Thermometers

Mercury only rises from the bulb

Every thermometer on the grid is a run of cells with a round bulb at one end, and you fill it the way a real one works: mercury starts at the bulb and climbs toward the stem in one unbroken column. It can't skip a cell, and it can't float in the middle with a gap beneath it. That fixed direction is the key to the entire puzzle. Know a cell partway up is filled, and every cell between it and the bulb must be filled too. Know a cell near the bulb is dry, and everything past it toward the stem is dry as well.

Start with the giveaways

The numbers around the edge say how much mercury each line holds: the left number counts filled cells across its row, the top number counts them down its column. You solve by choosing how far each thermometer fills so that every count comes out right at once. Start with the freebies. A line marked 0 is completely empty, and a line whose number equals its length is completely full, and both send information racing down every thermometer they cross. From there, lean on the bulb direction. The moment a cell beside a bulb is proven empty, the whole stem above it empties; the moment a stem-end cell is filled, the run back to the bulb fills.

A row worth working through

Here's the kind of reasoning the puzzle runs on. Picture a row of five cells marked 4, holding two thermometers laid end to end: a three-cell one with its bulb at the left edge, and a two-cell one with its bulb at the right edge. The row needs four filled cells between them. Run the cases. The short thermometer holds at most 2, so the long one must hold at least 2. The long one holds at most 3, so the short one must hold at least 1. That's already three certainties: the long thermometer's bulb and second cell are filled, and so is the short thermometer's bulb. Only one question remains, whether the last filled cell is the long one's tip or the short one's second cell, and a glance down those two columns usually settles it. No guessing happened. You squeezed the row between its minimums and maximums, and the clue did the rest. Nearly every hard step in Thermometers is this same squeeze wearing different clothes.

Habits of fast solvers

Alternate your scans: a pass across the rows, then a pass down the columns, because every cell you settle changes the arithmetic in the crossing line. Re-check a line the moment it reaches its number, since everything else in that line is instantly empty, and those empties drain whole thermometers toward their tips. Watch the near-full lines too: a row of six marked 5 has a single empty cell, so the moment any one cell in it is proven empty, every other cell in the row fills. The mistakes are just as predictable. Treating a clue as a minimum instead of an exact count. Filling from the tip because it looked convenient, which the bulb rule forbids. Pushing on after a wrong assumption instead of backing up, which turns one small error into a painted-over board. The clicks are cheap: click a cell to raise the mercury to it, click a filled cell to pull the level back down, click the bulb to drain the tube. Being wrong costs nothing but seconds on the timer.

All deduction, no dice

There's never a need to guess. Because each thermometer fills from one fixed end only, a single decision about its level can cascade across the grid, and reading a crowded row against a crowded column almost always shakes the next cell loose. Watch for thermometers whose bulb sits in a nearly-full line; they often have to fill to the brim, since any gap would fall short of the count. Thermometers belongs to the modern family of edge-count puzzles alongside Battleships and Tents: numbers on the border, objects in the grid, one exact arrangement that satisfies everything. Fill every thermometer to satisfy the numbers and the board is done, then go again for a faster time.