Research Notes

Combinatorial Design
& Metaheuristic Optimization

Exploratory articles on covering array theory, Bat-Lévy algorithms, and Monte Carlo methods — bridging discrete mathematics with accessible, real-world applications.

Featured Articles
C(v,k,t)
Theory • 8 min read

What Is a Covering Design C(v,k,t)? A Clear Introduction

From finite sets to block designs — understand the mathematical foundation behind covering arrays. We break down v (pool size), k (block size), and t (coverage strength) with concrete numerical examples and explain why this problem sits at the heart of combinatorial design theory.

Key topics: Steiner systems, block design, covering arrays, combinatorial optimization
Algorithms • 10 min read

How the Bat Algorithm Works: Echolocation Meets Combinatorial Optimization

Nature-inspired metaheuristics explained: echolocation pulses, frequency tuning, and Lévy flight random walks. Discover how the discrete Bat Algorithm adapts biological principles to tackle NP-hard covering problems — balancing exploration and exploitation for near-optimal solutions.

Key topics: Bat Algorithm, Lévy flight, metaheuristics, stochastic search, NP-hard
Simulation • 7 min read

Monte Carlo Verification: Testing Mathematical Guarantees Empirically

How do you verify a structural guarantee? Monte Carlo simulation with Fisher-Yates shuffle provides unbiased statistical evidence. Learn how 100,000 random draws confirm that a covering set truly eliminates 0-match and 1-match outcomes — and how to interpret match distribution tables.

Key topics: Monte Carlo, Fisher-Yates, statistical verification, probability tables
NP
Complexity • 6 min read

NP-Hard in Plain English: Why the Coverage Problem Defies Brute Force

C(49,6) = 13,983,816 combinations. Exhaustive search is impossible. Understand why covering design belongs to the class of NP-hard problems, what computational complexity really means, and why metaheuristics are not just clever — they're necessary.

Key topics: NP-hard, computational complexity, exhaustive search, P vs NP
5/34→90
Application • 9 min read

Pick 5 Lottery Coverage: From 5/34 to 5/90 — How Many Blocks for ≥2 Guarantee?

A practical walkthrough across pool sizes. See how block count scales from small (v=20) to large (v=90) pools for Pick 5, Pick 6, and Pick 7 systems. Compare Greedy vs Bat Algorithm performance and understand the cost-coverage trade-off with real generated examples.

Key topics: Pick 5, Pick 6, Pick 7, pool size scaling, Greedy vs Bat comparison
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More Articles Coming

New research notes on combinatorial design, metaheuristic optimization, and Monte Carlo methods are in preparation. Topics include: Lévy flight deep-dive, Steiner systems and block design history, Fisher-Yates shuffle analysis, and comparing Greedy vs metaheuristic approaches.

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