SudokuBoku

17 clue sudoku The minimum 9×9

Seventeen givens, the fewest a sudoku can have. Every puzzle here comes from the complete collection, graded by what it takes to finish.

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.
  • Each row, each column and each box takes the digits 1 to 8 once.
  • A box is four squares wide and two tall: two boxes across a row, four down a column.
  • No digit repeats inside any of the three.
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

Seventeens for the printer

A PDF of puzzles from the collection, each one rearranged into a form of its own so no two sheets look alike, with its level printed beneath it and the answers at the back. Sixty-four blanks is a lot of pencil work, so this builder stops at four to a page.

Difficulty

Seventeen is a floor, not a record

A clue is a digit printed on the grid before you begin, and a puzzle is proper when its clues leave exactly one way to finish. Ask how few clues a proper puzzle can carry and the answer has been settled since 2012: seventeen. Not seventeen as the lowest anyone has managed, but seventeen as a wall. Sixteen has been shown to be impossible, and the puzzles that sit exactly on the line have all been found and counted.

The impossibility was established by Gary McGuire, Bastian Tugemann and Gilles Civario, who did not reason their way to it so much as look. There are 5,472,730,538 essentially different completed grids, and their program examined every one for a set of sixteen squares that would pin it down alone. About 7.1 million processor-hours later, none of them had one. It remains a computation rather than an argument anyone can follow on paper, and that is worth admitting; but a computation that visits every case is a proof all the same.

How you search five billion grids for something that is not there

Every completed grid contains small groups of squares whose digits could be swapped around into a different completed grid — two 1s and two 2s at the corners of a rectangle spanning two boxes are the simplest case. A clue set that misses such a group entirely cannot tell the two grids apart, so a sixteen-clue puzzle would have to touch every one of them. The search listed those groups for each grid and asked which sets of sixteen squares hit them all; the few that did were then checked outright, and every one of those admitted a second answer.

Where the 49,158 come from

Gordon Royle, a mathematician at the University of Western Australia, kept the list for years, growing it from puzzles sent in and puzzles found by search until it stood at 49,151. Others took it on and pushed it to 49,157, and the last, number 49,158, turned up in 2019. Then the question was closed from the other side: a scan of every completed grid, run the way the sixteen-clue search was run, found nothing the list did not already hold. So what this page serves is not a sample of seventeens. It is all of them.

Each entry stands for a family rather than a single grid. Rename the digits, swap the three rows of a band among themselves, swap whole bands, do the same with columns and stacks, reflect the lot across the diagonal: every one of those moves turns a proper puzzle into a proper puzzle with the same solving path, and together they give one entry up to 1,218,998,108,160 forms. The list keeps a single form per family and chooses it by sorting — the version that reads earliest as a string of characters is the one written down. Reading earliest means opening with as many empty squares as possible, which is why the whole file begins with a run of dots and every stored puzzle has its clues crowded into the lower right.

The first puzzle in the collection, exactly as the list stores it: nothing in the top row, nothing in the first two squares of the second, and seventeen clues pushed towards the bottom right, because this is the arrangement of the puzzle that sorts before all of its equivalents.
The same puzzle after one of its 1,218,998,108,160 rearrangements — digits renamed, rows and columns shuffled within their bands and stacks, the grid reflected. It has the same answer under the renaming, the same solving path and the same difficulty, and this is the kind of form the board hands you.

The board never shows a stored form. Each press of New puzzle draws a rearrangement at random, so the grid in front of you has almost certainly never been printed anywhere. That could only be honest if difficulty survived the shuffle, so it was checked: two thousand puzzles were graded before and after a random rearrangement, and the band and the hardest step came out identical in every one.

A rearranged sudoku is the same sudoku. The list keeps one copy per family, and this board shows you any of the trillion it stands for.

Seventeen clues is not seventeen times harder

The reputation is fearsome. Search for these puzzles and you will read that they take hours, that they need forcing chains, that most people cannot finish one without a computer. Nobody writing that had graded them, so we did: every one of the 49,158 was replayed with the same ladder of human techniques that grades every board on this site, and sorted by the hardest move it needed.

All 49,158 puzzles, sorted by hardest technique

Easy naked singles only
0 · 0%
Medium hidden singles
21,905 · 45%
Hard subsets, locked candidates
19,742 · 40%
Expert wings, fish, colouring
5,398 · 11%
Evil almost locked sets, swordfish
814 · 1.7%
Extreme past every technique here
1,299 · 2.6%
The band is decided by the hardest move the solve needs, the rule the level buttons follow everywhere on this site. Nearly half of the collection never asks for anything beyond a hidden single, and only one puzzle in forty defeats every technique the ladder knows.

45 per cent of them fall to hidden singles alone. Add the puzzles that need nothing beyond a subset or a locked candidate and 85 per cent of the collection sits at Hard or below — the level where pencil marks stop being optional, and nothing more exotic is asked for. The wings, fish and colouring of Expert account for 11 per cent; the almost locked sets and swordfish of Evil for under two; and 1,299 puzzles, 2.6 per cent, run past the end of the ladder into chains or a considered guess.

The hardest step tells the same story from the other side. Hidden singles top the table because a puzzle that needs nothing else is graded by them; then come pointing pairs, then the skyscraper, and everything below that is scarce.

Hardest step needed Puzzles Share
Hidden Single 21,905 44.6%
Pointing Pair 14,353 29.2%
Skyscraper 2,817 5.7%
Naked Pair 2,383 4.8%
Hidden Pair 1,833 3.7%
Beyond the ladder 1,299 2.6%
XY-Wing 1,180 2.4%
Box/Line Reduction 1,117 2.3%
Everything else 2,271 4.6%

"Everything else" gathers the nine rarer endings: W-wings, almost locked sets, colouring, triples, the fish, and the two puzzles whose hardest move is a jellyfish.

So where does the reputation come from? Partly from the 1,299 that earn it. Mostly from what sparseness does to the experience rather than the logic. With sixty-four squares to fill and only three or four hidden singles on offer at the start, the opening is slow and unforgiving of a missed pencil mark, and on a Medium seventeen every one of the sixty-four steps is a placement, with no easy stretch to coast through. That feels harder than a Hard puzzle with twenty-six clues. Measured by the moves it asks for, it usually is not.

The number of clues measures how much is missing, not how hard it is to find.

Why none of them is easy

On this site Easy means one thing: the whole grid falls to naked singles, squares with a single digit left. Not one seventeen qualifies, and the reason is arithmetic. A square has twenty peers, and it becomes a naked single only once eight different digits are standing among them. Seventeen clues rarely aim that many at one place: 88 per cent of the collection offers no naked single at all before the first move, 5,719 puzzles offer exactly one, and 49 offer two.

One of the rare ones. The tinted square in row 8 sees the eight amber clues — 3 and 6 along its row, 7, 2 and 4 down its column, 1, 5 and 8 in its box — and those are eight different digits, so a 9 is the only thing it can hold. Nearly nine seventeens in ten open with no square like this anywhere on the board.

The opening move, then, is a hidden single: a digit with one home left in some house. Every seventeen but 49 has one available before anything is placed, and on average only 3.4 are on the board at once, which is why the first minutes feel like hunting for a needle. Just 35 puzzles offer no single of either kind at the start; on those the first move has to be an elimination, a pointing pair that thins the candidates before any digit can go down.

What seventeen clues have to look like

A set of clues this small cannot take any shape it likes, and one constraint can be reasoned out before looking. A proper puzzle can leave out at most one digit. Were two missing from the clues, take any solution and swap those two digits throughout it: every row, column and box still holds each digit once, no clue has been contradicted, and you hold a second solution. So a seventeen may omit one digit entirely, and nearly half of them do — 23,788 of the 49,158 use only eight. They run a shade harder than the rest, 13 per cent of them grading Expert against 9 per cent of the nine-digit puzzles, though not by anything like the margin the bare grid suggests.

Seventeen clues that use only eight of the nine digits: there is no 2 anywhere on this board, so every 2 in the answer has to be found by elimination. A puzzle missing two digits is impossible, which makes eight the floor as surely as seventeen is.

The rest of the shape was measured rather than reasoned:

Across the collection Puzzles Share
A row or column with no clue in it 32,228 66%
A whole box with no clue in it 4,596 9%
Two empty boxes 71 0.1%
Only eight digits used 23,788 48%
Fullest box holds 3 clues 42,590 87%
Fullest box holds 5, the most any has 6 <0.1%

An empty line is not a weakness. The puzzle is proper, so those nine squares are fully decided by clues that sit elsewhere; two thirds of the collection has one, and any of them can be turned so that the bare line runs down a column instead.

Five levels from one number

Because the whole collection is graded, the level buttons above the board mean here what they mean on the classic board: Medium serves a seventeen that falls to hidden singles, Hard one that needs a subset or a locked candidate, and so on up. Each band lives in its own set of files and a press of New puzzle fetches one slice of one band, so the page never downloads the four-megabyte collection to hand you one grid. The line above the grid names the actual hardest step of the puzzle you were given, not the band's general promise.

There is no Easy button, and not because it was left off: there is nothing to put behind it. Extreme, by contrast, has 1,299 puzzles behind it, grids the ladder cannot finish, where the extreme page's case about chains and testing applies in full. Evil has 814, decided almost entirely by almost locked sets, and the 5,398 at Expert are the place to meet a skyscraper in the wild, since it is the commonest expert-level ending in the collection by a distance.

Proved twice

Uniqueness was established two ways. The whole collection was counted outright once, every puzzle searched for a second answer and none found. The curation that builds the banks then proves it again on every run, by a shortcut worth spelling out: every technique on the ladder except the unique rectangle is direct logic, removing only what cannot be part of any solution, so a puzzle the ladder finishes without that one move has been shown to have exactly one answer simply by being solved. 47,425 of the 49,158 are proved that way; the 1,301 the ladder cannot finish, or finishes with a unique-rectangle step, are counted the slow way, and so is a one-in-a-hundred sample of the rest, to keep the shortcut honest.

Playing a seventeen here, and on paper

Sixty-four blanks make pencil marks a necessity rather than a habit, and the Fill button writes the complete set in one step so the opening scan can begin from a full picture. From there the natural rhythm is digit-first: choose a digit, ask each box in turn where it can still go, and place it wherever the answer is one square. The medium page teaches that scan in full, and a seventeen is the purest exercise in it there is, because for most of the collection it is the entire solve.

The share button copies the 81 squares into a link, so a rearranged form that has never existed anywhere can still be sent to someone and opened exactly as you saw it; a game reached that way is saved under its own key and never overwrites the one in progress here. The pack builder above puts the collection on paper, each grid freshly rearranged, with its band printed beneath it and the answers at the back. It stops at four to a sheet, because a grid with sixty-four squares to mark up wants the room.