Sri Lankan Math Geek

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`.

  Logarithmic functions often present interesting challenges in mathematics. In this blog post, we'll explore a problem involving points on the graph of a logarithmic function and how to find the positive difference between their ` x`-coordinates. Let's dive into the details and solve this intriguing problem. Points `A` and `B` lie on the graph `y=\log_2(x)`. The mid point of `AB` is `(6,2)`. What  is the positive difference between `x`-coordinates of `A` and `B` ? . . . . . . . . . . . . . . . . . . Show Answer Answer: `4\sqrt{5}` Explanation:

Geometry often presents fascinating problems that challenge our understanding of shapes and their relationships. In this blog post, we'll explore a problem involving a circle inscribed in a square and a smaller square inscribed within that circle. Let's dive into the details and find the area of the smaller square. A circle inscribed a in square  The radius of the circle is `r`. A smaller square is inscribed inside the circle with each of its sides touching the circle. If the area of larger square is `4r^2`. What is the area of the smaller square? . . . . . . . . . . . . . . . . . . . . . . Show Answer Answer: `2r^2` Explanation:  

When it comes to solving real-world problems using mathematics, understanding the relationship between speed, distance, and time is crucial. In this blog post, we'll explore how to calculate the length of a platform given the speed of a train, the length of the train, and the time it takes to cross the platform. A train is `120m` long and crosses a platform in `20` seconds at `54km/h`. What is the platform length?  . . . . . . . .. . . . . . . . . . . . . . . . . . . Show Answer Answer: ` 180m ` Explanation:

Mathematics often presents us with intriguing problems that require a blend of algebraic manipulation and logical reasoning.  Try to solve the following problem. If `x+\frac{1}{x}=3`, find `x^3+\frac{1}{x^3}`? This is very easy one. Even Grade 8 students also can do it. . . . . . . . . . . . . . . Show Answer Answer: 18 Explanation:  `\left( x+\frac{1}{x} \right)^3=x^3+3x+\frac{3}{x}+\frac{1}{x^3}` `\left( x+\frac{1}{x} \right)^3=x^3+3\left(x+\frac{1}{x}\right)+\frac{1}{x^3}` `\left(3 \right)^3=x^3+3\left(3\right)+\frac{1}{x^3}` `27=x^3+9+\frac{1}{x^3}` `x^3+\frac{1}{x^3}=27-9=18`

Have you ever stumbled upon a math trick so intriguing, it feels like magic? Today, we’re diving into an enchanting number trick that involves simple addition and subtraction. Follow these steps and get ready to be amazed! If you need to see the magic click here Step-by-Step Instructions: 1. Choose a number between 10 and 99.     - For example, let's take 74 2. Add the two digits of your number together.    - In our example: 7 + 4 = 11. 3. Subtract the sum you just calculated from your original number.    - So, 74 - 11 = 63. 4. Find your result in the diagram below. Note the symbol associated with that number. 95 ♣ 84 ☃ 73 ☃ 62 ♣ 51 ☃ 40 ☂ 29 ♦ 18 ♦ 7 ♥ 94 ☂ 83 ☂ 72 ♦ 61 ♠ 50 ♦ 39 ☃ 28 ♠ 17 ★ 6 ♥ 93 ♪ 82 ★ 71 ☀ 60...