🧮 10 Ways to Explore Pi & Euler’s Number on Raspberry Pi (2026)

While other languages might leave you guessing why #define PI isn’t built-in, your Raspberry Pi doesn’t just store constants—it breathes life into them. Imagine sitting at your desk, watching a tiny, credit-card-sized computer churn out millions of digits of $\pi$ in real-time, or visualizing Euler’s number ($e$) as a spiraling graph that dances across your screen. That isn’t science fiction; it’s the reality of computational mathematics on the Raspberry Pi.

At Why Pi™, we’ve seen students go from “Hello World” to calculating the Chudnovsky algorithm in a single afternoon. But why stop at just printing numbers? In this guide, we’ll walk you through 10 powerful methods to turn your Pi into a mathematical powerhouse. From Monte Carlo simulations that estimate $\pi$ using random darts to building a live OLED dashboard that displays the digits of $e$ as they are calculated, we cover every angle. We’ll even dive into the mind-bending insights of quantum entanglement and how the same math that defines a circle governs the fabric of the universe.

Ready to stop wondering why C leaves you hanging and start coding your own universe? Let’s turn that Raspberry Pi into the ultimate math lab.

Key Takeaways

• Algorithmic Mastery: Learn to implement high-speed algorithms like the Chudnovsky method and Taylor Series to calculate $\pi$ and $e$ with millions of digits of precision.
• Visual Learning: Transform abstract numbers into stunning visuals using Matplotlib and Turtle Graphics, making complex convergence rates easy to understand.
• Hardware Integration: Go beyond the screen by building GPIO-driven OLED dashboards and hosting your own Mathematical Constant API for real-time data access.
• Performance Tuning: Discover how to leverage NumPy and C++ to accelerate calculations, turning your Raspberry Pi into a supercomputer for number theory.
• Deep Insights: Explore the surprising connections between mathematical constants and quantum physics, including Bell’s Inequality and the nature of infinity.

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Table of Contents

• ⚡️ Quick Tips and Facts
• 📜 A Brief History of Pi, Euler, and the Tiny Computer Revolution
• 🛠️ Setting Up Your Raspberry Pi for Mathematical Exploration
• 🐍 1. Calculating Pi to Millions of Digits Using Python
• 🐍 2. Approximating Euler’s Number (e) with Iterative Algorithms
• 🐍 3. Visualizing Mathematical Constants with Matplotlib and Turtle
• 🐍 4. Generating Random Numbers to Estimate Pi via Monte Carlo Simulations
• 🐍 5. Exploring the Golden Ratio and Other Irrational Numbers
• 🚀 6. Pushing Performance: Using C++ and NumPy for High-Speed Calculations
• 📡 7. Building a Pi-Display Dashboard with GPIO and OLED Screens
• 🌐 8. Hosting Your Own Mathematical Constant API on the Raspberry Pi
• 🧠 Insights: Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality
• 🧠 Insights: Thinking Outside The Box Versus Knowing What’s In The Box
• 🧠 Insights: Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect
• 🧠 Insights: What Exactly is Dirac’s Delta Function?
• 🧠 Insights: AI Enriched Problem Solving in Mathematical Discovery
• 🧠 Insights: Remote Operated Gate Control System for Math Labs
• 🧠 Insights: Why Do Programming Languages Like C Not Define Constants Like Pi?
• 🧠 Insights: The High School Potato Paradox and Infinite Series
• 🧠 Insights: The Ant on a Stretchy Rope Puzzle and Harmonic Series
• 🧠 Insights: Undergrad Three Circle Problem and Geometric Constants
• 🧠 Insights: High School Area of Overlapping Squares and Pi Approximations
• 🧠 Insights: Undergrad Find the Number of Triangles and Combinatorial Math
• 🏆 Conclusion
• 🔗 Recommended Links
• ❓ FAQ
• 📚 Reference Links

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⚡️ Quick Tips and Facts

Before we dive into the code that will make your Raspberry Pi sing with mathematical precision, let’s hit the ground running with some high-impact facts and pro-tips from our engineering team at Why Pi™.

• Pi ($\pi$) is irrational: It never ends, never repeats, and your Pi won’t calculate it all in a lifetime. We’re talking about a number that has been calculated to over 10 trillion digits by supercomputers, yet your Raspberry Pi 5 can still crunch millions of them in seconds. 🤯
• Euler’s Number ($e$) is the base of natural growth: If you’ve ever wondered how bacteria multiply or how interest compounds, $e$ (approx. 2.71828) is the secret sauce. It’s the limit of $(1 + 1/n)^n$ as $n$ approaches infinity.
• The “Why Pi” Philosophy: We believe that mathematics is the language of the universe, and the Raspberry Pi is the most accessible translator we have. You don’t need a PhD to explore the infinite; you just need a 4GB board and a bit of Python.
• Monte Carlo Method: One of the coolest ways to estimate $\pi$ on your Pi is by throwing “darts” (random numbers) at a square. It’s a statistical approach that turns randomness into precision.
• Hardware Matters: While a Raspberry Pi Zero W can run these scripts, a Raspberry Pi 5 or 4 with a heatsink is recommended for heavy-duty calculations involving millions of iterations.

Did you know? The first 39 digits of $\pi$ are sufficient to calculate the circumference of the observable universe to the precision of a hydrogen atom. Yet, we keep calculating more. Why? Because curiosity has no limit.

For a deeper dive into the history of $\pi$ and its significance in our daily lives, check out our comprehensive guide on Why Pi™: The Story of Pi.

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📜 A Brief History of Pi, Euler, and the Tiny Computer Revolution

From Clay Tablets to GPIO Pins

The story of mathematical constants is as old as civilization itself. Long before the Raspberry Pi Foundation released their first board in 2012, humans were scratching approximations of $\pi$ onto clay tablets in Babylon.

• The Babylonians (c. 190–160 BC): They used a value of 3.125 for $\pi$. Not bad for a civilization that didn’t even have the concept of zero!
• Archimedes of Syracuse (c. 250 BC): The grandfather of approximation. He inscribed and circumscribed polygons around a circle, narrowing $\pi$ down to between 3.1408 and 3.1429. He did this with a ruler and a lot of patience.
• The Birth of $e$: Fast forward to the 17th century. Jacob Bernoulli was trying to figure out compound interest. He asked: “If I compound interest continuously, what is the maximum amount I can get?” The answer was a number that would later be named $e$ by Leonhard Euler.

The Modern Era: Why Pi™ and the Raspberry Pi

Why did we choose the Raspberry Pi for this journey? Because it democratizes computational mathematics.

In the past, calculating the first million digits of $\pi$ required a mainframe computer the size of a room. Today, a device the size of a credit card, costing less than a pizza, can do it. This shift allows students, hobbyists, and engineers to experiment, fail, and learn in real-time.

The Conflict: Some traditionalists argue that using a computer to calculate $\pi$ is “cheating” because it removes the human struggle. We disagree. The struggle isn’t in the arithmetic; it’s in the algorithm design. Understanding how the computer calculates $\pi$ is far more valuable than manually multiplying numbers.

The Video Perspective: A Visual Journey

To truly grasp the scope of these constants, we recommend watching the featured video summary embedded in our community discussions. It highlights how constants like the Golden Ratio ($\phi$), Pythagoras’ Constant ($\sqrt{2}$), and the Feigenbaum Constants ($\delta, \alpha$) are interconnected.

The video emphasizes that while $\pi$ and $e$ are the stars, constants like Apéry’s Constant ($\zeta(3)$) and Khinchin’s Constant ($K$) play crucial roles in chaos theory and number theory. It’s a reminder that mathematics is a web, not a ladder.

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🛠️ Setting Up Your Raspberry Pi for Mathematical Exploration

Before we write a single line of code, we need to ensure your Raspberry Pi is ready for the heavy lifting. You wouldn’t try to run a marathon in flip-flops, and you shouldn’t try to calculate $\pi$ on a Pi with a corrupted SD card.

1. The Hardware Checklist

Component	Recommendation	Why?
Board	Raspberry Pi 5 or 4 (4GB/8GB)	More RAM = faster large-number arithmetic.
Storage	High-endurance MicroSD (32GB+)	Frequent writes during calculations wear out cheap cards.
Cooling	Active Cooler or Heatsink	Calculating $e$ to 10,0 digits generates heat.
Power	Official USB-C Power Supply (5.1V/5A)	Stable voltage prevents crashes during long runs.
Peripherals	Keyboard, Mouse, Monitor	Essential for debugging and visualization.

2. Software Environment: The Python Advantage

We at Why Pi™ strongly recommend using Python for this exploration. It’s the lingua franca of data science and has libraries specifically designed for arbitrary-precision arithmetic.

Step-by-Step Setup:

1. Flash the OS: Use the Raspberry Pi Imager to install Raspberry Pi OS (64-bit). The 64-bit version handles large integers more efficiently.
2. Update Everything:

sudo apt update && sudo apt upgrade -y

3. Install Essential Libraries:
   We need numpy for array operations, matplotlib for plotting, and mpmath for arbitrary-precision math.

sudo apt install python3-pip python3-numpy python3-matplotlib
pip3 install mpmath

Pro Tip: If you are using a Raspberry Pi Zero 2 W, stick to lower precision calculations or use a headless setup (SSH) to save resources. The Zero is powerful, but it has limited RAM.

3. Choosing the Right IDE

While you can code in the terminal, we recommend Thony IDE (pre-installed on Raspberry Pi OS) for beginners. It allows you to see variable values in real-time. For advanced users, VS Code with the Python extension offers a more robust environment.

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🐍 1. Calculating Pi to Millions of Digits Using Python

Now, let’s get our hands dirty. How do we calculate $\pi$ on a Raspberry Pi? We have two main approaches: Series Approximations and Chudnovsky Algorithm.

The Naive Approach: Leibniz Formula

The Leibniz formula is simple but slow. It looks like this:
$$ \pi = 4 \times (1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \dots) $$

Why it’s problematic: It converges incredibly slowly. You need 50,0 iterations just to get 5 decimal places. On a Pi, this is a fun exercise, but not practical for high precision.

The Powerhouse: Chudnovsky Algorithm

This is the algorithm used by world-record calculators. It converges fast.
$$ \frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)! (k!)^3 (640320)^{3k + 3/2}} $$

The Code:
We use the mpmath library to handle the massive numbers. Standard Python floats only go up to 15-17 digits. mpmath lets us set the precision to 1,0, 10,0, or even 1,0,0 digits.

from mpmath import mp

# Set precision to 10 digits
mp.dps = 10

# Calculate Pi
pi_value = mp.pi

print(f"Pi to 10 digits: {pi_value}")

Performance Insight:
On a Raspberry Pi 4, calculating 1,0 digits takes milliseconds. Calculating 10,0 digits might take a few seconds. But if you try 1,0,0 digits, you’ll see the CPU fan spin up!

Question for you: If you could calculate $\pi$ to a million digits, what would you do with the data? Store it? Analyze the randomness? Let us know in the comments!

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🐍 2. Approximating Euler’s Number (e) with Iterative Algorithms

Euler’s number, $e$, is the heartbeat of calculus. It represents continuous growth.

The Definition: Compound Interest

Recall Jacob Bernoulli’s problem. The formula is:
$$ e = \lim_{n \to \infty} (1 + \frac{1}{n})^n $$

The Code:
Let’s try this in Python. We’ll start with $n=1$ and increase it to see how the value converges.

def calculate_e(n):
 return (1 + 1/n)**n

for in [1, 10, 10, 10, 10, 10]:
 print(f"n={i}: {calculate_e(i)}")

The Result:

• $n=1$: 2.0
• $n=10$: 2.5937…
• $n=10$: 2.7048…
• $n=10$: 2.7169…
• $n=10$: 2.7181…

Notice how it slowly creps toward 2.71828.

The Taylor Series Approach (Faster!)

A much faster way to calculate $e$ is using the Taylor Series expansion:
$$ e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \dots $$

This converges much faster than the compound interest formula. With just 20 terms, you get $e$ to 15 decimal places.

import math

def calculate_e_taylor(terms):
 total = 0
 for in range(terms):
 total += 1 / math.factorial(i)
 return total

print(f"e (20 terms): {calculate_e_taylor(20)}")

Comparison:

Method	Iterations for 10 digits	Speed on Pi 4
Compound Interest	~10,0,0	Slow
Taylor Series	~15	Instant

Why this matters: In engineering, efficiency is key. Choosing the right algorithm can save you hours of computation time.

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🐍 3. Visualizing Mathematical Constants with Matplotlib and Turtle

Math is beautiful, but numbers on a screen are boring. Let’s visualize these constants.

Visualizing Pi with Turtle Graphics

The Turtle library is perfect for drawing geometric shapes. We can draw a circle and measure its circumference to estimate $\pi$.

The Concept:

1. Draw a circle with radius $r$.
2. Calculate the circumference $C = 2 \pi r$.
3. Use the turtle to trace the circle and count the steps.

import turtle
import math

screen = turtle.Screen()
t = turtle.Turtle()
t.speed(0)

radius = 10
circumference = 2 * math.pi * radius

# Draw the circle
t.circle(radius)

# Display the calculated Pi
t.penup()
t.goto(-10, -150)
t.write(f"Calculated Pi: {math.pi}", font=("Arial", 16, "normal"))

turtle.done()

Visualizing Euler’s Number with Matplotlib

Let’s plot the convergence of $e$ using the Taylor Series.

import matplotlib.pyplot as plt
import math

terms = range(1, 25)
e_values = [sum(1/math.factorial(i) for in range(n)) for n in terms]

plt.plot(terms, e_values, marker='o', label='Convergence to e')
plt.axhline(y=math.e, color='r', linestyle='--', label='Actual e')
plt.title('Convergence of Euler\'s Number (e)')
plt.xlabel('Number of Terms')
plt.ylabel('Value')
plt.legend()
plt.grid(True)
plt.show()

What you’ll see: A curve that shoots up quickly and then flattens out, hugging the red dashed line of the actual value of $e$. This visual feedback is crucial for understanding convergence.

Fun Fact: The Golden Ratio ($\phi$) can also be visualized by drawing a series of squares that spiral outward. Try coding that next!

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🐍 4. Generating Random Numbers to Estimate Pi via Monte Carlo Simulations

This is the most “Raspberry Pi” way to do math. We use randomness to find a constant.

The Monte Carlo Method Explained

Imagine a square with a circle inscribed inside it.

• Area of Square = $(2r)^2 = 4r^2$
• Area of Circle = $\pi r^2$
• Ratio of Areas = $\frac{\pi r^2}{4r^2} = \frac{\pi}{4}$

If we throw random darts at the square, the ratio of darts landing inside the circle to the total darts thrown will approximate $\frac{\pi}{4}$.

The Code:

import random
import math

def estimate_pi(num_darts):
 inside_circle = 0
 for _ in range(num_darts):
 x = random.uniform(-1, 1)
 y = random.uniform(-1, 1)
 distance = math.sqrt(x**2 + y**2)
 if distance <= 1:
 inside_circle += 1

 pi_estimate = (inside_circle / num_darts) * 4
 return pi_estimate

# Run with 1 million darts
print(f"Estimated Pi: {estimate_pi(10)}")

Performance Note:

• 10,0 darts: Rough estimate (3.14 +/- 0.05)
• 1,0,0 darts: Better estimate (3.141 +/- 0.02)
• 10,0,0 darts: Excellent estimate, but takes time on a Pi.

Why use this? It teaches probabilistic algorithms. In the real world, we often use Monte Carlo simulations for everything from weather forecasting to financial modeling.

Challenge: Can you optimize this code to run faster on the Raspberry Pi? (Hint: Look into NumPy vectorization).

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🐍 5. Exploring the Golden Ratio and Other Irrational Numbers

While $\pi$ and $e$ are the celebrities, the Golden Ratio ($\phi$) is the rock star of aesthetics.

The Golden Ratio ($\phi$)

Defined as $\frac{1 + \sqrt{5}}{2} \approx 1.61803$.
It appears in nature (sunflowers, nautilus shells), art (Mona Lisa), and architecture (Parthenon).

Calculating $\phi$ on Pi:
We can use the Fibonacci Sequence. The ratio of consecutive Fibonacci numbers converges to $\phi$.

def fibonacci_ratio(n):
 a, b = 0, 1
 for _ in range(n):
 a, b = b, a + b
 return b / a

print(f"Golden Ratio (Fibonacci 20): {fibonacci_ratio(20)}")

Other Irrational Numbers to Explore

• $\sqrt{2}$ (Pythagoras’ Constant): The diagonal of a unit square.
• $\sqrt{3}$ (Theodorus’ Constant): Found in equilateral triangles.
• Euler-Mascheroni Constant ($\gamma$): The difference between the harmonic series and the natural log. It’s a mystery—no one knows if it’s rational or irrational!

Why explore these?
Each constant opens a door to a different branch of mathematics. $\sqrt{2}$ leads to geometry, $\gamma$ leads to number theory, and $\phi$ leads to fractals.

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🚀 6. Pushing Performance: Using C++ and NumPy for High-Speed Calculations

Python is great, but it’s interpreted. For high-speed calculations, we need to get closer to the metal.

Why C++?

C++ is compiled, meaning the code is translated directly into machine language. On a Raspberry Pi, C++ can be 10x to 10x faster than Python for numerical loops.

The Trade-off:

• Python: Easy to write, slow execution.
• C++: Harder to write, blazing fast execution.

Using NumPy in Python

If you don’t want to switch to C++, NumPy is your best friend. It uses optimized C code under the hood.

Comparison:

• Standard Python Loop: 10 million iterations = ~5 seconds.
• NumPy Vectorized: 10 million iterations = ~0.05 seconds.

Code Example (NumPy):

import numpy as np

# Vectorized Monte Carlo
n = 10
x = np.random.uniform(-1, 1, n)
y = np.random.uniform(-1, 1, n)
inside = np.sum(x**2 + y**2 <= 1)
pi_est = 4 * inside / n
print(f"NumPy Pi: {pi_est}")

Recommendation:
For learning and protyping, stick with Python. When you need to calculate $\pi$ to 10 million digits or run a simulation in real-time, switch to C++ or use NumPy.

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📡 7. Building a Pi-Display Dashboard with GPIO and OLED Screens

Why just calculate $\pi$ when you can display it live? Let’s build a dashboard that updates the value of $\pi$ and $e$ in real-time on a small OLED screen.

Hardware Needed

• Raspberry Pi (any model)
• 0.96″ OLED Display (SSD1306 driver)
• Breadboard and Jumper Wires

The Setup

1. Connect the OLED to the GPIO pins (I2C interface).
2. Install the smbus and Adafruit_GPIO libraries.
3. Write a script that calculates the next digit of $\pi$ and updates the screen every second.

The “Wow” Factor:
Imagine a device on your desk that constantly churns out new digits of $\pi$. It’s a conversation starter and a testament to the power of the Raspberry Pi.

Tip: Use a Raspberry Pi Pico for this if you want a standalone device without the overhead of the full OS. The Pico is perfect for simple display tasks.

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🌐 8. Hosting Your Own Mathematical Constant API on the Raspberry Pi

Turn your Pi into a server that serves mathematical constants to the world (or your local network).

The Concept

Create a simple web API using Flask (a lightweight Python web framework).

• Request: GET /api/pi?digits=10
• Response: {"pi": "3.14159..."}

Why do this?

• Educational: Teach students how APIs work.
• Practical: Use it as a backend for a mobile app or a smart home display.
• Fun: Challenge friends to guess the 10th digit of $e$.

Code Snippet (Flask):

from flask import Flask, jsonify
from mpmath import mp

app = Flask(__name__)

@app.route('/api/pi')
def get_pi():
 mp.dps = 10 # Default 10 digits
 return jsonify({"pi": str(mp.pi)})

if __name__ == '__main__':
 app.run(host='0.0.0.0', port=50)

Security Note:
If you expose this to the internet, add authentication. You don’t want someone flooding your Pi with requests for 1,0,0 digits and crashing it!

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🧠 Insights: Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Wait, what? We’re talking about Pi, and now we’re talking about quantum entanglement?

Bear with us. The connection is mathematics. The same mathematical rigor used to calculate $\pi$ is used to prove that quantum mechanics is non-local.

The Insight:
Bell’s Inequality is a mathematical test. If the universe followed classical physics, the correlation between entangled particles would be limited. But experiments show they violate this limit.

• The Math: It involves trigonometric functions (sine and cosine) which are deeply tied to $\pi$.
• The Pi Connection: The probability amplitudes in quantum mechanics often involve $e^{i\pi}$.
• Why it matters: It shows that mathematics is not just a human invention; it describes the fundamental fabric of reality.

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🧠 Insights: Thinking Outside The Box Versus Knowing What’s In The Box

The Box: The standard algorithms (Leibniz, Taylor Series).
Outside the Box: Using Monte Carlo simulations or quantum algorithms.

The Lesson:
Sometimes, the most efficient way to solve a problem isn’t the most obvious one.

• Inside the Box: Calculating $\pi$ term by term.
• Outside the Box: Using a random number generator to estimate it.

Application:
In engineering, we often get stuck in “inside the box” thinking. The Raspberry Pi encourages us to experiment and find “outside the box” solutions.

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🧠 Insights: Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

This is a deep dive into physics, but the math is the same.

• Kinematic: Describes motion without forces.
• Dynamic: Describes forces causing motion.

Entanglement is a structural property of the universe, not a force. It’s like the geometry of a circle (defined by $\pi$) vs. the force pushing a ball.
The Takeaway: Understanding the structure (math) is often more important than the mechanism (physics).

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🧠 Insights: What Exactly is Dirac’s Delta Function?

The Dirac Delta Function ($\delta(x)$) is a “function” that is zero everywhere except at $x=0$, where it is infinite, and its integral is 1.

• The Math: It’s a distribution, not a function.
• The Pi Connection: It’s used in Fourier Transforms, which decompose signals into sine and cosine waves (involving $\pi$).
• Why it matters: It’s essential for signal processing on the Raspberry Pi (e.g., audio processing).

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🧠 Insights: AI Enriched Problem Solving in Mathematical Discovery

The Future:
Can AI discover new mathematical constants?

• Current State: AI is used to find patterns in the digits of $\pi$.
• Future: AI might prove theorems or find new relationships between constants.
• The Pi Role: Your Raspberry Pi can run local AI models (like TinyLLaMA) to explore these questions.

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🧠 Insights: Remote Operated Gate Control System for Math Labs

Imagine a remote lab where students can control a Raspberry Pi to run calculations.

• The Setup: A Pi connected to a motorized gate (simulating a physical experiment).
• The Math: The gate opens based on the calculated value of $\pi$.
• The Lesson: Combining hardware control with mathematical computation creates a powerful learning tool.

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🧠 Insights: Why Do Programming Languages Like C Not Define Constants Like Pi?

The Controversy:
Why doesn’t C have a built-in PI constant?

• Reason 1: C is a low-level language. It trusts the programmer to define constants.
• Reason 2: Different precision requirements.
• The Workaround: You define it yourself: #define PI 3.141592653589793.
• The Lesson: Control is a feature, not a bug. It forces you to understand the precision you need.

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🧠 Insights: The High School Potato Paradox and Infinite Series

The Paradox:
10 lbs of potatoes are 9% water. If they dry to 98% water, how much do they weigh?

• The Math: It involves infinite series and ratios.
• The Result: They weigh 50 lbs! (The solid part stays 1 lb, which is now 2% of the total).
• The Connection: This is a classic example of how intuition fails and mathematics prevails.

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🧠 Insights: The Ant on a Stretchy Rope Puzzle and Harmonic Series

The Puzzle:
An ant crawls on a rope that stretches every second. Will it ever reach the end?

• The Math: It involves the Harmonic Series ($1 + 1/2 + 1/3 + \dots$), which diverges to infinity.
• The Answer: Yes, the ant will eventually reach the end, but it might take a long time!
• The Connection: This shows the power of infinite series and their counter-intuitive nature.

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🧠 Insights: Undergrad Three Circle Problem and Geometric Constants

The Problem:
Three circles of equal radius intersect. What is the area of the overlapping region?

• The Math: Involves trigonometry and $\pi$.
• The Solution: Requires calculating the area of circular segments.
• The Lesson: Geometry is the foundation of computational geometry used in computer graphics and robotics.

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🧠 Insights: High School Area of Overlapping Squares and Pi Approximations

The Problem:
Two squares overlap. What is the area?

• The Math: Simple geometry, but can be extended to Monte Carlo methods.
• The Connection: Just like the circle in the square, we can estimate areas using random sampling.

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🧠 Insights: Undergrad Find the Number of Triangles and Combinatorial Math

The Problem:
How many triangles can be formed from a set of points?

• The Math: Combinatorics ($nCr$).
• The Connection: Combinatorics is the basis of cryptography and data science.
• The Pi Role: The Raspberry Pi can brute-force these calculations to find patterns.

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🏆 Conclusion

(Note: As per your instructions, the Conclusion section is omitted here and will be written in the next step.)