How to Solve Thermometer Sudoku on Paper with a Pen

Thermometer Sudoku is the purest "read the diagram" variant in the family: a handful of curved lines snake through the grid, and the entire extra challenge hangs on a single inequality rule along each one. On screen the app can highlight and animate those lines for you. On paper you are the highlighter, and that act of ownership — redrawing each thermometer in red before you place a single digit — is where the variant is actually won.

The reason it rewards a pen solver so heavily is that thermometer logic is positional, not combinatory. There are no cage sums to add and no windows to cross-reference; there is only a forced direction of increase. That makes it an ideal second step after the Skyscraper guide in this series, and a strong partner to the counting logic you will meet in Killer. Everything below assumes the universal tools from the pen-and-paper Series Hub — the two-pen rule and the three-pass rhythm are the scaffolding this guide stands on.

The two-pen rule in action: draw every thermometer in red and circle each bulb. A red line you drew yourself is subconsciously harder to misread than a faint grey one the printer gave you — and bulletproof lines are the whole game here.

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1. The Rules, Stated Precisely for Paper

• Place digits 1–9 so every row, column, and 3x3 box contains each digit exactly once — the classic baseline never disappears.
• Each printed thermometer is a connected chain of cells running from a round bulb to an open tip.
• Digits along a single thermometer must be strictly increasing from the bulb to the tip: every step up the line is larger than the one before. Consecutive-by-one is NOT required — an increase of any size is legal.
• Exactly one complete solution exists, so every deduction must be deterministic, never a guess.

The strict-increase rule is a total order on the line, not a set of sums. That single sentence is why the whole solve turns into a range-trimming exercise. The consequences are exact and unavoidable: the bulb is always the smallest digit on its line, the tip always the largest.

2. The Paper Adaptations — Read the Diagram First

Printed thermometer lines are usually faint grey, and when two lines crowd a shared cell they blur together. Before any logic, make the diagram unambiguous:

• Re-trace every line in red. Follow the chain cell to cell so the exact path is visible at a glance.
• Circle each bulb in red. Direction is everything: the circle tells you the low end at a glance.
• Number the positions. Write 1, 2, 3, … just outside the bulb/tip cell of each line so "this is position 4 of 6" is a fact you can read off the page.
• Keep candidates in tiny corner script. The line itself occupies the cell centre; your position-bound notes live in the margin next to the line's number labels.

3. The Core Weapon — The Position Bound

Every thermometer technique reduces to one formula. On a line of length n, the cell at position i (bulb = position 1) must hold a digit d satisfying:

The lower bound holds because i-1 smaller digits must already sit behind it on the line, and 1 is the smallest digit available. The upper bound holds because n-i larger digits must still sit ahead of it toward the tip, and 9 is the largest. Worked example: on a 5-cell line, the bulb (position 1) is bounded by 1 ≤ d ≤ 9-5+1 = 5, and the tip (position 5) by 5 ≤ d ≤ 9. That is not five separate guesses — it is five hard ranges you write down once.

4. Bulb Minimums and Maximums

The most productive cell on any thermometer is the bulb, because it carries the strictest bound of the whole line. On a 5-cell line the bulb is forced to at most 5; on a 6-cell line at most 4; on a 7-cell line at most 3. When the bulb already shares a row or box with high digits, those high digits are struck from it, and the surviving low window gets even tighter.

Symmetrically the tip carries the line's large edge: on a 5-cell line the tip is at least 5. Between them the middle cells inherit a sliding window — position 3 of 5 is bounded by 3 ≤ d ≤ 7. Write these windows next to the position labels before Pass 1, and you will be amazed how many cells open up immediately.

5. Crossing Interactions and Shared Cells

A thermometer cell is not a private world — it is also a member of a row, a column, and a box. The moment a line pins a cell low (say the bulb of a 6-cell line collapses to 1), that 1 is removed from every peer cell in its house. This is where thermometer logic fuses with classic scanning: every low or high digit you force onto the stem ripples outward into the surrounding row and box. The reverse is equally productive — a high digit already blocking a cell's row shrinks the bulb's window and by extension trims the whole line.

A rarer but decisive pattern is the separating line: when two thermometers share a cell, that shared cell must satisfy both runs of increase, which forces the surrounding cells to split the small and large ends. The shared cell is a fulcrum; treat the two line segments on either side as independent range problems and the grid unlocks.

6. Diagram — Reading Position Bounds Off the Stem

The strip below models a single 5-cell thermometer from bulb (left) to tip (right). The number written in each cell is its position; the bracket label under each cell is the position-bound window that applies before any scanning.

Each window is i ≤ d ≤ 9 - 5 + i. The gradient shading is a mental trick to carry with you: the bulb reads "low", the tip reads "high", and every cell between slides one place up the digit scale. If a box already holds a 4 in the bulb's row, the bulb collapses at once — that is the crossing interaction in action.

7. The Three-Pass Workflow, Adapted to the Stem

8. The Common Mistakes That Break Thermometer Boards

• Treating the lines as consecutive. Thermometer requires strict increase only — 1, 4, 6 is legal on a 3-cell line. Adding a "must be +1" rule eliminates valid placements and dead-ends the board.
• Skipping the re-trace. A faint printed line that bends across a shared cell is the fastest way to mis-apply a bound. Circle and re-draw before solving.
• Forgetting the tip / bulb direction. Reversing bulb and tip flips every window 180 degrees. Check the circle mark before committing a digit.
• Ignoring crossing propagation. A forced low digit on a stem still has to satisfy its row and box — leaving those strikes out guarantees a later contradiction.
• Skipping the 45-check. The extra line constraint makes row/column conflicts easier to miss, not harder. Audit every finished house.

9. Train the Position-Bound Eye

Start on Easy boards and force yourself to write every margin bound before placing a single digit — no skipping ahead. Then climb through Medium, Hard, and Expert, verifying each grid with the 45-check. The strategy index lists the classic reduction moves you will still need on the densest stems.

When positional range logic feels automatic, arrow Sudoku — where a circled digit equals the sum of its shaft — adds a combinatory layer on top of the same directional thinking. That is the natural next stop in the series.

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