Scanning

Start With the Fullest Numbers

Not all nine digits are equal at any moment in a solve. The ones already scattered most across the board are the easiest to finish — so those are the ones to chase first.

Why the count matters

Every time a digit appears on the grid, it closes a row, a column and a box to itself. A number placed six times has already sealed off most of the board, so its three remaining cells are boxed in by constraints from every side. A number placed only twice, by contrast, still has the run of the grid and almost nothing pinning it down. Chasing the well-represented digit is simply chasing the easier win.

How to use it

Before you start sweeping, glance across the board and note which one or two numbers appear most often. Begin your hidden-single hunt there. Take that digit box by box: in each box still missing it, strike out the rows and columns that already carry it, and watch a single empty cell survive far more often than it would for a sparse number. Place it, and its count climbs again, which frequently makes the very next-fullest digit solvable too.

The snowball effect

This is why the tip pays off out of proportion to how obvious it sounds. Finishing off a nearly complete number does not just fill a few cells — it tightens every unit those cells belong to, which pushes the next digit closer to completion. Work the fullest numbers in a rolling order and the grid snowballs: each digit you close brings another to the brink, and placements arrive in a satisfying chain instead of one hard-won cell at a time.

When every number looks sparse

Early on a hard board, no digit may appear more than two or three times, and this tip seems to offer little. It still does. Even a slight edge — the number with three placements rather than two — is the better bet, because it has more lines already closed. And the moment you make a few placements, some digit will pull ahead of the pack and become your new anchor. The technique is not only an opening move; it is a compass you re-check every time the board changes.

A quick example

Say the 8 already sits in six boxes and the 2 in only two. Hunting the 2 means wading through a grid where it could still go almost anywhere. Hunting the 8 means visiting just three boxes, each so hemmed in by existing 8s that a single cell usually remains. You place two or three 8s in the time it would take to rule out one cell for the 2 — and each 8 you place chips away at the 2's freedom for later, when it is finally worth your attention. Multiply that small saving across a whole board and it becomes the difference between a solve that flows and one that grinds. Every cell you place from the fullest number tightens three more units, and those tightened units are where your next easy digit will surface — so the order you work in quietly decides how much of the grid you get almost for free.

Fold it into your scan

Treat the count as the running order for your systematic scan: fullest digit first, sparsest last, re-sorting as placements land. It costs nothing and steers your effort toward the cells most likely to give. Try it on Medium Sudoku, where a handful of well-populated numbers usually carries the opening, then keep the habit on the classic grid at Sudoku AZ.