Number Loom: when a box reserves a number for one row
A six-by-six teaching board shows how two possible positions inside one box can rule out an entry elsewhere, even before either position is solved.
Worked examples use the Standard board unless a different size is stated. Hard and Expert may change the board dimensions or amount of information to track.
A useful restriction need not place a number
Sometimes Number Loom offers no immediately forced entry, but a group of candidate positions still tells you something. If every possible location for a number inside one outlined box lies in the same row, that number must occupy that row inside the box. It cannot also appear elsewhere in the row, even though you do not know which of the box’s squares will receive it.
This is a relationship between two groups, rather than a guess about one square. It works because each box and each row must contain the number exactly once. Check every empty square in the box before using the restriction. Missing a third candidate in another row would invalidate the conclusion.
Read a concrete six-by-six position
Here is a separate Hard-size teaching board, with 0 marking an empty square and slashes separating rows: 0 0 3 4 5 0 / 0 5 0 1 2 0 / 2 3 0 0 0 0 / 0 0 0 2 3 4 / 0 0 0 6 0 0 / 0 1 2 3 4 0. Boxes are two rows tall and three columns wide, as in Hard mode.
Inspect the box covering rows 3 and 4, columns 4 through 6. Its fourth-row squares already contain 2, 3 and 4. In row 3, column 4 cannot take 1 because column 4 already contains 1 in row 2. The only possible positions for 1 within this box are therefore row 3 column 5 and row 3 column 6.
Carry the restriction outside the box
Both surviving positions lie in row 3. That row’s 1 must consequently be inside this right-hand box. Row 3 column 3, which sits in the neighbouring box, cannot be 1. Before considering the box relationship, that square’s existing row, column and box values would have allowed 1 as a local candidate.
Nothing here tells you whether column 5 or column 6 gets the 1. Do not fill either merely to express the deduction. Instead, keep the eliminated candidate in your reasoning and reconsider the other numbers possible in row 3 column 3. Another constraint may now force an entry, or you may need to inspect a different group first.
Separate candidate work from board entries
This implementation has a number keypad and Clear, rather than a pencil-mark mode. Candidate lists can be held mentally or written separately on paper. Filling a tentative number is an actual board entry, not a private annotation. Undo restores the previous entry, while a duplicate marker only reports a conflict already present in a row, column or box.
The same box-to-line reasoning applies to a column when all surviving box positions share that column. Expert’s boxes are three by three instead of two by three, so identify their boundaries again rather than carrying Hard’s geometry across unchanged. A useful deduction removes a possibility for a stated reason; it need not immediately produce a finished row.
An original guide for this site's implementation. Found an unclear step? Let us know.