Exhaustive transversal count

How to use it

The last resort within transversal's Symbol tables pass: the exact verdict from checking every possible completion of the table, reached only when the cheaper checks above stay silent.

Whole-table version of the covering check. For one letter, its unplaced pairs must land one per still-free row and one per still-free column — a transversal. List the letter's leftover pairs and their homes, then enumerate the ways to complete the table (rows × columns, each pair in an admissible cell). A placement appearing in no completion is impossible; a pair with a single possible home is forced.

Work the most-constrained symbol first — the one with the fewest completions — and check digits as well as letters. Each table is small enough to enumerate directly when the symbol is half-placed; between tables, ordinary techniques consume whatever the blocks freed.

Why it is sound

By the transversal lemma, in every valid solution the cells of a fixed letter form a permutation matrix (one cell per row and per column) carrying every digit exactly once. Hence the not-yet-placed pairs must occupy cells in pairwise distinct rows and columns, each cell admitting its pair. A placement supported by no such assignment occurs in no valid solution, so blocking it removes only impossible configurations. The digit side is identical. The technique is sound but incomplete: past a certain table size, checking every completion stops being practical, so it's skipped.