Factorio's Space Age expansion introduced quality tiers — five levels from normal to legendary — that turn every crafting chain into a probabilistic pipeline. Quality modules give each craft a chance to bump output up a tier, but the odds of hitting legendary in one shot are vanishingly small (0.0248% at max). The intended design answer is upcycling loops: craft, check quality, recycle the losers at 25% ingredient recovery, repeat until you get what you want. Planning these loops by intuition is somewhere between difficult and impossible. Simon Sapin's insight is that the entire quality system is a matrix multiplication problem. Each crafting step with quality modules can be represented as a 5×5 transition matrix, where rows are input quality tiers and columns are output probabilities. Chain multiple steps — mining ore, smelting plates, assembling products — and you simply multiply the matrices together. The output is a probability vector telling you exactly what fraction of your production lands at each tier. The transition matrix Tquality(q) is upper triangular with a clean structure: the diagonal holds (1-q), the superdiagonal holds 9q/10, and each subsequent entry shrinks by a factor of 10, with the final column catching the remainder. At q=0 you get the identity matrix — no quality change — which is a satisfying edge case that confirms the formalism works. This is textbook Markov chain theory applied to a video game, and it's the right tool. The article walks through three quality strategies in increasing complexity. "Gambling" is the simplest: craft once with quality modules and keep what you get, no recycling. "Washing" loops self-recycling items (like ore) through the recycler with quality modules until they either reach the desired tier or are destroyed. The recycler's 75% destruction rate per pass means most items die, but the survivors climb. The math for washing involves a geometric series of matrix powers — each pass through the recycler is another multiplication by the transition matrix scaled by the 25% survival rate. What makes this more than a math exercise is the tool Sapin built on top of it. Existing Factorio planners like Factoriolab struggle with the looping nature of quality upcycling because the feedback makes simple ratio calculations blow up. Matrix methods handle loops natively — the steady-state distribution falls out of the algebra. The calculator lets players input their module configuration and recipe chain, and get back exact machine counts and yield expectations. The article is unfinished — it cuts off mid-explanation of the washing loop formalism — but what's here is a clean, well-motivated derivation that any player with linear algebra basics can follow. The progression from "probabilities as scalars" to "probabilities as vectors" to "crafting steps as matrix multiplication" is pedagogically effective. Sapin is teaching you to see factory games as applied math, which is exactly how a certain kind of player already experiences them. This sits in a long tradition of players building external tools that become essential infrastructure for their games. EVE Online's market spreadsheets, Kerbal Space Program's delta-v calculators, Dwarf Fortress's Dwarf Therapist — the pattern is always the same: a game's internal complexity exceeds what intuition can handle, and someone with the right technical background builds the bridge. Sapin's contribution is notable for how cleanly it maps game mechanics to a well-understood mathematical framework, making the solution both exact and extensible.
