SudokuBoku

Arrow sudoku Circles and paths 9×9

Each arrow's digits add up to the digit in its circle — a total you have to work out at the same time as the digits making it.

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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

Circles and paths for the printer

A PDF of fresh arrow puzzles, built by your own browser. Rings, shafts and heads print in mid-grey beneath black digits — the same relationship the screen keeps — and the answer key repeats the layout, so a finished sheet can be checked arrow by arrow as well as square by square.

Difficulty

The total is one of the squares

An arrow grid is an ordinary 9×9 with circles drawn on some squares and a path leading out of each one. The rule is a single sentence: the digits along a path add up to the digit in its circle. Rows, columns and boxes carry on exactly as they always have.

Compare that with killer sudoku, the site's other arithmetic board, and one word does all the work. A killer cage carries a printed total. It is a fact you are given, and cage reasoning is a matter of spending a known budget. An arrow's total is a square of the puzzle, unsolved like every other square. Until you know the circle you don't know what the path is worth; until you know the path you can't pin the circle. Every arrow is a statement about both ends at once, and it is worth reading in whichever direction happens to be easier at the time.

In practice that makes arrows feel less like arithmetic and more like leverage. A single digit landing anywhere on an arrow tightens everything else on it immediately, including a circle three squares away that nothing in its own row, column or box had touched.

What a path's length already tells you

Before any digit is written, the shape of an arrow constrains it. Each square on a path holds at least 1, so a path of L squares needs a circle of at least L. And where the path squares all see one another — sharing a row, a column or a box — they must differ, so the least they can total is 1+2+…+L.

Run those two numbers out and the board gets surprisingly opinionated. A two-square arrow whose pair share a house can never have a circle below 3. A three-square arrow lying along one line has a circle of 6, 7, 8 or 9 and nothing else — and no square on that path can exceed 6, because the other two need at least 3 between them.

Three arrows before a single digit is written. The two-square path along row 1 already forces its circle to 3 or more; the three-square path down column 6 forces its circle to 6, 7, 8 or 9, and caps every square on it at 6. The bent path at the bottom escapes both limits by leaving the box it started in.

The four-square theorem

Push the same arithmetic one step further and something absolute falls out. Four different digits total at least 1+2+3+4, which is 10, and no circle can hold 10. So no four-square arrow can ever contain four different digits. Every one of them repeats a digit somewhere, which means every one of them has to leave the house it started in — a repeat inside a single row, column or box is illegal, so the path is obliged to bend.

That is not a measurement or a rule of thumb; it is arithmetic, and it is the reason the paths on this board stop at four squares. A five-square arrow needs three repeats and reduces to a scattering of 1s and 2s, which reads as a puzzle about counting rather than a puzzle about sudoku.

A four-square arrow from a puzzle this page generated, seed and all. Its digits total nine, and — as they must — two of them are the same. The amber squares are the pair: R6C2 and R7C3 hold 1 apiece, which is legal only because the bend has carried the path into a different box.

Repeats are allowed, and people forget it

This trips up solvers arriving from killer, where a digit may never appear twice in a cage. An arrow imposes no such thing: two squares on one path may hold the same digit whenever ordinary sudoku permits it. The theorem above depends on that being true — take repeats away and four-square arrows would simply be impossible.

How the hint thinks

Most solvers work an arrow with rough bounds: the most this path could be, the least it could be, squeeze from both ends. Our hint button doesn't estimate. When you ask it about an arrow it walks every way that path could still be filled — respecting the pencil marks in each square, the distinctness sudoku demands between squares that see each other, and a total the circle could actually hold — and then throws away any mark that no surviving filling uses.

That is exhaustive rather than approximate, and it is affordable for one reason: a path is at most four squares and a total is at most nine, so the number of fillings to check is in the hundreds, not the millions. The bound on path length buys the exactness. The eliminations it produces are often sharper than a bounds argument gives — a mark can sit comfortably inside the plausible range and still belong to no legal filling at all.

Fewer arrows, harder puzzle

Difficulty here is measured the way it is measured everywhere on this site: the puzzle is replayed with human techniques and named after the hardest step the replay needed. Doing that across a few hundred generated grids turned up the same inversion the thermometer board found, and rather more strongly.

Arrows do not make a puzzle harder. Each one is a window onto squares you have not reached, and the more windows there are the faster the grid falls open. Measured at twenty-four givens, going from two or three arrows up to four or six drops the chance of a puzzle needing an advanced technique from about one in twenty to about one in a hundred. So the levels here thin the arrows out as they climb, instead of adding them.

Drawing another arrow on an arrow puzzle is a way of making it easier.

Why Expert has more clues than Hard

Follow that logic to the end and the clue ladder stops being a ladder. Hard digs to twenty-four givens with two or three arrows. Expert digs to twenty-six — two more clues — with exactly two arrows and no more. It looks backwards written down, and it is the only board on this site where a harder level starts with a fuller grid.

The reason is what Expert has to promise. It claims a puzzle needing a wing, a fish or a colouring chain, which requires the technique ladder to reach that depth and still finish. Strip the arrows down at twenty-four givens and a third of the grids simply run past every standard technique into chain territory, where no honest label fits. Two arrows and twenty-six givens is the narrow ridge between the arrow sum doing all the work and nothing doing any of it. Only about one dig in fourteen lands there, which is why generation sifts a hundred candidates to find one.

Every level is also held to the rule that the arrows must matter. Each candidate puzzle is re-checked with its arrows ignored, and if the bare givens still pin a single answer the puzzle is discarded rather than served — the arrows were decoration. From Hard upward we ask for more: the graded solve must actually use an arrow sum at some point, so the levels that promise the variant's own move measurably contain it.

Repeats, refusals, and getting it onto paper

The technique this board will not use

Unique rectangle arguments turn on the claim that a certain pattern would give the grid two solutions; the second solution they picture comes from swapping a pair of digits inside a rectangle, and on this board that swap can change what an arrow totals. If the alternative is illegal then the pattern was never deadly and the deduction proves nothing, so the hint engine declines it across the whole grid — the same refusal it makes on cages and thermometers.

Everything else stands. Select any square on an arrow and the whole arrow lights up alongside the usual row, column and box; the highlight is deliberately not claiming they are peers, because they aren't, only that they all answer to one total. The Fill button writes pencil marks that already know what each circle can afford. A screen reader is told which end of which arrow a square is on, and how far along.

Paper needs nothing special from this variant, which is a relief after the last one: rings, shafts and heads are ink, so a printed arrow grid says what it is. They print mid-grey under black digits, and the answer sheets repeat the layout so a finished booklet can be checked one arrow at a time. The share button hands over the givens and the layout together — the same clues under different arrows are a different puzzle, so the arrows travel in the link, and a shared grid is proved to have exactly one solution under them, on a budgeted search, before this page will play it.