Scanning and cross-hatching

What it is

Scanning is the habit of sweeping your eyes across the grid looking for places where a digit is forced. The best-known form is called cross-hatching. You pick one digit and one box, then imagine drawing lines through every row and column that already contains that digit. The lines “hatch out” cells in the box. If only one empty cell survives, the digit belongs there.

It is called cross-hatching because the imaginary lines cross each other, like the shading on an old engraving.

Scanning needs no notes and no pencil. On Easy puzzles you can often solve the whole grid this way.

How to spot it

Cross-hatching works best on boxes where:

• the digit you are looking for is missing, and
• the same digit appears in the neighbouring boxes, in the same band of rows or the same stack of columns.

The boxes come in groups of three. Boxes 1, 2 and 3 share rows 1 to 3. Boxes 1, 4 and 7 share columns 1 to 3. If two boxes in a row of boxes already have a 4, the third box must place its 4 in the one row the other two have not used. That alone narrows things down to three cells, and the columns usually finish the job.

A good routine is to take the digits 1 to 9 in turn and, for each digit, visit every box that is missing it. With practice this becomes quick.

A worked example

Let us look for a 7 in box 1, which covers rows 1 to 3 and columns 1 to 3.

First, scan across the rows. There is a 7 in row 1, at column 6 (in box 2). There is also a 7 in row 2, at column 8 (in box 3). So box 1 cannot have its 7 in row 1 or row 2. It must be in row 3.

Now look at the three cells of row 3 inside box 1:

• Row 3, column 1 is already filled with a 4.
• Row 3, column 2 is empty, but column 2 has a 7 further down, in row 8.
• Row 3, column 3 is empty, and column 3 has no 7 anywhere.

Only row 3, column 3 survives. The 7 goes there.

Notice how the two 7s in boxes 2 and 3 did most of the work. Once you have placed this 7, look again at boxes 4 and 7. The new 7 in column 3 might be just what you need to place a 7 in one of them.

Scanning a row or column

You can also scan a single row or column. Pick a row with only a few empty cells and ask, for each missing digit, where it could go. If row 6 is missing a 3 and three of its four empty cells are blocked by a 3 in their column or box, the fourth cell must hold the 3. This is the same reasoning as cross-hatching, just aimed at a line instead of a box.

A common mistake

People often scan only the rows, or only the columns. Cross-hatching needs both. In the example above, the rows narrowed the 7 down to row 3, but it was the column check that picked out the exact cell. If you stop halfway, you will miss placements that are sitting right in front of you.

The other trap is not going back. Each digit you place can open up new placements nearby. After every placement, glance at the boxes that share its row and column.

When you will need it

Scanning is the backbone of every Easy puzzle on sudoku.ie, and it stays useful at every level. Even on an Expert Saturday, most of the cells are filled by scanning. The advanced techniques only step in when scanning runs dry. When it does, you are ready for pencil marks and hidden singles.