SudokuBoku

8×8 sudoku Sixty-four squares · six levels

Boxes four wide and two tall, the digits 1 to 8, and the whole technique ladder: the smallest grid that keeps every level of the full one.

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Puzzle —
Rules
  • Fill every row, every column and every 3×3 box with the digits 1 to 9.
  • A digit can appear once in each of those — never twice.
  • There is exactly one solution, and logic alone will reach it.
  • Rows and columns each take the digits 1 to 9, once apiece.
  • The boxes are irregular shapes, drawn with the heavy outlines.
  • However a region bends, no digit repeats inside it.
  • The colour patches are cages; the small number is what a cage's digits add up to.
  • No digit repeats within a cage — nor in any row, column or 3×3 box.
  • There are no given digits: the sums alone pin down the whole grid.
  • Everything from classic applies: rows, columns and 3×3 boxes take 1 to 9.
  • The two tinted diagonals must each hold 1 to 9 exactly once as well.
  • All the classic constraints hold — rows, columns and 3×3 boxes take 1 to 9.
  • Each of the four shaded windows must also contain 1 to 9 exactly once.
  • The windows sit a step in from the corners, overlapping four boxes apiece.
  • Rows, columns and 3×3 boxes work exactly as they always have.
  • On top of that, two squares a chess knight's move apart may not hold the same digit.
  • Select a square and its knight partners are ringed with a dashed outline.
  • Every row, column and 3×3 box holds 1 to 9, exactly as in classic.
  • Along each grey path, digits strictly climb from the round bulb to the far end.
  • They don't have to be consecutive — each square just beats the one before it.
  • Rows, columns and 3×3 boxes take 1 to 9, exactly as in classic.
  • The digits along each arrow add up to the digit in the circle it comes from.
  • Digits may repeat along an arrow — as long as no row, column or box objects.
  • Fill every row, column and 3×3 box with the digits 1 to 9, as usual.
  • Between squares in the same box you'll find a sign; its point aims at the smaller digit.
  • Every comparison inside every box is shown — a hundred and eight of them.
  • The higher levels hand you fewer digits, not fewer signs. Expert gives you four.
  • Fill every row, column and 3×3 box with the digits 1 to 9, as usual.
  • A white dot between two squares means their digits differ by one.
  • A black dot means one digit is exactly double the other. 1 and 2 are both, and always show black.
  • A boundary with no dot is a clue too — those two digits are neither one apart nor double and half. Every boundary is marked or deliberately bare.
  • Expert hands you no digits at all. The dots and the gaps are the whole puzzle.
  • Fill every row, column and 3×3 box with the digits 1 to 9, as usual.
  • A square with a ring behind it holds an odd digit — 1, 3, 5, 7 or 9.
  • A square with a frame behind it holds an even one — 2, 4, 6 or 8.
  • Most squares are marked with neither, and those are genuinely unrestricted.
  • Every square switches to the fully marked board — one level, gentler than it looks, and two puzzles sharing a grid rather than one.
  • Every row, column and box holds five odd digits and four even ones — so once five squares of a house are known odd, the rest of it is even, marked or not.
  • Solve the 9×9 sudoku as usual — every row, column and box takes 1 to 9.
  • Ten ships are hidden on the same squares: one of four, two of three, three of two and four single squares.
  • Ships are straight, and no two of them touch — not even at a corner.
  • The numbers beside and below say how many ship squares each row and column holds.
  • A ship listed with digits sits on exactly those digits, in order — so the grid is your map.
  • Five 9×9 grids interlock: four corners and a centre, each corner sharing one 3×3 box with it.
  • Every grid obeys ordinary sudoku on its own — rows, columns and boxes each take 1 to 9 once.
  • A tinted box belongs to two grids at once, and only together do the five have one solution.
  • Fill every row, every column and every box with the digits 1 to 6.
  • The boxes are three squares wide and two tall.
  • Nothing repeats in any row, column or box.
  • Put 1, 2, 3 and 4 into each row, each column and each 2×2 box.
  • No digit may appear twice in any of them.
  • Every square can be worked out — no guessing needed.
  • Each row, each column and each box takes the digits 1 to 8 once.
  • A box is four squares wide and two tall: two boxes across a row, four down a column.
  • No digit repeats inside any of the three.
Keyboard shortcuts
1–9
Place a digit, or toggle a pencil mark
Arrows
Move around the grid
N
Notes mode on or off
Backspace
Clear the square
Ctrl + Z
Undo · Ctrl + Y redo

Eight by eight, on paper

An 8×8 grid prints comfortably at every density from one to a sheet to six, and all six levels are on the menu. Each grid is built here in your browser and proved to have one answer before it is drawn.

Difficulty

Sixty-four squares, and a box that is half a row

An 8×8 sudoku keeps the contract and changes the furniture. Each row, each column and each box holds the digits 1 to 8 once, and there is exactly one way to finish. The furniture is the box: eight does not split into equal squares any more than six does, so the boxes here are four squares wide and two tall, two of them side by side across a band and four stacked down a stack. A square therefore has seven row-mates, seven column-mates and three box-mates it shares neither with — 17 peers out of 63, or 27%. That sits between the crowded 6×6, where a square watches 12 of 35 (34%), and the full grid's 20 of 80 (25%). By that measure this board is nearer the classic one than its missing row suggests.

The tinted box owns half of its top row and a quarter of its first column; the amber squares are the rest of each. What the box learns about a digit along a row it can hand to one other box. Along a column, to three.

Everything the shape does to the solve follows from that figure. A box here owns half of every row it crosses and a quarter of every column, so a deduction that runs along a row has one neighbour to land on and a deduction that runs down a column has three. The two directions are different kinds of move on this grid, and the next section measures how different.

Two by four, or four by two?

Most pages describe these boxes as 2×4, rows first; we draw them four wide and two tall and say so width first, as the 6×6's 3×2 boxes are described. Turn the page through a right angle and the two are the same puzzle. The only thing that flips is which direction the numbers below call a row.

The move a wide box makes twice as often, and the one it cannot make at all

Locked candidates are the first move above the singles, and the one this box shape bends most. A pointing pair is a digit confined to one line inside a box, which strikes that digit from the rest of the line outside the box. Along a row the confinement is easy to come by, because a box has only two rows: a digit whose places in the box happen to avoid one of them is confined to the other, and the strike lands on the four squares of the band's other box. Along a column it needs the digit's places to share one of the box's four columns, which is rarer, and the strike reaches six squares down the stack. Over 5,000 puzzles dug to the floor, the ladder found 1,706 pointing pairs that ran along a row and 856 that ran along a column — 2 to one, which is the box's own proportions.

24 digits into a Hard puzzle. In the top-right box the 8 has only the three tinted squares left, all in row 1, so 8 cannot sit in the amber squares — the rest of that row, over in the other box. Nothing cheaper is available here: at this position the pointing pair is the first move the ladder offers.

The other locked-candidate move runs the opposite way: a digit confined to one box within a line, struck from the rest of that box. Down a column it is a real deduction on this grid, and it fired 642 times in the same run. Along a row it fired 0 times in 5,000 puzzles, and that is not chance. A row crosses two boxes. If a digit's places in a row all lie in the left box, the right box has no place for that digit in this row, so its places there all lie in its other row — which is a pointing pair in the right box, striking exactly the squares the row move would have struck, and the ladder reaches pointing pairs first. When the right box has a single square left for the digit, a hidden single gets there before either. Two boxes to a band leaves nothing for row-wise claiming to do. The same is true on the 6×6, and the move only returns on the 9×9 because a band there holds three boxes.

With two boxes to a band, whatever a row could claim, the neighbouring box has already pointed at.

Where the ladder comes back

The 6×6 page makes the case that a small grid steps over the middle of the technique ladder: six-square houses run out of room before a pair or a locked candidate can be the hardest thing in a puzzle. Eight is where that stops being true, and the measurement is the same one, run the same way. We dug 5,000 puzzles to the floor and replayed each against the ladder that grades every board on this site. 1,580 of them, 32%, needed something beyond the two singles. A locked candidate fired in 26% of puzzles, a naked pair in 11%, a skyscraper in 8% and an almost locked set in 8%. 17 of the 18 rungs a classic grid can use fired at least once. The one that never did was the jellyfish, which the technique trainer declines to offer on the full grid for the same reason. And 216 puzzles, 4.3%, defeated the ladder outright.

Which rung is the hardest step matters more than which rungs appear, because the hardest step is what a level is. At the floor the commonest hardest step above the singles is the pointing pair (5.7%), then the almost locked set (5.6%), the skyscraper (4.9%), the XY-wing (3.7%) and the W-wing (2.7%). Every band the classic board offers is a band this grid produces on its own, in numbers large enough to serve, which is why the selector above has six chips where the 6×6's has three. The difference is not that eight is harder than six. It is that eight is big enough for the middle of the ladder to get a turn.

Nineteen clues, whatever you ask for

The digger removes clues from a finished grid one at a time and keeps any clue whose removal would let a second solution in. On this size it stops at about nineteen. Ask it for eighteen and it leaves 19.4 on average; ask for twelve and it leaves 19.4. The fewest it has ever left is 16, 3 times in 5,000, and the commonest count is 19. The table gives what each clue target buys, and the last three rows are the point: below twenty, the number you ask for stops being a number you get.

Asked for Left Easy Medium Hard and above
32 32 99% 1% 0%
28 28 89% 10% 1%
24 24 54% 39% 7%
22 22 28% 57% 15%
20 20.1 12% 62% 26%
18 19.4 7% 59% 34%
12 19.4 9% 60% 31%

Two thousand digs per row, graded by the hardest technique each needed. "Left" is the average number of clues the digger actually kept.

That floor decides how the six levels are built. Easy digs to 30 clues and Medium to 22, where a naked single and a hidden single are respectively the likeliest hardest step. Everything from Hard upward digs to the floor and is sorted by the grader alone: the same nineteen-or-so clues are a Hard puzzle when a subset or a locked candidate is the hardest thing in them, Expert when a wing or a fish is, Evil when an almost locked set or a swordfish is, and Extreme when the ladder cannot finish. From Hard up, the clue count is no part of the difficulty on this board. That is unusual enough to say plainly, and it is the honest consequence of a digger that will not go lower.

For scale, the completed grids of this size have been counted, though not by us — the space is far past anything our solver can walk. There are 29,136,487,207,403,520 of them, of which 1,673,187 are essentially different once relabelling, reflection and the swapping of rows, columns, bands and stacks are set aside; the count is Ed Russell's, checked by Kjell Fredrik Pettersen. We counted the 6×6's 28,200,960 ourselves, and the 9×9's 6.67 sextillion is Felgenhauer and Jarvis's. Eight sits nine orders of magnitude above six and five below nine.

Six levels, and why the top two are generated rather than stored

On the classic board, Easy to Expert are dug on demand while Evil and Extreme come from curated banks, because a 9×9 dig costs tens of milliseconds and an Evil puzzle is a rare outcome of one: waiting for it live would mean a spinner. Here a dig costs 1.1 milliseconds and a full grade 0.6, so a rare outcome is cheap to wait for. Evil turns up in 5.6% of floor digs and Extreme in 4.3%; the board tries up to 120 digs for the one and 170 for the other, which leaves each with a miss rate under one in a thousand and a worst case of about a quarter of a second. It is the only board on the site whose Evil and Extreme puzzles are made in front of you, and from Hard upward the line above the grid names the technique that particular puzzle needs, from the grader itself, rather than repeating the band's name.

On an Extreme puzzle the Hint button will eventually have nothing to offer, exactly as it does on the 9×9 Extreme page, which says what to do at that point better than this one can. The moves behind the other bands are taught on the level pages, and all of it transfers to this grid unchanged, with the one exception the second section above is about:

  • Medium — the hidden single, and the digit-first scan that finds it.
  • Hard — pencil marks, subsets and locked candidates.
  • Expert — X-wings, skyscrapers, the wings and colouring.
  • Evil — swordfish and almost locked sets.

Playing it here, and on paper

The number pad is two rows of four, the shape of the box, and the pencil marks inside a square take the same two-by-four, so a mark sits where its digit's box would put it. Keys 1 to 8 place a digit and 9 does nothing. Your game is saved under its own key and survives a visit to any other board, and a share link carries the sixty-four squares themselves: the receiving page re-proves that the puzzle has one answer, under this board's geometry, before it shows anything. The pack builder above prints the same grids as vector artwork at every density from one to six a sheet, with the heavy rules following the four-by-two boxes and the answer key at the back, and all six levels on its menu.