Templates

Templates is a close cousin of Pattern Overlay Method that uses the same enumerated placement patterns but with one extra step. Where POM unions all patterns and eliminates cells absent from the union, Templates examines pairs of templates — one for each of two different digits — and looks for incompatibilities. If digit a placed by template T1 conflicts with every possible placement of digit b under any of b's surviving templates, T1 itself is impossible and can be eliminated from the search space, which in turn shrinks the union for digit a.

How it differs from Pattern Overlay

POM works one digit at a time. Each digit's elimination set is computed from its own template enumeration alone. Templates introduces cross-digit reasoning: a template for digit a is invalid if it places a in cells where every template for digit b would have to place b conflictingly. Many template-pairs that are individually consistent for their own digit fail when combined.

The procedure: for each pair (a, b) of digits, for each template T_a of digit a, check whether at least one template T_b of digit b is compatible (no two cells in the same row, column, or box hold both a and b at conflicting positions). If no compatible T_b exists, T_a is eliminated from the search. Repeat across all digit pairs, iterating until convergence.

Why it's heavier than POM

The combinatorics get expensive fast. POM does at most 9 enumerations; Templates does up to 36 pairwise checks (the number of digit pairs), each of which compares every template of one digit against every template of another. On a hard puzzle with several hundred templates per digit, Templates can fire when POM doesn't — but the runtime cost is several orders of magnitude higher.

Most software solvers don't run Templates by default. They reserve it for the small set of puzzles where POM exhausts itself and a brute-force solver would otherwise be the next step. Manual solvers don't run Templates at all — the pairwise enumeration is impractical even for software, let alone by hand.

The relationship to advanced chain reasoning

Templates' cross-digit incompatibility check is conceptually similar to certain AIC patterns that span multiple digits, and to the cross-digit colour propagation in 3D Medusa. The difference is the level of abstraction: AIC and Medusa work at the level of named patterns and chain links, while Templates works at the level of full grid placements. In theory the same eliminations are reachable through cleaner chain logic; in practice some are easier to find via enumeration than via pattern recognition.

For most solvers, the Templates entry is here for completeness more than as a daily tool. Its cousin POM is the more commonly cited technique of the two.