SudokuBoku

Anti-knight sudoku Knight's move 9×9

Nothing here is shaded, caged or outlined — the extra rule lives between squares, so the board draws it for you as you go.

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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.
Keyboard shortcuts
1–9
Place a digit, or toggle a pencil mark
Arrows
Move around the grid
N
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Backspace
Clear the square
Ctrl + Z
Undo · Ctrl + Y redo

Knight-move grids for the printer

A PDF of fresh anti-knight puzzles, built by your own browser. Because these grids print exactly like ordinary ones, every sheet carries the rule across its header with a small five-by-five key beside it — the answer pages too, so a booklet left on a table still says what game it is.

Difficulty

A rule you cannot point at

Anti-knight sudoku is an ordinary 9×9 with one sentence bolted on: two squares a chess knight's move apart may not hold the same digit. A knight moves two squares one way and one square the other, so from most squares there are eight such partners, scattered in a ring that no row, column or box has any opinion about.

That makes it the odd one out among the variants here, and not for the reason you might expect. Every other extra rule on this site can be drawn. Killer has its cages, jigsaw its outlines, hyper its shaded windows, thermo its glass tubes; even the two X-sudoku diagonals, invisible on paper, are at least a shape. A knight's move is a relation between two squares and nothing else. There is no set of nine to shade, no path to trace, no total to print. Look at one of these grids cold and you will see a classic sudoku with too few clues.

The weaker rule was the easier one to build

In a technical sense this is a thinner kind of constraint than the others. Every rule the site had shipped before it said these squares hold 1 to 9 between them — a house, nine at a time. The knight's move says only not those two, over and over. Which turns out to be the older idea: our solver has always kept, for each square, a list of the squares it may not match. Feeding the knight's move in as more entries on that list was the whole implementation. Nothing about pencil marks, hints or grading needed telling.

One hundred and fifty-two refusals

Count the knight pairs on a 9×9 and there are 224 of them. But 72 of those join two squares that already share a box — a knight's move of one row and two columns can easily land inside the same 3×3 — and a rule that repeats something the board already said adds nothing. Strip those out and the variant contributes exactly 152 new refusals, laid over the 810 the classic rules already impose.

They fall very unevenly. On a classic grid every square sees precisely 20 others, which is why a classic sudoku feels the same everywhere. Here the count runs from 20 to 28, and where a square sits decides how hard the extra rule presses on it.

Every knight partner of the centre square. All eight land outside its box, so R5C5 answers to 28 other squares where the same square on a classic grid answers to 20. No square on this board is watched more closely.

The four squares the rule forgets

Now put the same picture in a corner. From R1C1 a knight can reach only two squares on the board — R2C3 and R3C2 — and both of them are inside its own 3×3 box, which had already forbidden them. The corner's peer count stays at 20. The four corners of an anti-knight grid are, as far as the extra rule is concerned, playing classic sudoku.

Both moves a knight can make from the top-left square stay inside the tinted box, which forbade them anyway. The same is true at all four corners — the only squares on the board the variant leaves completely alone.

Between those extremes the board grades smoothly: eight squares next to the corners pick up a single new partner, twenty pick up two, twenty pick up four, twenty-eight pick up six, and one — the centre — picks up all eight. The average square answers to 23.75 others rather than 20. If you want somewhere gentle to start a hard grid, start in a corner; if you want the square where a placement does the most work, it is always R5C5.

The move the knights make possible

Locked candidates on a classic board work like this: corner a digit into two or three squares of a box, and if they all sit on one line, the rest of that line loses the digit. The reasoning never mentions the box or the line specifically — it only needs one of these squares holds the digit and this other square can see all of them. On this board, seeing has a second meaning, and the same argument runs through it.

So: corner a digit into two, three or four squares of any house, then look for squares a knight's move from all of them. They cannot hold that digit either. Our hint button calls this the knight's move interaction and names it whenever it fires, exactly as it names an X-wing.

The tinted squares are the last two homes for some digit in the top-left box. R5C2 is a knight's move from both, along the two paths drawn, so whichever of them takes the digit, R5C2 cannot have it. Mixed cases count too: a square that sees one home down a column and the other by a knight's move is ruled out just the same.

The reason this matters more than a novelty deserves is that the elimination reaches places nothing else does. In the figure, R5C2 shares no row, no column and no box with the square at R3C3 — on a classic grid the two have no relationship whatsoever. The knight's move is the only thing connecting them, and it is enough.

What the measurements said

Difficulty here is decided the way it is decided everywhere on this site: a puzzle is replayed by the same technique ladder a person would use, and its band is named after the hardest step that replay needed. Which meant that once the knight rule was in, we could simply ask what it does to the shape of a solve. The answer was not what we expected.

Over two hundred puzzles dug to twenty-two clues, the single most common hardest step was the knight's move interaction itself — it topped out four solves in ten, and turned up somewhere in half of all of them. Meanwhile the classic middle of the ladder collapsed. Naked pairs appear in 5% of these grids against 15% of comparable classic ones; triples all but vanish. The knight links keep killing candidates before a pair has time to form, and the move that kills them is the variant's own.

The knight's move doesn't extend the technique ladder. It hollows out the middle and stands there instead.

One consequence of that is worth stating plainly, because it decides what the Expert button on this page means. The wings, the fish and the colouring chains — the whole Advanced tier — thin out along with the subsets. Dig a hundred grids to twenty givens and 45% top out at an intermediate move, 22% never get past a hidden single, 18% defeat the standard ladder outright, 4% need its very deepest rungs, and just 11% land in the band between. Expert is the scarce level on this board rather than the deep one. We still ship it, because generation is quick enough to sift sixty candidates for one, but the label is doing a slightly different job here than it does on a classic grid.

Grids that need their knights

The failure mode for any bolted-on rule is that the clues left behind would have pinned the same single answer without it, so a solver who never learns the rule wins anyway. Every puzzle here is checked against that before it is served: strip the knight's move, count the solutions the bare classic grid admits, and if the answer is one, throw the puzzle away. Not downgrade it, not keep it in reserve — discard it.

The check bites at the generous end, as it does on windowed grids. Leave forty givens on the board and 37% of our candidates survive classic rules unaided, which is what ruled out using the classic Easy clue target. At thirty-four givens the failure rate falls to 9%, and from twenty-eight downward it effectively stops being the binding check — 99 digs in every hundred we measured carried their knights. From Hard upward we ask for something stronger still: the graded solve must actually fire the knight's move interaction, so the two bands that promise the variant's signature move measurably contain it.

No safe corner for the rectangle

There is one technique the hint engine refuses to use here at all. Unique rectangle arguments run if this pattern held, the puzzle would have two solutions, and it doesn't — but the second solution they imagine is made by swapping a pair of digits inside a rectangle, and on this board that swap happens under the eyes of every knight neighbour of all four corners. It may not be legal, which means the pattern may not be deadly, which means the deduction has a hole in it. On an X grid we could refuse it near the diagonals and allow it everywhere else. Here there is no safe corner, so it stands down across the whole board.

Playing it here, and the sheet with no ink

Because there is nothing to shade, the board earns its keep the moment you touch a square: its row, column and box light as usual, and the knight partners light with them under a dashed ring. That ring is the entire user interface of this variant. Everybody forgets the rule at least once — usually about two thirds of the way through, when the grid has started to feel like ordinary sudoku again — and a ring you can glance at is worth more than a rule you have to recite. Screen-reader users get the same fact spoken: a square announces itself as a knight's move from the current selection.

The sheet with nothing drawn on it

Paper is the harder problem, and the only place this variant costs something. Every other board on the site arrives on the page carrying its own rules in ink. A printed anti-knight grid is typographically identical to a printed classic one, so a sheet that says nothing is a sheet with no solution. Ours put the rule across the header of every page — puzzles and answers alike — with a small five-by-five key showing one square and the eight it forbids. It is also why this board stays out of the mixed booklet on the printable page: in a pack that rotates through five boards, that line would have to be repeated under every single grid.

The share button hands over the givens and the variant name, and nothing else needs to travel — the eight offsets are the same on every anti-knight board that has ever been set. What does travel with a link is the check: a shared grid is proved to have exactly one solution under the knight rule before this page will play it, on a search with a step budget, so no address anyone invents can leave the tab spinning.