SudokuBoku

Greater than sudoku Comparisons 9×9

Each sign points at the smaller of the two squares it sits between. Scattered over a clued grid, or every one of them on a bare 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.
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

Signed grids for the printer

A PDF of fresh greater-than puzzles. The chevrons print in full black rather than the grey the other boards give their structure, because here they are not structure — they are the clues, and on the harder sheets they are very nearly all of them. One or two to a page: signs need rules wide enough to sit on.

Difficulty

An ordering that branches

Between two neighbouring squares you may find a sign. Its point aims at the smaller digit and its open side at the larger — the convention every printed version of this puzzle uses. Beyond that nothing changes: rows, columns and boxes still take 1 to 9 with nothing repeated. How many signs there are, and whether any of them span two boxes, depends on which of the two styles below you are playing.

This is the second rule on the site with a direction to it, and the first one is the right thing to measure it against. A thermometer is an ordering along a path, so its reasoning runs in one line: count up from the bulb, count down from the tip, meet in the middle. Signs are an ordering on a graph. A square can be under two neighbours and over two others at once, and the chains running through it fork, rejoin and arrive from four directions. You cannot walk it; you have to let bounds settle across it.

What the arrows say before you write anything

Here is the deduction that makes this variant work, and the useful thing about it is that it costs nothing to apply. Look at a square and count the longest chain of arrows running down from it. If three squares sit below it in a chain, those three need three different digits all smaller than this one, so this square cannot be less than 4. Count the chain running up from it and the same argument caps it from above.

That is true of an untouched grid. Before a single digit is entered, every square on the board already has a range. On a board carrying the full set of signs the chains reach surprisingly deep: counting the squares of one generated grid, 20 had nothing below them, 25 had one, 19 had two, 12 had three and 5 had four — and a square with four below it is pinned to 5, 6, 7, 8 or 9 before you have looked at a row.

Count the arrows beneath a square and you have its floor, on an empty grid, for free.

One box of a real generated board, with its twelve comparisons and no digits at all. Follow the arrows down from the top-left square of the middle row and four squares lie beneath it, so it cannot be smaller than 5 — and nothing has been written in yet. It turns out to be 9.

The board applies that much for you: the pencil marks you get from the Fill button already know the chain lengths. What it deliberately does not do for you is read the arrows against digits you have actually placed. That is a move, it has a name here, and the levels are built on the difference.

Four levels, and what separates them

Grading works the way it works everywhere on this site — the puzzle is replayed with human techniques and named after the hardest step the replay needed, and both styles are measured with the same ruler. The ladder for this board has four rungs, and the middle two are both about signs:

A comparison is one step: this square carries a sign against a neighbour that already holds a digit, so it must beat that digit or fall short of it. An inequality chain is more than one step: bounds carried through squares that are all still empty, two of them tightening each other without either being known. Splitting those two apart mattered more than it sounds. Run them as a single rung and the chain swallows every puzzle on the board — measured, three of the four level labels would have described nothing at all.

The level is how much of the grid you get

In the dense style the difficulty dial is not the signs at all. Every level shows the same 108. What changes is the number of given digits, and it changes hard: Easy hands you 34, Medium 30, Hard 12 — and Expert hands you four. A grid of eighty-one squares, four of them filled in, and a hundred and eight arrows to get the rest out of. The scattered style moves both dials together, from forty digits and thirty signs down to twenty-four and eighteen, because moving either one alone runs out of road.

Measured across generated grids, that ladder is clean: at 12 givens 95% of puzzles need a genuine chain, and at 4 givens one in ten also needs a classic pattern on top once the arrows have said everything they can. Going the other way, at 34 givens most puzzles fall to singles alone, because the chain lengths have already done the narrowing.

Two ways to use one rule

The switch beside the number pad picks between two genuinely different puzzles, and the difference is not decoration. Scattered is the familiar one: a normally-clued sudoku with a couple of dozen signs placed across it, including ones that reach between boxes. Every sign is the dense form — all 108 comparisons inside the boxes, on a grid emptied out until the signs have to carry it.

Cross-box signs belong to the sparse style and only to it. When a grid carries every in-box comparison, a sign spanning two boxes tells you little you could not already work out. When it carries twenty, those are the ones tying one box's reasoning to the next, and about a fifth of the signs on a scattered board are of that kind.

What load-bearing means, and what it doesn't

On a scattered board the signs are dug down to a target rather than to the bare minimum, which means some are individually redundant — you could rub one out and the puzzle would still have one answer. That deserves saying plainly rather than glossing, because it sounds like the thing this site refuses to do.

It isn't, and the distinction is the one that applies to a forty-clue Easy sudoku: most of those givens are redundant too, and nobody calls them decorative. What has to be load-bearing is the rule, and that is checked on every puzzle at every level — strip the signs off entirely and the grid must stop having a single answer, or it is discarded. From Hard upward we ask for more again: the graded solve has to use an inequality chain, not merely be capable of one.

Digging signs all the way to the minimum was tried and rejected on measurement. It leaves four to fifteen of them, and about two thirds of the resulting puzzles land beyond our technique ladder entirely — a minimal puzzle is a hard puzzle, in signs exactly as in digits. What does survive climbs steadily with difficulty, though, and that climb is the most honest summary of the two styles there is.

One classic technique is switched off in both styles. Unique rectangle arguments claim a pattern would give the grid two answers; the second answer they picture comes from swapping a pair of digits, and a sign can forbid that swap outright. Where the pattern is not deadly the deduction proves nothing, so the hint engine declines it across the whole board — the same refusal it makes on cages and thermometers.

Reading a board with almost nothing on it

A dense grid with four digits on it is intimidating in a way a normal sudoku never is, and the way in is not to look for a digit. It is to look for the longest chain you can find and work out what it forces at each end. Chains of four are the ones worth hunting: they cap five squares at once and they are visible as a run of arrows all pointing the same way.

The hint button walks the same ladder the grader does, so it will offer you a comparison before a chain and a chain before anything classical — and it names which it is giving you, so the two stop blurring together. The share button hands over the givens and the signs together, because the same four digits under different arrows are a different puzzle entirely; a shared link is proved to have one answer under its own signs, on a budgeted search, before this page will play it. A link whose arrows run in a circle is refused outright rather than loaded and discovered to be impossible twenty minutes in.