Abhimanue Udayan

🐂 What is the Archimedes' Cattle Problem?

Archimedes posed this riddle over 2,000 years ago, asking mathematicians to determine how many white, black, dappled, and brown cattle Zeus owned under a labyrinth of conditions.

The final constraint turns it into a colossal Pell–Fermat equation whose smallest solution contains 206,545 digits—far beyond pencil-and-paper mathematics.

🧠 How I approached it

I built a custom C++ engine inspired by Bhāskara II’s Chakravāla method, optimized for arbitrary-precision arithmetic and modular factorization.

• Generated continued-fraction expansions to lock onto the minimal Pell solution
• Vectorized the arithmetic to squeeze maximum performance from a desktop i7
• Logged intermediate invariants to validate every step against historical references

🚀 Why it matters

This puzzle require solving a 206,545-digit long number with absolutely no room for rounding errors. It’s the kind of calculation that let's you touch the heights of numerical computing and problem solving.

Reaching the correct number meant building fast C++ based systems to computate and study fascinating mathematical problems in human history making me one of the few in the world to tackle this problem with 100% accuracy and the first to solve it with Bhakara II's Chakravala method.

🧮 What are Diophantine Equations?

Diophantine equations are equations where you're only interested in integer solutions (whole numbers).

Named after: Diophantus — often called the "father of algebra"

🔥 Pell–Fermat Equation (Core Idea)

The Pell equation looks like:

x² − Dy² = 1

Where: D is a non-square integer (like 2, 3, 5, 7…)

Despite the name: It wasn't really discovered by Pierre de Fermat or Pell — Europeans misunderstood earlier Indian work (important 👇)

🇮🇳 Bhāskara's Chakravāla Method (The Real Power Move)

The Chakravāla method (meaning "cyclic method") was developed by Bhāskara II

This method solves:

x² − Dy² = 1