The half-finished grid problem

The classic hard-Sudoku stuck moment: you've placed maybe forty digits, the grid is two-thirds full, and the rest won't budge. Every empty cell still has three or four candidates, none of the obvious moves are working, and the gap between where you are and a finished puzzle feels much larger than the actual number of empty cells suggests.

This is the half-finished-grid problem, and it has a small handful of specific causes. Diagnosing which one you've hit is most of the work; once the cause is named, the move usually follows.

Why the second half is harder than the first

The first half of a hard Sudoku is mostly carried by naked and hidden singles, with a handful of pairs. These moves are the same ones easy and medium puzzles use, just slightly thinner on the ground. By the time you've placed thirty or forty digits, you've used all the moves your easy-puzzle toolkit gave you, and the rest of the grid is asking for techniques the easier puzzles never required.

The second half is where mid-level techniques live: naked and hidden pairs and triples past their obvious appearances, pointing pairs, box-line interactions, X-wings, and increasingly the chains and uniqueness patterns of expert-tier solving. The pause at sixty per cent isn't because the puzzle has gotten arbitrarily harder; it's because you've crossed the threshold where the easy-tier moves stop sufficing.

The honest framing: the second half is a mid-level puzzle, and you're tackling it after already solving an easy puzzle's worth of moves. That's why it feels so much harder than your remaining work suggests.

The diagnostic tree

When you're stuck on a half-finished grid, walk through these in order.

Have you missed a hidden single in the units affected by your last few placements? Every digit you place creates new constraints in the affected row, column, and box; those constraints often surface a hidden single in the surrounding cluster. Beginners and intermediates frequently miss these because they're scanning ahead instead of re-scanning the units the placement just changed. Re-scan deliberately; the hidden single is often visible immediately.

Are there naked or hidden pairs in the most-pencil-marked region? The cluster with the most candidates is usually where the next pattern lives. Read the marks as a shape rather than as a list. Two cells sharing exactly two candidates is a naked pair; two digits sharing exactly two candidate-cells is a hidden pair. Either one eliminates candidates from the surrounding cells, often producing several singles in a chain.

Are there pointing pairs? When two of a digit's candidate-cells in a single box sit in the same row or column, the digit can't appear elsewhere in that row or column outside the box. Pointing pairs are slightly harder to spot than naked pairs but live in the same neighbourhood; the same scan finds them.

Is there an X-wing? Probably not on a medium puzzle, often on a hard or expert. Look for a digit that has exactly two candidate-cells in two different rows, and check whether those candidate-cells share two columns. Same logic rotated for column-based X-wings.

Have you made an error? If none of the above produces a move and you've been at it for ten minutes, consider that a wrong placement somewhere earlier may have made the grid unsolvable. Scan every placed digit for conflicts with its row, column, and box. Check pencil marks for stale candidates that should have been removed. If your interface has an error-check button, this is the right moment.

What not to do

Don't guess. The half-finished grid is the canonical place where the temptation to guess is strongest, and it's also the place where guessing does the most damage — a wrong guess thirty placements deep cascades into a tangle that's harder to unwind than the whole rest of the puzzle. Our piece on what to do instead of guessing covers the longer answer.

Don't restart unless you've confirmed an error. Restarting a hard puzzle because the second half feels stuck wastes the work of the first half and trains a "stuck = restart" habit that makes future hard puzzles harder. The diagnostic above almost always finds the move.

Don't switch to an easier puzzle as a shortcut. If you're genuinely stuck and have walked through the diagnostic, an easier puzzle might be the right call for the day. But not as a flinch response to the first stuck moment — only after the diagnostic has run and produced nothing.

When the diagnostic doesn't find anything

Occasionally — maybe one puzzle in twenty — the diagnostic finds no move and the grid genuinely needs a technique you don't have yet. That's a valid outcome. Take note of where the puzzle stopped, look up the next move (most digital interfaces will tell you, or you can come back to the puzzle after learning the technique elsewhere), and add the technique to your toolkit.

The pattern that matters: most half-finished grids are stuck because of one of the standard mid-level techniques applied somewhere you haven't scanned, not because of an exotic technique you've never heard of. Trust the diagnostic. Most of the time it's right, and the move is closer than the stuck feels.