Dual-transversal cover

How to use it

This is transversal's Dual cover pass — line cover run on a pair's coupled letter and digit tables at once.

Pick a leftover pair, say F1, and draw its two small cover tables: the cells of letter F (one per row and column) and the cells of digit 1 (one per row and column). The tables meet in exactly one entry — the F1 cell.

In each table, a line with a single coverer locks that entry; an entry with a single cell takes its row and column from the others. A cell forced in one table cannot serve a different entry in the other. A proposed F1 home that leaves either table with a homeless line is impossible: block it. It is line cover run twice, on the pair's two faces at once — cheap to check, and it sees separations that the single-face version misses.

Why it is sound

Fix a candidate pair Ld. The cells of letter L and of digit d are both transversals, and their unique intersection is exactly the Ld cell. In their two small cover tables, a row or column with one coverer locks that entry, and an entry with one cell takes its row and column. A forced cell in one table is unavailable to a different entry in the other. Thus if a proposed Ld home makes either table homeless, that home is impossible. The procedure is a deterministic cover chain, not an enumeration of transversals.