Where to look first on a fresh grid

A fresh Sudoku grid is mostly empty, and the first move on a fresh grid is rarely random. The strongest openings come from a small handful of habits — looking for specific things in a specific order, before you've placed a single digit. Most beginners waste the first two minutes scanning at random and find nothing; an experienced solver finds three or four placements in the same time, just by knowing where to look.

If you've internalised the naked and hidden singles, this piece is the next step: the discipline of the opening scan, the moves that come before any of the moves you've practised.

Count the digits

Look at the givens — the digits already placed when the puzzle starts. Count which digit appears most often. On most easy puzzles, one or two digits will already have five, six, or seven placements; the others might have only two or three.

The digit with the most placements is your first target. The reason is constraint density. Each placement of a digit eliminates an entire row, column, and box from being a possible home for the digit elsewhere. A digit that's been placed seven times has only two empty cells left where it could legally go. Often, both of those cells are constrained enough that you can find the seventh and eighth placements with a quick scan of the remaining boxes. Place those, and now another digit moves to the top of the count.

This is the hidden-single technique applied at the digit level rather than the unit level. Same logic, faster yields on a fresh grid.

Find the densely-given regions

Some boxes start with five or six givens; some start with one or two. The boxes with the most givens are the cheapest to solve early, because every empty cell in them has only three or four candidates instead of seven or eight.

A useful first scan: find the most-given box, then check each of its empty cells for naked singles. Then do the same for the second-most-given box. By the time you've worked through three or four heavily-constrained boxes, the grid has often shifted enough that the lighter boxes have become solvable too.

The same logic applies to rows and columns. A row with seven givens has two empty cells that probably each have one or two candidates — easy naked singles waiting to be found. The row scan is sometimes faster than the box scan on puzzles where the givens cluster horizontally.

The "where can this digit go" sweep

Pick a digit that's been placed three or four times. For each row of three boxes — boxes 1-2-3, then 4-5-6, then 7-8-9 — check whether the digit appears in two of the three boxes. If it does, the third box must contain the digit in the row that the existing two placements don't cover. That narrows the digit's home in the third box to one of three cells, and a quick check of the rest of the box's constraints often picks the exact one.

A worked example: suppose the digit 5 is placed in box 1 (in row 1) and box 3 (in row 3). It must appear in box 2, but not in row 1 (already in box 1) and not in row 3 (already in box 3). So in box 2 it must be in row 2. That narrows the placement to three cells, and any already-placed digit in row 2 of box 2 narrows it further. Often that's enough to pick the exact cell on the first pass.

This is the box-line interaction trick — no pencil marks, just visual scanning of three boxes at a time. It catches a lot of opening moves that single-cell scanning misses.

When to stop scanning and start placing

The opening scan isn't free. After about a minute of staring at the same grid, you start re-deriving the same observations and the marginal yield drops to zero. Place what you've found, then re-scan the grid in its new state — every placement opens new constraints somewhere else.

A useful rhythm: thirty to sixty seconds of scanning, place what you find, repeat. By the third or fourth scan-place cycle on an easy puzzle, you're usually past 50% solved and the rest follows easily.

The opening scan is the cheapest move in your toolkit. It costs nothing in pencil marks, sharpens the eye for harder patterns later, and produces placements that would otherwise require half a minute of pencil-marking to find. Most experienced solvers do this so habitually they don't notice they're doing it. Once you've practised the discipline, you won't either.