Mathematics

← Login Counter | Series Overview | Sieve of Eratosthenes → The Problem: Computing Pi by Throwing Darts Imagine a square dartboard with a circle inscribed inside it. Throw random darts at the square. The ratio of darts landing inside the circle to total darts thrown approaches π/4. Why? Mathematics: Square side length: 2 (from -1 to 1) Square area: 4 Circle radius: 1 Circle area: π × 1² = π Ratio: π/4 Throw 1 million darts, multiply by 4, and you’ve estimated π. More darts = better estimate. This is Monte Carlo simulation: using randomness to solve deterministic problems. ...

← Mandelbrot Set | Series Overview The Problem: The Simplest Unsolved Math Problem The Collatz conjecture (3n+1 problem) is deceptively simple: Start with any positive integer n If n is even: divide by 2 If n is odd: multiply by 3 and add 1 Repeat until you reach 1 The conjecture: Every positive integer eventually reaches 1. Example (n=12): 12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 The mystery: This has been verified for numbers up to 2^68, but never proven. It’s one of mathematics’ most famous unsolved problems. ...

← Sieve of Eratosthenes | Series Overview | Collatz Explorer → The Problem: Rendering Fractals in Parallel The Mandelbrot set is defined by a simple iterative formula: Start with z = 0 Repeatedly compute z = z² + c If |z| exceeds 2, the point escapes (not in the set) Color each pixel by iteration count The beauty: Each pixel is completely independent. Perfect for parallelism! The challenge: Some pixels escape in 5 iterations, others take 1000+. This creates load imbalance—some workers finish instantly while others grind away. ...

← Monte Carlo Pi | Series Overview | Mandelbrot Set → The Problem: Finding Primes with Filters The Sieve of Eratosthenes is an ancient algorithm for finding prime numbers. The concurrent version creates a pipeline of filters: each prime spawns a goroutine that filters out its multiples. The Algorithm: Generate sequence: 2, 3, 4, 5, 6, 7, 8, 9, 10, … Take first number (2), it’s prime, filter all multiples of 2 Take next number (3), it’s prime, filter all multiples of 3 Take next number (5), it’s prime, filter all multiples of 5 Repeat until desired count The Beauty: Each prime creates its own filter. Numbers flow through a pipeline of increasingly selective filters. What passes through all filters must be prime. ...