SudokuBoku

Thermo sudoku Thermometers 9×9

Grey paths cross the grid, and along each one the digits must climb away from the round bulb — a rule about order, not difference.

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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

Mercury for the printer

A PDF of fresh thermometer puzzles, drawn by your own browser. The paths print as rounded grey tracks with a filled dot at each bulb, and the answer key at the back repeats them, so you can trace a finished grid climbing every run of glass.

Difficulty

A rule with a direction

Every constraint you have met on this site so far says the same thing in a different shape: these squares are all different. Rows say it, boxes say it, a killer cage says it with a sum attached, a hyper window paints it. A thermometer says something new. Along the path, from the bulb at one end to the tip at the other, each digit must be strictly greater than the one before it. Not merely different — greater. The rule has a direction, and the little round bulb tells you which way it points.

The digits don't have to be consecutive; 1, 4, 5, 9 climbs a four-square thermometer perfectly well. And because "greater than" chains through every step, a thermometer quietly bans repeats along its whole length — even where the path bends through squares that share no row, column or box. Select a thermometer square on the board above and the entire run lights up: those are all, in effect, peers of one another, a fact your classic instincts don't volunteer.

What the glass knows before you write anything

Here is the arithmetic that makes this variant solvable at all. Take any square on a thermometer and count its position — say it is the third square of a five-square path. There are two squares below it that must each be smaller, so it can't be less than 3; there are two above that must each be larger, so it can't be more than 7. Before a single clue is read, every square on the glass carries a range set purely by where it sits: position and length are themselves clues.

A five-square thermometer, empty. No digit has been placed anywhere, yet each square already carries the range its position forces — the bulb can't exceed 5, the tip can't drop below 5, and the middle square is pinned to 3–7. Longer glass squeezes tighter: every square on a five-path has exactly five possibilities.

Work the formula through and something pleasing falls out: on a thermometer of length L, every square — bulb, tip, or anywhere between — has exactly 10 − L possibilities. A three-square path leaves seven candidates per square; a seven-square path leaves three. Each square you extend a thermometer by costs every square on it one candidate, which is why length, not count, is what to check when you size up a fresh grid.

Every square added to a thermometer deletes one candidate from every square already on it.

The glass that solves itself

Push the formula to its limit: a nine-square thermometer leaves 10 − 9 = 1 possibility per square. It must read 1, 2, 3, 4, 5, 6, 7, 8, 9 from the bulb up — eighty-one constraints and no puzzle left. Our generator caps its paths well short of that, for exactly this reason.

One digit moves the whole column of mercury

Placing a digit on a thermometer is nothing like placing one in a row. A 6 in a row removes 6 from eight other squares and is done. A 6 on a thermometer pushes on every square of the path, and it pushes by distance: the square directly below the 6 must be 5 or less, but the square two steps below can't exceed 4, and three steps below can't exceed 3. Climbing costs at least one per step, so a digit's influence grows the further along the glass you look. The same wave runs upward from it, one minimum per step.

One placed digit — the tinted 6, four squares up a six-square thermometer — and the ranges it forces everywhere else. Reading away from the 6, each step down loses one from the ceiling and each step up adds one to the floor; the tip's range was 6–9 by position alone and the single placement drags it to 8–9.

The board's hint engine treats this reading as its opening move, and it has a sharper one behind it that needs no placed digit at all. Suppose pencil marks alone say a bulb holds 4 or 5. Then the next square up is at least 5, the one after at least 6 — the lowest surviving mark in each square sets a floor for the square above it, and the highest sets a ceiling for the square below. Walk the path once up and once down and the ranges tighten from both ends at once, like mercury settling. The hint button calls this the thermometer squeeze, and on our medium grids it is the hardest move you'll need: marks squeezing marks, with nothing written in ink anywhere on the glass.

Thin glass is the dangerous glass

When we tuned the four levels we expected the usual dial: more thermometers, harder puzzle. Measurement said the opposite, and it wasn't close. Give a grid five or more paths and the ranges above do so much narrowing that singles and squeezes carry you from the first square to the last — at twenty-six clues under a five-to-seven thermometer layout, only three graded digs in a hundred needed anything a hard label could honestly mean. Dense mercury is generous mercury: every path is information, and enough of them solve the puzzle for you.

So this page's difficulty dial turns the other way. Easy serves four to six thermometers over thirty-eight clues and lets the glass do the work. Hard and Expert serve two or three short paths over twenty-three or twenty-four clues — just enough mercury to demand the squeeze, too little to replace the subsets, wings and fish that those levels are named for. If a thermo grid ever looks sparse to you, don't relax: the setter took the help away, not the challenge.

Proofs, refusals, and where the paths travel

Every grid served here passes two mechanical proofs before you see it. First, the thermometers must be load-bearing: we strip them off and count how many solutions the bare clues would admit under plain classic rules, and if the answer is one, the paths were decoration and the candidate is thrown away. That floor bites hardest at Easy, where the clues are generous — roughly half of all easy digs fail it and are discarded. Second, from Hard upward the graded solve must actually fire the squeeze at least once, so the levels that promise the variant's signature move demonstrably contain it. Both checks run on every puzzle, not on a sample.

One refusal follows from the rule itself. The unique-rectangle family of techniques argues "this pattern would allow two solutions, so avoid it" — but an ordering constraint can break the second solution on its own, exactly as cage sums do on the killer board, so the hint engine here never offers a unique rectangle anywhere on the grid. Slower, and sound.

The share button folds the thermometers into the link itself — each path travels as its bulb and the turns it takes — so the same glass reaches whoever you send it to, and is re-proved unique before it is trusted. The pack builder above puts the tracks on paper, where the squeeze has to happen in your head.