SudokuBoku

Hard sudoku

Twenty-six clues, and the first level where the deductions stop fitting in your head. This is where you start writing things down.

Easy Medium Hard Expert Evil Extreme

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

The moment hard begins

Every hard puzzle here still opens with singles. You'll place eight or ten digits without much trouble, and then the grid will go quiet — no square with one option, no digit with one home. That silence is the actual start of the puzzle.

What breaks it is a technique that operates on possibilities rather than placements. To use one you need the possibilities visible, which means candidates written into the squares. Our hard puzzles are verified to require at least one such technique, and verified not to need the grid-spanning patterns that expert demands.

Writing candidates without drowning in them

Fill every empty square with every digit still legal there, and a hard grid becomes a wall of small numbers. Two things make it manageable.

First, don't fill them in at the start. Solve the singles first — every digit you place removes candidates you'd otherwise have written and then crossed out. The board above has a Fill button that writes them all in one step, and the right time to press it is when the singles have run dry, not before.

Second, read for squares with exactly two candidates. Bivalue squares are the raw material of almost every technique above this level, and once you start noticing them the wall resolves into a much smaller set of interesting places.

Subsets: when squares club together

Suppose two squares in the same row both hold exactly the candidates 3 and 7. You don't know which takes which. You do know that between them, they will consume both digits — so no other square in that row can be a 3 or a 7. Cross them off. That's a naked pair, and it extends: three squares between them holding only three digits behave the same way.

The mirror image catches more people out. If, in a row, the digits 3 and 7 can only appear in two particular squares, then those two squares are the 3 and the 7 — whatever else they were listing can be deleted. That's a hidden pair, and it's the same relationship read from the other end.

Locked candidates: the workhorse

This one probably unlocks more hard puzzles than everything else combined, and it needs no arithmetic at all.

Look at a box and pick a digit. If every square in that box that could hold the digit sits in the same row, then the digit is definitely somewhere on that row inside that box. You don't know which square. You don't need to — it means the digit cannot appear on that row anywhere else, outside the box. Cross it off along the rest of the row.

A digit's only two candidates in the first box sit on the top row. Whichever square takes it, that row is claimed inside the box — so the six squares beyond it cross the digit off.

Run it backwards too: if within a row a digit can only sit inside one box, remove it from the rest of that box. Same idea, opposite direction. Between them these two account for most of the "how on earth was I supposed to see that" moments at this level.

Two mistakes worth avoiding

Stale candidates. If you write marks by hand and then place a digit, you have to remove that digit from every square it now sees. Miss one and you'll make a deduction from information that isn't true any more. The board here does this for you.

Guessing to get moving. A hard puzzle never requires it, so a guess here is always a shortcut past something findable. If you're properly stuck, the hint will name the technique that applies and show you the squares — which is a far better trade than picking a digit and hoping.

Both of these get easier with drilling rather than reading. The technique trainer hands you a position where a naked pair is the only move on the board, and does the same for hidden pairs and locked candidates — the three that turn hard puzzles from a grind into a routine.

The step after this

At hard, every technique is local: it lives inside one row, one column, one box. Expert is where patterns start spanning the whole grid and you have to look at four squares in different corners at once. That's a genuine change in what you're doing with your eyes, not just a harder version of this.