SudokuBoku

Odd-even sudoku Rings and frames 9×9

A ring means the digit is odd; a frame means even. Most squares carry neither, and that restraint is the whole design.

00:00 Mistakes 0
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

Marked grids for the printer

A PDF of fresh odd-even puzzles. The marks print as outlines rather than the filled grey pads most published sheets use, because an outline puts no ink underneath the digit and a home printer having a bad toner day can then only lose one of the two. One or two to a page.

Difficulty

A rule that talks to one square at a time

The grid is an ordinary 9×9. Some of its squares carry a shape drawn behind the digit: a ring means an odd digit lives there — 1, 3, 5, 7 or 9 — and a frame means an even one, so 2, 4, 6 or 8. Squares with neither are unrestricted, and on this board that is most of them.

Every other extra rule on this site says something about a relationship. A cage adds its squares up, a dot compares two of them, a knight's move forbids a match, a thermometer imposes an order. Each needed new machinery, because "these squares stand in this relation" is not something a sudoku solver already knows how to think about.

A parity mark relates a square to nothing whatsoever. It shortens one list of candidates and it is finished. That makes it the cheapest rule here by a distance — no new geometry, not a line of change to the solver — and also the most dangerous, for a reason that took a measurement to see.

One asymmetry is worth pocketing first. There are five odd digits and four even ones, so a ring leaves five candidates standing where a frame leaves four: a frame is always the stronger clue. That lopsidedness is what the rest of the board is built on.

Shade every square and you have built two puzzles

The obvious way to set this variant is to take a finished grid and mark all 81 squares from its own parities. You get a handsome board — 45 rings and 36 frames, necessarily — and you do not get a harder puzzle. You get a different and smaller one, twice. The switch beside the grid builds it, so none of what follows has to be taken on trust.

Here is why. Under a complete map, odd digits can only stand in rings and even digits only in frames. Every house holds all nine digits, so its five rings must hold a rearrangement of 1, 3, 5, 7 and 9, and its four frames a rearrangement of 2, 4, 6 and 8. Neither of those sentences mentions the other set. Rows, columns and boxes all split the same way, and what you are left with is two puzzles laid over each other that never once interact: a five-digit one on 45 squares, and a four-digit one on the other 36.

The top-left box of a solved grid with all nine squares marked — the arrangement the Every square switch produces. The five rings hold 5, 9, 7, 1 and 3; the four frames hold 4, 2, 6 and 8. Shuffle the ring digits among the rings however you like and the frames never notice, because nothing in the rules connects the two groups once every square's parity is known.

When every square is marked, a given inside a ring can never help you place a digit inside a frame. Half your clues are inert against half the grid.

The halves, counted

That is a claim you can check rather than take on trust, and the engine tests do. Take the rings of one solved grid and the frames of a completely unrelated one under the same map, splice them together, and see whether the result is a legal sudoku. Two thousand splices later, all two thousand were legal and none were not. The halves really are independent.

Counting them exhaustively puts a number on the damage. Across six real maps the ring half admitted between 549,120 and 1,728,000 fillings, and the frame half between 1,032 and 4,224. Because they are independent those multiply, so a fully marked grid with no digits at all has somewhere between half a billion and seven billion answers. A rule that looks like it is constraining everything turns out to have thrown away the thing that makes a sudoku a sudoku, which is that all nine digits are competing for the same squares.

Counting a house

So the board you get by default marks a subset, and its marks are dug rather than copied off the answer. What that buys is a move which a complete map cannot contain, because a complete map has nothing left to work out.

Start from the lopsided count: five odd, four even, in every row, every column and every box. Now count the squares in one house that could still take an even digit. There have to be at least four. If there are exactly four, those four squares are the even ones — there is no room for the count to fall anywhere else — and every other square in that house is odd, whether or not anybody drew a ring on it.

Row 5 of a Medium board from this page, carrying four frames and no other mark. A row has exactly four even digits to give, so those four frames have used them all up — and the five tinted squares are odd. Nothing has been placed anywhere on the grid at this point; the deduction is pure bookkeeping, and its answers turn out to be 1, 7, 3, 9 and 5.

Notice what the output of that move is. It is not a digit, and it is not even an ordinary elimination — it is a parity for a square nobody marked. Which means it feeds itself: a square worked out to be even in row 3 is a known even when you come to count column 7, and column 7 may run out of room because of it.

The same count, one step deeper

A square can also be shut out of a parity by ordinary sudoku. If the digits already placed around some square have knocked out 2, 4, 6 and 8, that square is odd as surely as if it had a ring on it, and it should be counted as one. Take the census that way — off the pencil marks rather than off the ink — and it finds houses the first version walks past.

The two are graded apart, and that split is not tidiness. Measured as a single rung the candidate-aware version swallowed nearly every board it was shown, which would have left three of the four level labels describing the same thing. Kept separate, Medium is the count you can do by looking and Hard is the count you have to earn — and the harder one turns out to be the scarcest thing on this board, because a grid that can be squeezed once can usually be squeezed until singles take over.

How much shading is too much

Now the two halves of this article collide. Marks make censuses possible, and censuses derive parities, and derived parities are exactly what the decomposition needs in order to happen. So a setter who marks generously is not sliding up a difficulty scale. They are sliding toward the two-puzzle board, and they get there long before all 81 squares carry a shape.

The number that matters is therefore not how many marks are drawn but how many squares end up with a knowable parity once the census has run itself out. Over 300 maps at each density:

Marks drawn Squares with a knowable parity Maps that settle the whole grid
10190%
20400%
28540%
32610%
36663%
407113%
457739%
507965%
568194%

Each mark is worth about two squares, and the exchange rate barely moves. By 50 marks two boards in three have quietly become two puzzles — with 31 squares still showing nothing at all, which is why you cannot spot it by looking.

That table is an acceptance test rather than a curiosity. Every board this page generates is checked against it, and any layout whose census settles all 81 squares is thrown away before a single digit is dug. Levels run from 18 marks up to 26, which keeps every one of them comfortably clear of the cliff.

The digits move the other way, and they stop moving sooner than you would expect. Easy hands over 26 of them; Medium about 17; Hard about 16 and no fewer, however low the generator is told to aim. Below that the marks simply run out of slack — a shape trims one square's list, and there is a floor to how much of a grid a few dozen trimmed lists can carry between them. So Expert is not the level with the fewest digits: it lands in the same fifteen-to-seventeen range as Hard, on fewer marks, because what makes it Expert is the classic ladder standing behind the shapes rather than the shapes themselves.

The other board, and what it is good for

Search for this variant and you will find both kinds in roughly equal number, which is why the switch beside the grid offers both. It is worth saying plainly what changes when you move it, because the difference is not cosmetic and not a difficulty setting.

Marked everywhere, this puzzle has one level. Measured over thirty digs at each of five clue counts from 26 digits down to 18, the complete map grades easy between nine and ten times out of ten, and the counting move above fires on none of them — not rarely, none, at every density. That is the theorem showing up as a measurement rather than an argument: where every parity is already known there is nothing left to count. A level chooser on that board would be four labels for one puzzle, so it disappears when you flip the switch.

What it is genuinely good at is a different thing, and a nice one. It hands you twenty-two digits — a clue count that would make an ordinary sudoku punishing — and then lets you solve it with nothing harder than "this square has only one digit left". It looks far more difficult than it is. That makes it the gentlest board on this page and a good place to send someone who finds a normal grid daunting, and it is why it ships rather than being described and withheld.

One thing it is not is a shortcut past the standards the rest of the site keeps. Its digits are dug against the same uniqueness proof, and it faces the same test every variant here faces: rub the marks out and the puzzle must stop having a single answer. At twenty-two digits it passes that on every board measured. It is exempt from exactly one check — the one that keeps the scattered board out of this state — and from nothing else.

Reading this board

Published odd-even puzzles almost always print the marks as filled grey pads with the digit sitting on top. This board draws outlines instead, and that came out of a contrast measurement rather than a preference. A fill has to composite with every state a square can be in — selected, related, matching, hinted, wrong — and at the lightest wash that still read as a mark, entry ink fell to 4.34 against a plain dark cell and 2.93 against a selected one. An outline touches no ink at all, so every digit on the board keeps exactly the contrast it had before the shapes arrived.

The pencil marks pull inwards a little inside a marked square, for a reason worth the fifteen per cent: at full width the four corner marks sit right on a ring, and the corner marks are 1, 3, 7 and 9 — the four digits a ring is most likely to be about. Get one wrong and the board says which shape you broke and what it wanted, rather than sounding a buzzer; a parity slip is the commonest mistake here and it is also the most fixable.

One technique the hint engine will not use

Unique rectangle arguments say that a certain pattern would leave the grid with two answers, so the pattern cannot be there. The second answer they imagine comes from swapping a pair of digits between two corners — and swapping digits of opposite parity is precisely what a mark on either corner forbids. If the alternative was never legal the pattern was never deadly, so the engine declines the deduction across the whole grid, exactly as it does on cages, thermometers and dots.

Sharing a board sends its map along with the digits, because the same givens under a different set of shapes are a different puzzle and usually a different level — and that holds across the switch too, since a shared link carries all 81 marks or the couple of dozen, whichever it was built with. Anything that arrives is re-checked before this page will play it: the map has to be satisfiable at all — six rings in one row is a well-formed link and an impossible board, and catching that is counting rather than searching — and the pair then has to be proved to have exactly one answer, on a budgeted search that cannot be made to hang. The pack builder above puts the same boards on paper, where the census is if anything easier, since counting four frames along a row is something the eye does better than the mouse.