Easy is the place to learn the reading itself, since the givens settle so much that each sum has only a few arrangements left.
Hard hides six sums and takes six more digits away, and still falls to reading and singles nineteen times in twenty. Expert hides ten sums, and the lines without one are where it gets hard: a board that feels stuck usually has a hidden-sum line crossing a sum-read line, and locked candidates are the move. Every board still has exactly one answer, and none needs guessing.
What goes wrong on a Sandwich Sudoku
- Adding the bread. The 1 and the 9 are never part of the sum. A 1 and a 9 with a 5 between them make 5, not 15.
- Assuming the order. The 1 can come after the 9. Every width of gap can sit in the line both ways round, and the sum does not say which.
- Forgetting the outside. The digits a sum leaves out of the gap must go outside it. A sum of 35 puts every digit from 2 to 8 between the ends; a sum of 5 across two cells is 2 and 3, so 4, 5, 6, 7 and 8 all go outside the bread.
- Reading a hidden sum as 0. On Hard and Expert some lines show no number. A blank is not a zero; it is a line with no sum rule at all.
- Reading a sum once. Every digit placed in the line, and every 1 or 9 placed in a line that crosses it, changes which arrangements survive. Re-read it each time.
Where does Sandwich Sudoku come from?
The rule was in print years before the name. Prasanna Seshadri published it on 19 February 2013 as “Between 1 and 9 Sudoku” for the Daily Sudoku League, in these words: “the clues outside the grid are the sum of the numbers that are between the digits 1 and 9 in that row or column.” It reached a wide audience in March 2019, when the YouTube channel Cracking the Cryptic posted “A sudoku with only 3 givens? How is that even possible?”, a sandwich puzzle whose sums carried almost the whole grid. On 6 May 2019 Alex Bellos wrote it up in the Guardian as “Sandwich sudoku — a new puzzle goes viral”, the Guardian began printing sandwich sudokus that June, and a Sandwich Sudoku app from Cracking the Cryptic and Studio Goya followed in August.
What do the buttons do?
Select a cell, then a digit from 1 to 9; the page opens on Medium. Selecting a cell highlights its row, column and box. Under the grid:
- Notes turns pencil marks on and off. A correct digit clears itself from the notes of its row, its column and its box; a wrong one leaves your notes alone, and the sums are yours to re-read.
- Erase clears the selected cell, and pressing the same digit twice does the same.
- Undo takes back the last change to a cell, though not the notes a correct digit cleared elsewhere, and not the Mistakes count.
- Hint fills one cell with its answer, picked at random rather than the cell you could work out next.
- Solve asks first, then fills the grid and stops the timer.
- New Game deals another board at the level you are on; the level buttons deal one at theirs.
Digits 1–9, the arrow keys, Backspace or Delete and Ctrl+Z all work from a keyboard. The board is saved in this browser after every move, sums and timer included, so it is waiting when you come back. There is a new board every day on the daily page, from the same generator.
Where to go next
Sandwich Sudoku is the sum variant with its sums outside the grid. Arrow Sudoku draws them inside, a circle that is the total of the path leading away from it, and Killer Sudoku puts a sum in every cage and takes the givens away. Skyscraper Sudoku keeps the clues round the edge but counts what you can see from them rather than adding anything up, and Thermo Sudoku drops the arithmetic for an order along a line.