Mathemacula

The 15 Puzzle: A Century of Beautiful Mathematics Mathematics & Puzzles The 15 Puzzle: A Century of Beautiful Mathematics A simple grid of sliding tiles hid one of the most elegant theorems in combinatorics — and sparked a global obsession that shut down businesses, distracted ship captains, and drove mathematicians to their desks for decades. Group Theory Combinatorics Permutations Algorithms In the winter of 1880, something strange swept through Boston. Shop owners arrived to find their clerks standing motionless behind counters, pushing small numbered squares around wooden trays. Pilots navigated poorly. Legislators neglected bills. A Baltimore newspaper editor, having stepped out for a noon lunch, was discovered by his staff long past midnight, still pushing little pieces of pie around on a plate. "T...

Imagine walking into a room with 22 strangers. What are the odds that two of you share the same birthday? Most people would guess low, after all, there are 365 days in a year. But surprisingly, the probability is over 50%. This counterintuitive result is known as the Birthday Paradox , and it’s a classic example of how human intuition often fails when it comes to probability. What Is the Birthday Paradox? The Birthday Paradox (also called the Birthday Problem) refers to the surprising fact that in a group of just 23 people , there's a greater than 50% chance that at least two people share a birthday. With 50 people , that probability jumps to 97% , and by 60 , it’s nearly certain at 99.4% . This isn’t about someone sharing your birthday; it’s about any two people in the group sharing a birthday. That distinction is key. Why Does It Work? The paradox arises from the way probabilities compound. In a group of 23 people, there are 253 unique pairs (calculated as ( \frac{23 \cd...

In the early 18th century, the city of Königsberg (now Kaliningrad, Russia) was a bustling Prussian port built around the Pregel River. The river split the city into four distinct land areas — two riverbanks and two islands — connected by a total of seven bridges . The citizens of Königsberg enjoyed strolling through their city, and a particular challenge became a favourite topic of conversation: Is it possible to take a continuous walk that crosses each of the seven bridges exactly once? You could start anywhere, end anywhere, and walk in any order — but you could not cross the same bridge twice. Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges (Image: Wikipedia) The layout Imagine standing on one of the riverbanks. From where you are, you can see bridges leading to both islands, and from each island, more bridges leading to the other island and to the opposite bank. The arrangement is symmetric enoug...

  Mathematics isn't just about numbers; it's a profound way to explore and represent patterns that resonate with beauty. Recently, I stumbled upon a fascinating way to blend mathematical equations with graphical representation using Python. Let me introduce you to two gorgeous 3D visualizations, both rooted in a simple yet intricate equation. import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Parameters n = 800 A = 1.995653 B = 1.27689 C = 8 petalNum = 3.6 # Create grid r = np.linspace ( 0 , 1 , n ) theta = np.linspace ( -2 , 20 * np.pi , n ) R , THETA = np.meshgrid ( r , theta ) # Calculate coordinates x = 1 - ( 1 / 2 ) * (( 5 / 4 ) * ( 1 - np.mod ( petalNum * THETA , 2 * np.pi ) / np.pi ) ** 2 - 1 / 4 ) ** 2 phi = ( np.pi/ 2 ) * np.exp ( -THETA / ( C * np.pi )) y = A * ( R** 2 ) * ( B * R - 1 ) ** 2 * np.sin ( phi ) R2 = x * ( R * np.sin ( phi ) + y * np.cos ( phi )) X = R2 * np.sin ( THETA ) Y = R2 * np.cos (...

Have you ever come across a meme that perfectly blends humor with a touch of mathematics? One such meme that caught my eye recently goes like this: At first glance, it might seem like just another funny meme, but there's a clever twist that makes it stand out. Let's break it down and appreciate the humor behind it. Explanation: `i`  in mathematics usually refers to the imaginary unit, where `i = − 1 i = \sqrt{-1} In `\mathbb{Z}_17`, `16=-1` Then `sqrt{16}=4=\sqrt{-1}=i` Have you come across any other math-related memes that made you laugh? 

Trigonometric identities often present fascinating challenges that require a blend of algebraic manipulation and understanding of trigonometric properties. They allow us to transform and simplify expressions, making it easier to solve equations and understand the relationships between different trigonometric functions.   If `sin(x)+cos(x)=\sqrt{3}-1`, then what is `sin(2x)`? . . . . . . . . . . . . . . . . . . . Show Answer Answer: `3-2\sqrt{3}` Explanation:  `(sin(x)+cos(x))^2=(\sqrt{3}-1)^2` `\underbrace{\sin^2(x)+\cos^2(x)}_1+\underbrace{2\sin(x)\cos(x)}_\sin(2x)=3-2\sqrt{3}+1` `1+\sin(2x)=3-2\sqrt{3}+1` `\sin(2x)=3-2\sqrt{3}`  

Mathematics often presents intriguing problems that involve the properties of digits within numbers. In this blog post, we'll explore a problem where we need to find all two-digit numbers \( N \) that satisfy a specific equation involving the product and sum of their digits. Let's dive into the details and solve this problem step-by-step. Let ` P(n) ` and ` S(n) ` denote the product and sum, respectively, of the digits of a positive integer ` n `. Determine all two-digit numbers ` N ` that satisfy the equation: ` P(N) + 2S(N) = N `  For example, `P(23)=2\times 6` and `S(23)=2+3=5`.   . . . . . . . . . . . . . . . Show Answer Answer: ` 14, 36, 77 ` Explanation:   Let ` N = 10a + b `, where ` a ` and ` b ` are the tens and units digits of ` N `, respectively. Here, ` a ` and ` b ` are integers such that ` 1 \leq a \leq 9 ` and ` 0 \leq b \leq 9 `. Then , The product of the digits ` P(N) ` is `P(N) = a \cdot b` The sum of the digits ` S(N) ` is:     ...

In mathematics, finding the maximum value of a function is a common and intriguing problem. In this blog post, we'll explore how to determine the maximum value of the function   `E=\frac{48}{p^2-2p+3}`.   where `p`  is a real number. Let's dive into the details and solve this problem step-by-step. What is the maximum value of `E` if `p` is real number? `E=\frac{48}{p^2-2p+3}`. . . . . . . . . . . . . . Show Answer Answer: 24 Explanation:  First, observe the numerator is constant. Then, observe that quadric expression, The maximum value of E, we need to minimize the denominator. (E is inversely proportional to     denominator Let's say denominator `\phi(p)=p^2-2p+3`. It has critical point of' `\phi'(p)=2p-2=0\implies p=1`. denominator has minimum when `p = 1`. `E` has maximum where `p = 1`.