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Puzzle Quarry

Counting a Line to Its Limit

The first technique that looks at a whole line instead of a neighbourhood — and the first one that can fill five cells from a single observation.

The quota, and what happens when it runs out

Every line holds exactly half of each symbol: three each in a 6×6, four each in an 8×8, seven each in a 14×14. The moment a line contains that many of one symbol, it has no more of that symbol to give, and every empty cell remaining in it must take the other. One count, and the line is finished.

This is a different shape of reasoning from the pair techniques and it is worth noticing the difference rather than absorbing it silently. A double looks at two adjacent cells and ignores everything else in the line; counting ignores adjacency completely and looks only at totals. Solvers who never make that switch stall at exactly the point medium puzzles begin.

The payoff scales with size. On a 6×6 a count typically places one or two cells; on a 14×14 it can place five or six at once, and each of those creates fresh pairs for the cheap techniques to work on. That is why the rhythm is count once, then sweep — not count, count, count.

Before — row 1 already holds three 1s
123456
11011
20
30
41
51
60
After — the rest of the row is decided
123456
1001011
20
30
41
51
60
Row 1 reads _ _ 1 0 1 1. Three of the six cells in a 6×6 row must be 1s, and three 1s are already there, so the row has none left to give: both empty cells take a 0. Note that no pair or gap anywhere on this board would have found it — the count is the only route.

cells the reasoning read the cell it decides what it wrote

When to spend the effort

Not while pairs remain. Counting a fourteen-cell line is real work compared with spotting two neighbours, and doing it early means counting lines that a free sweep was about to change anyway. The signal that it is time is a complete pass — every row for touching pairs, every column, then both again for gapped pairs — that places nothing at all.

When you do count, count everything rather than the line that looks promising. A line one symbol short of its quota is visually indistinguishable from a line three short until you tally it, and the tally on six or eight cells takes a second. Guessing which line to count is how the technique gets a reputation for being tedious.

The near-quota case is worth watching too. A line with its quota all but reached is not yet decided, but it is one placement away from being decided, so a symbol dropped into it by some other deduction will very often finish it immediately. Noting which lines are close is the cheapest form of planning this puzzle offers.

Why it is exactly one level up

A medium grid on this site is one the deduction solver finishes using pairs and counting and nothing else. That is not an approximation of difficulty, it is the definition: the solver applies the cheapest technique available at every point, and if the hardest rung it ever needed was counting, the grid is medium.

Which means a medium grid is guaranteed to contain at least one moment where the pairs genuinely run out. If yours has not stalled yet, you have not reached that moment — keep sweeping, because a count taken early is a count wasted.

Where this sits on the ladder

Common questions

What is the counting technique in a binary puzzle?

Tallying one line against its quota. Each line holds exactly half of each symbol, so once half the cells hold one symbol, every remaining empty cell in that line must take the other.

When should I start counting rather than looking for pairs?

After a complete sweep for both touching and gapped pairs places nothing. Counting costs more attention per line, and counting a line that a free sweep was about to change is wasted work.

Does counting work on columns as well as rows?

Identically. The quota rule applies to every line in both directions, and columns are the ones players forget to tally — a column count is often the move that restarts a stalled grid.

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