SudokuBoku

Kropki sudoku Dots and gaps 9×9

A dot between two squares tells you how their digits relate. So does a boundary with no dot on it — and there are far more of those.

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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
Notes mode on or off
Backspace
Clear the square
Ctrl + Z
Undo · Ctrl + Y redo

Dotted grids for the printer

A PDF of fresh kropki puzzles. Both kinds of dot print in full black — filled against hollow is the entire notation, and a grey ring at this size is a smudge — with each one knocking a small white gap out of the rule it sits on. One or two to a page: below that the dots stop separating.

Difficulty

Two dots, and a hundred boundaries without one

A kropki grid is an ordinary 9×9 with marks on the rules between neighbouring squares. A white dot means the two digits differ by one. A black dot means one is exactly double the other. Rows, columns and boxes behave as they always have.

Stop there and you have a mildly decorated sudoku. The rule that makes it a puzzle is the third one, and it is about the marks that aren't there: a boundary with no dot on it means neither relationship holds. Those two digits are not one apart, and neither is double the other. Every one of the 144 boundaries on the grid has been considered and answered — most of them answered "no".

A bare rule is not the absence of a clue. It is a clue that happens to be drawn as nothing.

That inverts the usual arithmetic of a variant. Cages, thermometers and circles all add ink you can point at, and a square nowhere near any of it is simply unconstrained. Here the marked boundaries are the minority by better than two to one, and a player who reads only those is working with the smaller half of the grid.

Why 1 and 2 always show black

There is one pair the two rules disagree about. 1 and 2 differ by one, so they want a white dot; 2 is double 1, so they want a black one. Something has to break the tie, and this board always draws black — the common published convention, and the only choice that leaves the notation carrying its full weight.

Follow it through. If black always wins that pair, then a white dot can never be sitting on 1 and 2, so a white dot rules the digit 1 out of both of its squares. Had the tie gone the other way, white would have meant "one apart, and possibly the 1/2 pair", which is a weaker statement and one you could not act on. A tie-break that looks like housekeeping turns out to buy a real deduction on every white dot on the board.

Digit Across a black dot Across a white dot
12—
21, 43
362, 4
42, 83, 5
5—4, 6
635, 7
7—6, 8
847, 9
9—8

Anything not in this table sits across a bare boundary. Read the dashes and two facts fall out on their own: 5, 7 and 9 can never touch a black dot, and 1 can never touch a white one.

What a dot rules out before you look at anything

Those dashes are worth more than they look. A square with a black dot on any side of it has already lost 5, 7 and 9 — three of its nine candidates gone, from the shape of the board alone, before a single digit is read. A square touching a white dot has lost the 1.

Push it one step and the argument has to get careful. A square with black dots on both sides of it — left and right, or above and below — needs two different black partners, and only 2 and 4 have two. So it is a 2 or a 4, and nothing else. But black dots above and to the left do not force that, because those two neighbours share neither a row nor a column and are perfectly free to hold the same digit. The count has to be taken one axis at a time.

The top-left box of a board this page generated, with every dot its solution implies. Two black, three white — and seven boundaries deliberately bare. Check one of the bare ones: 2 and 8 sit side by side in the middle row with nothing between them, and that is right, because 8 is four times 2 and the rule says exactly double.

The chains that double are four long at most

Black dots chain, and the chains are short. Doubling from 1 you can reach 2, then 4, then 8, and then you are off the grid; from 3 you reach 6 and stop. That is the whole of it — two chains, one of four digits and one of two.

So a run of black dots is enormously informative. Three squares in a row joined by two black dots can only be a stretch of 1-2-4, 2-4-8 or their reverses. Four squares joined by three black dots have exactly one reading in each direction. And there is no run of five: six different digits doubling one into the next do not exist, so a puzzle that appears to show one is not a hard puzzle, it is a broken one.

The longest doubling run there is, taken from a generated grid: four squares in row 6 reading 1-2-4-8. Its three black dots admit exactly two fillings — this one and 8-4-2-1 — so a single digit anywhere along it settles the other three at once.

A grid with nothing written on it

Here is the consequence of all this, and it is the reason the variant exists. Fill a sudoku, mark every boundary honestly, then rub out all eighty-one digits. What is left is, almost always, still a puzzle with exactly one answer.

Measured over sixty generated grids, the complete dot layout recovers the grid it came from fifty-seven times. The three that fail have a second grid consistent with the same marks, and generation throws those away rather than shipping them. So the Expert level here does something no other board on this site can: it hands you an empty grid and the dots, and that is the entire puzzle.

The levels are graded the way everything here is graded — the puzzle is replayed with human techniques and named after the hardest step the replay needed — and the ladder for this board separates two moves that look identical on paper. Reading a dot against a digit already placed is one rung. Reading a blank against a placed digit is the next one up, not because the arithmetic is harder but because noticing an absence is. That split is what the level labels rest on: across the measured range, boards needing a blank read climb from one in ten to four in five as the givens disappear.

Playing it here

The dots are drawn on the rules where they belong, filled against hollow so the two kinds separate by shape before colour. Nothing is drawn on the bare boundaries — a hundred marks announcing "nothing here" would bury the forty-odd that say something — so the reading those blanks need is left to you, which is the correct place for it.

Where the board does help is when you get one wrong. A digit that breaks a relation is reported as breaking that relation, naming the square across the boundary and what it required, rather than simply buzzing. The Fill button's pencil marks already know the dots and the blanks around each square, and the hint button walks the same ladder the grader does, so it will offer a dot before a blank and a blank before a chain — and say which it is giving you.

One technique the hint engine will not use

Unique rectangle arguments claim that a certain pattern would leave the grid with two answers, so the pattern cannot be there. The second answer they picture comes from swapping a pair of digits inside a rectangle — and on this board a dot, or a bare boundary, can forbid that swap outright. If the alternative was never legal then the pattern was never deadly and the argument proves nothing, so the engine declines it across the whole grid, exactly as it does on cages and thermometers.

Sharing a board sends its layout with it. That matters more here than anywhere else on the site: an Expert puzzle's givens are eighty-one blanks, so a link carrying only the digits would carry nothing at all. Whatever arrives is re-proved to have a single answer under its own dots, on a budgeted search, before this page will play it — and the pack builder above puts the same boards on paper, where the blanks are just as loud and there is no Fill button to do the first pass for you.