Mathematical formulation of a resource-collection board game.
4 dimensions · 20 cards · 56 objects · 125 paths per loop
Each card is a vector y in {0,1,2,3}^4 with sum in {2,3,4,5}. There are exactly 101 feasible cards. Our deck selects 20 that balance coverage, spread, and isotropy. With 4 dimensions, we can visualize directly: Emerald/Ruby/Sapphire as 3D axes, Obsidian as color (dark → bright).
Select a tier then hit Random Walk to simulate a loop. The gain vector builds up as 3 cards are collected — if it dominates the object cost (g ≥ c componentwise), you can afford it.
Each loop: 15 of 20 cards are dealt into a 5×3 grid (5 locations, 3 days). The player chooses one location per day, yielding 5^3 = 125 possible paths. The gain is the sum of 3 collected cards in R^4.
The gain vector follows a discrete distribution from the Minkowski sum of 3 day-slices. Analytical moments: E[g] = 3y̅ and Σ_g = (51/19)Σ_Y. The distribution is far from Gaussian.
The survival function S(c) = P(g ≥ c) determines object rarity. With information, players optimize their path: S_1(c) uses day-1 knowledge, S_2(c) uses day-1+2. The power law S_k ≈ S_0^γ_k links all three.
On a log-log scale, S_k vs S_0 is remarkably linear. This means designing objects requires only calibrating P(random) — the informed probabilities follow automatically via the exponents γ_1 ≈ 0.82 and γ_2 ≈ 0.61.
Each object's cost decomposes into per-dimension rarity contributions. The total rarity budget R(c) = -log(α) determines the tier. Generalists spread budget evenly; specialists concentrate it.