How to solve large thermometer puzzles

What actually changes on big boards

We measured a sample of puzzles from our own generator. The last column shows how much of each board can be filled using only one-line-at-a-time reasoning (the method from the squeeze and the ceiling) before you need to combine lines.

Measured on 6 puzzles per row; figures vary from puzzle to puzzle.

Two lessons follow from this. First, single-line reasoning gets you a long way, but on the biggest boards it often stalls before the end. Second, there are a lot of thermometers to keep track of, so the challenge is as much organisation as insight.

Phase 1: The opening sweep

Go once through every row, then every column, applying the five opening moves: extreme clues, the overlap rule and small-clue ceilings. Don't follow chains yet. Just mark what each line gives and keep moving. This gives you a scatter of certain cells across the whole board, which tells you where the puzzle is "soft".

Phase 2: Work in regions

Pick the area with the most certain cells, usually a corner or an edge, and finish it before moving on. Regions near the edge are easier because edge thermometers have fewer neighbours. Working locally keeps the counts you need in your short-term memory instead of spread across the board.

Phase 3: Keep a worklist

Every time a cell is decided, its row and column (and, through flow, possibly other lines) change. Keep those lines on a mental "to check" list and process them one by one. This is how our solver works, and it is far more efficient than re-reading all 30 lines of a 15×15 board after each move.

Phase 4: When single lines run dry, combine them

If no line gives anything on its own, look at two neighbouring lines together. Add their clues, then work out the least and most mercury that the thermometers covering both lines can supply. A thermometer lying across both lines is counted once, with its combined contribution. This is the bridging technique, and on large curved puzzles it is often the key step.

If that fails, use a careful supposition: choose a cell with an obvious consequence, assume it, and follow it through until a line breaks. Undo back to that point and mark the opposite value. Our puzzles never need deeper guessing than this.

Use the tools

• Show remaining counts (Settings) turns each clue into a countdown, which removes most of the arithmetic.
• Highlight rule breaks flags gaps and over-full lines the moment they appear. A slip on a 15×15 board can otherwise cost twenty minutes.
• Click a clue to finish a line that is already complete or has exactly enough open cells.
• Pause and come back: progress, time and hints are saved automatically in your browser.
• Smart hints name the cell and the lines that decide it, so a hint teaches you something rather than just giving the answer.

Questions