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Mathematics - An Art of Thinking

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Paul Thurston, American mathematician In a previous blog , I spoke a lot about what mathematics isn’t. However, I don’t think I spoke quite enough on what mathematics truly is. Here's how I began that blog. "When we’re young we begin by learning the steps to add – we’re given the rules and we must learn to repeat them – no questions asked. Why does carry forward work the way it does? Well, no questions asked." However, I believe that the essence of truly understanding mathematics lies in the questions you ask – and the first one, the most fundamental one, is also probably one of the hardest questions you would and could find – What really is Mathematics? Something that differentiates explaining what mathematics is from explaining a lot of other concepts is that mathematics was not really ever invented . If a young student angry about excessive math h
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Kőnig-Egerváry theorem

Graph theory has been my most favourite thing to learn in maths and and in this blog i hope to spread the knowledge about Kőnig's theorem. It is advised that the readers are aware about basic graph theory terminologies including bipartite graphs. Before going on to the theorem i would like to go on about matchings and vertex cover which we are going to use in the theorem  Matchings A matching $M$ of a graph $G$ is a subset of the edges of a graph in which no two edge share a common vertex (non adjacent edges). For example :- The Matching $M$ over here is edge $\{ 3 - 5 , 1-2 , \}$ or $\{ 1 - 2 , 4 - 3 \}$ etc .  Maximum Matching is a matching that contains the largest possible number of edges for instance in the above example the maximum matching is 2 edges as there cannot be a subset of non adjacent edges having greater than 2 edges (Readers are advised to try so they can convince themselves) A Perfect Matching  is a matching that matches all vertices of the graph or in other sen

An introduction to Graph theory

Graph Theory is a branch of mathematics and computer science that deals with modelling various relationships using vertices and edges. Graph theory as a field is crucial today, and is used everywhere we go - from searching the internet to artificial intelligence, and from DNA sequencing to Google Maps. The power of graph theory emerges from its simplicity. Despite being a concept that is fairly simple to understand, it is an active field of research with hundreds of open problems which mathematicians work on today.  Introduction The Königsberg Bridge Problem The city of Königsberg is located on the Pregel river in Prussia. As shown in the image (a) below, the river divided the city into 4 landmasses which were connected by seven bridges. The citizens often wondered if it was possible to start from home, travel through the city crossing every bridge exactly once and return home. Exercise: Find a round trip crossing each bridge exactly once, or try to prove that no such trip exists. The

Basic Combinatorics

Hello people its Rishad again. Yes I'm still alive. Now I am back here with another post (after an eternity).  This post is just about very fun questions from combinatorics. This post is not going to have really "high level" content. For some this might be just fun puzzle solving (experienced olympiad people) while for those who are starting off and who have little exposure to olympiad combinatorics will have exposition to some really fun mathematics. The questions are just sort of puzzles and not much mathematical knowledge is required to solve them.  Some warmup before we go to solving the main set of questions. I will just be providing the answers to these questions you have to do these questions on your own.  Q1 . A number of bacteria are placed in a glass. One second later each bacterium divides in two, the next second each of the resulting bacteria divides in two again, et cetera. After one minute the glass is full. When was the glass half-full? ANSWER : 59 SECOND

Some Wandering Through Orthic Triangles

 Hi, I am Emon and I it's been long since I posted last. Today I will try to give you some ideas on how to work with a special type of triangle, known as " Orthic Triangles ". In this post, I will mainly focus on problem-solving, but still, let me first give you some ideas on what exactly it is and what properties does it have... Definition (Orthic Triangle). Let $ABC$ be a triangle and let $D, E, F$ be the foot of the perpendiculars from $A, B, C$ to $BC, CA$ and $AB$, respectively. Then, $\triangle DEF$ is known as the orthic triangle of $\triangle ABC$. Lemma $1$ (Orthic Triangle). If $\triangle DEF$ is the orthic triangle of $\triangle ABC$ with orthocenter $H$, then the following conditions are satisfied : $(i)$ $AEHF$ is a cyclic quadrilateral with circumdiameter $AH$. $(ii)$ $BCEF$ is a cyclic quadrilateral with circumdiameter $BC$. $(iii)$ $H$ is the incenter of $\triangle DEF$. Lemma $2$. $\angle ABE = \angle ADE$ and $\angle ACF=\angle ADF$. (We can prove this w

A phase I'll never forget...

 It was the night of 8th August 2021, when we had the closing ceremony of CAMP (A math camp organized by Ritam Nag, Aatman Supkar and Rohal Goyal). My eyes were a bit wet realizing that it is ending and I won't have those 3 - 4 hours of fun every night which I had the whole past week. Thinking that time about all the memories I made in the camp from joining the game nights to getting frank with my seniors, it was probably one of the best experiences I had in any of the math related thing I did. I was fascinated by the organization of this camp and was just wondering if I was good enough to ever do something like this in future, but I never really got enough courage or time to do anything like this until recently. Fast forwarding to the time after INMO results, me and many of my friends were very sad that IMOTC has been cancelled by HBCSE and we couldn't meet each other for another year. Well, jokingly in a meet around that time, Sunaina told me to ask Rohan if they can organize

Top 10 problems of the week!

This week was full Number Theory and algebra biased!! 😄 Do try all the problems first!! And if you guys get any nice solutions, do post in the comments section! Here are the walkthroughs of this week's top 5 Number Theory problems! 5th position (1999 JBMO P2):  For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. Walkthrough:  It doesn't require a walkthrough, I wrote this here, cause it's a cute problem for the person who has just started Contest math :P a. What is $A_0$? b. Find out $A_1$. c. Show that $\boxed{7}$ is the required answer! 4th position (APMO, Evan Chen's orders modulo a prime handout):   Let $a,b,c$ be distinct integers. Given that $a | bc + b + c, b | ca + c + a$ and $c | ab + a + b$, prove that at least one of $a, b, c$ is not prime. Walkthrough:  Fully thanks to  MSE  ! (Also one should try MSE, it has helped me a lot, ofc it's more tilted to Coll

Whose Game?

 In this blog post, we will talk about Combinatorial games, how to quantify positions, and analyse games. Most of this is based on the beginning chapters of the comprehensive book Winning Ways for your Mathematical Plays by Berlekamp, Conway and Guy. What is a Game? You might be familiar with multiple 'games', all with greatly varied meanings. For example, a  game could refer to a wild animal hunted for sport (This post isn't about those!). We will talk about games that are defined by the following (restrictive) rules: There are just 2 players, often named Left and Right. There are several well-defined positions and a starting position. There are clearly defined rules specifying the moves each player can make from a position, and the resulting positions are called options. Left and Right move alternately in gameplay. In the normal play convention, the player unable to move loses. In misere  play, the last player to move loses. The rules ensure that the game ends and one p