Arya’s Algorithms

The Fragile Thread of Memory and the Self Sometimes I wonder: if all my memories were stripped away in a single moment, who would remain? Would I still be “me,” or would the person I call myself vanish with the past? Memory feels like a fragile thread tying together every moment of my existence. And yet, when I look closely, I am not sure how strong that thread really is. Memory as the Architect of Identity When I say “I,” it is usually a bundle of memories that speaks. I remember the face of my mother when I was a child, the classroom where I first solved a difficult math problem, the smell of rain during a walk home from school. These recollections are not just events—they are bricks in the house of identity. Without them, the house collapses. But then, memory is slippery. Neuroscientists remind us that each time we recall something, we do not retrieve a file from a cabinet—we reconstruct it, reshaping the past in the present. My childhood memory may not be what truly happened...

The Pit of Integer Partitions: When Numbers Break Apart There are math problems that are easy to state but impossible to tame. Integer partitions are one of the purest examples. The problem sounds like child’s play: In how many ways can you write n n as a sum of positive integers, order irrelevant? For example, with n = 4 n=4 : 4 = 4 , 3 + 1 , 2 + 2 , 2 + 1 + 1 , 1 + 1 + 1 + 1. 4 = 4, \quad 3+1, \quad 2+2, \quad 2+1+1, \quad 1+1+1+1. So there are 5 partitions of 4. Simple, right? That’s what I thought. Then I fell into the pit. Step 1: First Steps Let’s compute the partition numbers p ( n ) p(n) : p ( 1 ) = 1 p(1)=1 . p ( 2 ) = 2 p(2)=2 : 2 , 1 + 1 2,1+1 . p ( 3 ) = 3 p(3)=3 : 3 , 2 + 1 , 1 + 1 + 1 3,2+1,1+1+1 . p ( 4 ) = 5 p(4)=5 . p ( 5 ) = 7 p(5)=7 . p ( 6 ) = 11 p(6)=11 . The sequence is: 1 , 2 , 3 , 5 , 7 , 11 , 15 , 22 , 30 , 42 , … 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, \dots Already I was tempted to guess a formula. I failed. Step 2: The Wro...

  Black Hole Revelations - Where Physics Breaks Down and New Laws Emerge Black holes represent the universe's most extreme laboratories, where gravity becomes so strong that space and time themselves break down. My journey into these cosmic monsters began with a simple question: what happens when you fall into a black hole? The answer led me through some of the deepest paradoxes in modern physics and to the frontiers of our understanding about information, entropy, and the nature of spacetime itself. The classical picture seemed straightforward enough. Karl Schwarzschild found the exact solution to Einstein's field equations for a spherically symmetric mass just months after general relativity was published. The Schwarzschild metric describes spacetime geometry around any non-rotating mass, from planets to black holes. The critical difference lies in whether the object's surface lies inside or outside the Schwarzschild radius rs = 2GM/c². But black holes aren't just ...

  The Introvert-Extrovert Dance: Understanding Different Social Energy Styles The party is in full swing. In one corner, someone is holding court with an animated story, gesturing wildly while a growing crowd laughs and adds their own comments. Across the room, two people are having an intense one-on-one conversation about philosophy, completely absorbed in each other's ideas. Near the kitchen, someone is helping the host with dishes, grateful for a task that allows them to contribute while taking a break from socializing. Later that evening, the storyteller will feel energized and ready for more social connection, while others will be completely drained and need hours of solitude to recharge. This scene illustrates one of the most fundamental differences in how humans experience social interaction: the distinction between introversion and extroversion. Yet despite decades of research and popular psychology, these concepts remain widely misunderstood, often reduced to simplistic...

  The Alpha Complex: Understanding Dominance, Leadership, and the Drive to Be on Top They walk into rooms like they own them, speak with unwavering confidence, and somehow always end up in charge of group decisions. But behind the alpha's commanding presence lies a complex psychology of dominance, insecurity, and an relentless drive to maintain their position at the top. Here's what really drives those who seem naturally born to lead. You know them instantly. They're the ones who take charge when everyone else is standing around confused, who speak first in meetings and somehow get others to follow their lead, who seem to navigate social hierarchies with an instinctive understanding of power dynamics. They're the natural leaders, the decision-makers, the ones others look to when crisis hits and someone needs to take control. But the psychology of "alpha" behavior is far more complex than the confident exterior suggests. Behind that commanding presence often...

  The Expanding Canvas - Dark Energy and the Accelerating Universe The discovery that our universe's expansion is accelerating ranks among the most shocking revelations in physics history. For decades, cosmologists debated whether the universe would expand forever or eventually collapse back on itself. Nobody seriously considered that cosmic expansion might be speeding up, driven by a mysterious form of energy that makes up 70% of everything that exists. My journey into dark energy began with Type Ia supernovae, the "standard candles" of cosmology. These stellar explosions occur when white dwarf stars accrete material from companion stars until they reach the Chandrasekhar limit and explode. Because they all involve roughly the same mass of material undergoing similar nuclear burning, Type Ia supernovae have remarkably consistent peak luminosities. The distance-brightness relationship seemed straightforward. If you know an object's intrinsic luminosity L and measur...

Walking Through the Dream While Awake Last night, as I drifted into sleep, I found myself in the middle of a street I’ve never walked before. The lamps glowed like amber suns, the air smelled faintly of rain, and the ground seemed to ripple under my steps. And then—like a sudden spark—I realized: I am dreaming. In that moment, the world shifted. The street became pliable, as if waiting for me to mold it. The buildings bent slightly toward me, curious, like living things. The awareness of dreaming did not wake me up, as it sometimes does. Instead, it rooted me more firmly in that unreal place. For the first time in weeks, I was lucid. The Awakening Within Sleep Lucid dreaming fascinates me because it feels like a paradox: being awake while asleep. Most nights, I surrender to dreams like a leaf carried downstream—pulled by currents I cannot resist, watching scenes unfold without question. But in lucidity, I awaken inside the current. Suddenly, I am not just a passenger but a pilot,...

Getting Stuck: The Stone Game That Wouldn’t Let Me Go One quiet evening, I found myself thinking about a very old type of problem in computer science and mathematics: two-player impartial games . The idea is simple — players take turns making moves until no move is possible, and the one who can’t move loses. I thought: Let me write some code to analyze one of these games. How hard could it be? The Setup I picked a variation of the Stone Game : There’s a pile of N stones. Two players alternate removing stones. On each turn, a player can remove 1, 3, or 4 stones. The player who takes the last stone wins. A classic "take-away" game. My goal was to write a program that, given N , tells me whether the first player has a winning strategy. Attempt 1: Brute Force Recursion I started with recursion. Define a function win(n) that returns True if the current player can force a win from a pile of size n . def win(n): if n == 0: return False f...

The Pit of Lattice Paths: Counting Without Crossing Some math problems are honest about their difficulty. You look at the Riemann Hypothesis, and it immediately feels intimidating. But lattice paths? Counting walks on a grid? That sounds like something for children. That’s what I thought, at least. Then I stepped in. Step 1: The Innocent Beginning The basic question: How many shortest paths are there from ( 0 , 0 ) (0,0) to ( n , n ) (n,n) if you can only move right ( R R ) or up ( U U )? At first this feels like shuffling cards. You need n n rights and n n ups. That’s 2 n 2n moves in total, arranged in some order. So the answer is just ( 2 n n ) . \binom{2n}{n}. For n = 3 n=3 : ( 6 3 ) = 20 \binom{6}{3} = 20 . Easy! Problem solved. …Or so I thought. Step 2: Adding a Wall What if I require the path to never go above the diagonal ? That is, at every point, the path must satisfy y ≤ x y \leq x . This tiny change sends the problem spiraling. Take n = 3 n=3 : there...