Difficulty is partly human and contextual, but a Competition catalog needs a repeatable ordering. Wang World therefore defines Complexity from a specified solver rather than from file size, number of colors, or one player’s completion time.
The forced-mrv-v2 model
The solver maintains candidates for every unfilled decision group. It repeatedly selects the position with the fewest legal options—the minimum remaining values heuristic. If only one option remains, that placement is forced and costs zero. If several options remain, the solver explores them in a deterministic seeded order.
What is counted?
Complexity counts only branches that end in contradiction and require backtracking. Routine candidate checks and forced propagation are free. The measurement is averaged across deterministic seeds and normalized with a fixed scale. The same puzzle definition always receives the same value under the same algorithm version.
Rules change the search space
Rotations create oriented candidates but share prototype inventory. Linked positions form one atomic decision and consume as many copies as the group has cells. Walls remove cells while adding edge constraints. Wildcards create broadly compatible but scarce candidates. Shuffle fixes the complete multiset. These effects are included directly, not added later as bonuses.
How to interpret the number
A value near zero means propagation reaches a solution with little or no failed branching. A larger value means the solver repeatedly follows plausible choices into contradictions. Two puzzles with similar Complexity can still feel different: one may demand a deep insight, another many small inventory deductions.
From Complexity to player Score
Competition Score rewards solving difficult puzzles quickly: Wang World divides Complexity by elapsed minutes and rounds the result. Complexity is a property of the puzzle and algorithm; Score combines that property with one timed performance.
See Competition for timing and leaderboard rules, or the generation pipeline for how candidates reach the catalog.