Play Battenburg Sudoku Online

What is Battenburg Sudoku?

Battenburg Sudoku is ordinary sudoku with small markers on some of the corners where four squares meet. Fill the grid so that:

• every row, column and 3×3 box holds 1 to 9 exactly once;
• and where a marker sits, the four squares around it form an odd/even checkerboard — the two on one diagonal odd and the two on the other even — while where no marker sits, they do not.

The marker is drawn as a tiny pink-and-yellow checkerboard, like a slice of Battenberg cake. It says nothing about which diagonal is odd; either way round counts. Every board shows all of them: if four squares form a checkerboard in the answer, their corner is marked.

What does a marker tell you before you place anything?

Think in parity before digits. Around a marker, one known parity fixes all four squares, because a checkerboard has only two ways to fall. Write that into your pencil marks straight away:

A blank corner works the other way round and needs more to go on: once three of its four squares are known, the fourth must spoil the checkerboard if the three would otherwise make one.

Read every corner like that, then count. Every row, column and box holds five odd digits and four even, so once five squares of a house are known to be odd, the rest are even, and once four are known even, the rest are odd. Across 40 boards per level, reading the markers and counting parities strikes a median of 42 pencil marks on Easy and 77 on Expert before any digit is placed, fixes the parity of about 30 to 40 squares that the rows, columns and boxes left open, and settles 11 squares outright on Easy, 4 on Expert.

Do the markers 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 with the markers, then checks the same givens as ordinary sudoku 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 markers and more than one without them. As ordinary sudoku, Easy givens have a median of three answers and up to 31, Medium a median of 12, and Hard and Expert more than 200 on every board we tried. A solver that ignores the markers stalls on every board at every level.

This page’s generator already got that right before this rewrite; its check was complete, and 160 of 160 old boards passed the same test. What it got wrong was speed. It only tested a marker once all four of its squares were filled, so its search wandered: an Expert board took 73 milliseconds at the median and up to a third of a second on a desktop, and in one smaller run over a second, all of it while the page loaded. The rewrite reads the markers as it goes and deals every level in about 2 milliseconds, 10 at worst.

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. Reading the markers and counting parities is the variant’s own move, and the solver does it for free before each step.

40 boards per level from this page’s generator, fixed seed; medians, and shares rounded. Markers vary a lot from board to board, from 3 to 21 of the 64 corners. The solver knows naked and hidden singles, locked candidates, naked pairs and triples, and the X-Wing; one Expert board used locked candidates, which is why that row does not reach 100. “Beyond our solver” boards still have exactly one answer. Easy comes in a few givens under 38 when that is what it takes to make the markers load-bearing.

This is a variant where the levels really do step up. Naked and hidden singles, once the markers are read, finish 39 of 40 Easy boards, 37 of Medium and 34 of Hard. Expert is a big step: half its boards fall to singles, and the other half go beyond everything our solver knows, even with every marker read and every house counted. They all have exactly one answer, and none needs guessing, but an Expert board here is a real puzzle. This page used to say a board shows 15 to 30 markers; measured, it is 3 to 21, with a median of 10 to 13.

Which level should you play?

Start on Easy even if ordinary sudoku is easy for you. The first thing to learn is the blank corners, and Easy gives you enough digits to see three squares of a corner at once. Medium and Hard are the same skill with fewer givens; nearly every board still falls to singles once you read the markers.

Expert is the step: half of its boards go beyond singles. If an Expert board stalls, stop looking at digits: go round the corners and mark every square whose parity you can prove, then count odd and even squares house by house.

What goes wrong on a Battenburg Sudoku

• Ignoring the blank corners. About fifty of the 64 corners have no marker, and every one of them forbids a checkerboard. Most of the board’s information is there.
• Assuming a marker says which diagonal is odd. It does not. It says the two diagonals differ; one known square decides which way round.
• Reading a marker from the wrong four squares. A marker belongs to the four squares that share its corner, two above it and two below. It says nothing about the squares beside those.
• Counting parities wrongly. Five odd and four even per house, never the other way: there are five odd digits, 1, 3, 5, 7 and 9, and four even ones.
• Waiting for digits. Most of this puzzle is solved in parities first. A square known to be odd has only five candidates, and a house with its parities settled splits into two small puzzles.

Where does Battenburg Sudoku come from?

Bram de Laat invented it. He published the first one on his blog, Para’s Puzzle Site, on 9 February 2013, as his entry in that week’s Daily Sudoku League, calling it “a Sudoku type I came up with”. The name, he wrote, “comes from the Battenburg markings” — the checked pattern on British police cars — “although it stays more in my mind for Battenberg cake, which is a checkerboard pattern”. His rule is the one this page uses: wherever two odd and two even digits form a checkerboard, a mark is given.

What do the buttons do?

Select a square, then a digit from 1 to 9; the page opens on Medium. Selecting a square highlights its row, column and box, and a wrong digit is marked the moment it goes in, including one that breaks a marker. 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.
• Erase clears the selected square, and pressing the same digit twice does the same.
• Undo takes back the last change to a square, though not the notes a correct digit cleared elsewhere, and not the Mistakes count.
• Hint fills one square with its answer, picked at random rather than the square 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, so it is waiting when you come back.

Where to go next

Battenburg Sudoku gives you parity a corner at a time. Odd-Even Sudoku gives it all at once, every square shaded odd or even, and is the gentler way into parity reasoning. Kropki Sudoku also marks the lines between squares, with dots that say two neighbours are one apart or one is double the other, and Killer Sudoku swaps parity for sums.