Arya’s Algorithms

  The Hidden Ocean Beneath Thought All day long, my mind feels like a busy city. Thoughts move through me like traffic—decisions, worries, plans, half-remembered melodies. But sometimes, when I pause, I sense something deeper. Beneath the chatter of consciousness lies an ocean I rarely touch. It is vast, hidden, and strangely alive: the unconscious. The Vastness Beneath the Surface Freud once compared the mind to an iceberg: only a small tip visible above the water, the rest submerged. Conscious thought, the part I identify as “me,” is just the glittering tip. The rest—the urges, fears, forgotten memories, unspoken desires—lie beneath, shaping me invisibly. And yet, I often forget this. I live as though my conscious thoughts steer the ship. But in truth, the ship is pushed by currents I barely understand. When I speak, sometimes words come faster than my awareness. When I dream, stories unfold without my permission. Even my instincts—fight, flight, hunger, attraction—are deci...

The Pit of Tiling: Dominoes, Trominoes, and the Madness of Covering Boards Some math problems look like games. Tiling problems are the purest of these: you have a shape (a board), you have some tiles (dominoes, trominoes, etc.), and you want to cover the board without overlaps or gaps. It feels like a puzzle you’d get in a children’s magazine. That’s what I thought when I first tried to tile a rectangle with dominoes. Then I fell into a pit that connected geometry, combinatorics, algebra, and computational complexity. Step 1: Dominoes on a Chessboard — The Happy Start Take a standard chessboard, 8 × 8 8 \times 8 . Can you tile it with dominoes, each covering two adjacent squares? Yes — and easily. Just line them up in rows. What about smaller boards? A 2 × n 2 \times n board: clearly possible, just lay them end to end. In fact, the number of ways is exactly the Fibonacci numbers! (each tiling ends with a vertical domino or two horizontals). A 3 × n 3 \times n board? A...

The Pit of the Banach–Tarski Paradox: How to Double a Ball There are problems that feel like they shouldn’t exist. They don’t just resist intuition — they spit in its face. The Banach–Tarski paradox is one of them. The statement is so wild you’d expect it to be a joke: You can take a solid ball in 3D space, cut it into finitely many pieces, and reassemble those pieces — using only rotations and translations — to form two balls identical to the original. Not squashed, not stretched. Just rotated and moved. One ball becomes two. When I first heard this, I thought: impossible. Surely someone is cheating. Surely this is wordplay. Surely this is some optical illusion. Then I stepped into the pit. Step 1: My First Reactions “Conservation of volume!” My physics instincts screamed at me. Volume can’t just double. “Maybe the pieces overlap?” But the theorem insists: no overlap, no stretching. “Maybe the pieces are infinitely many?” No. Just five pieces. Everything I tr...

  Social Hierarchies: The Invisible Power Structures in Our Daily Lives Walk into any social gathering and watch carefully. Notice who speaks first, whose opinions carry more weight, who gets interrupted and who doesn't, whose jokes get the biggest laughs, and who seems to naturally command attention without even trying. Within minutes, you'll observe something both fascinating and uncomfortable: the emergence of invisible social hierarchies that organize human interaction in ways we rarely acknowledge or discuss. These hierarchies aren't necessarily about formal authority or official titles. A CEO might defer to a local community leader at a neighborhood barbecue. A confident teenager might dominate conversation at a family dinner while successful adults listen. A soft-spoken expert might quietly influence major decisions while louder voices go unheeded. Understanding these dynamic, context-dependent power structures is crucial for navigating social interactions effecti...

Problem Pit: The Deceptive Double Integral The Problem Evaluate the double integral: $$\iint_R \frac{x-y}{(x+y)^3} \, dx \, dy$$ where $R$ is the region bounded by $x = 0$, $y = 0$, $x + y = 1$, and $x + y = 2$. This looked like a standard double integration problem. The region seemed clear, the integrand manageable. What could go wrong? As it turns out, this problem would take me through coordinate transformations and some surprising insights about symmetry. Setting Up the Region First, let me understand the region $R$: - $x \geq 0$ (right of y-axis) - $y \geq 0$ (above x-axis) - $x + y \geq 1$ (above the line $x + y = 1$) - $x + y \leq 2$ (below the line $x + y = 2$) The vertices of this trapezoidal region are: - $(1,0)$ where $x + y = 1$ meets $y = 0$ - $(0,1)$ where $x + y = 1$ meets $x = 0$ - $(2,0)$ where $x + y = 2$ meets $y = 0$ - $(0,2)$ where $x + y = 2$ meets $x = 0$ So $R$ is a trapezoid with these four vertices. First Approach: Direct Integration...

  Getting Stuck: The Threads That Wouldn’t Behave Concurrency has always fascinated me. The idea that multiple tasks can run “simultaneously” — or at least give the illusion of simultaneity — feels like wizardry. But when I tried to actually implement something non-trivial, I got stuck in ways I never expected. The Task I wanted to simulate a bank with multiple accounts. Each account supports deposit and withdrawal. Multiple users (threads) perform transactions at the same time. Sounds simple. Attempt 1: Naive Multithreading I started with Python’s threading module: import threading class Account: def __init__(self, balance=0): self.balance = balance def deposit(self, amount): self.balance += amount def withdraw(self, amount): if self.balance >= amount: self.balance -= amount return True return False account = Account(100) def transaction(): for _ in range(1000): account.depo...

  The Pit of Fractal Integrals : Measuring the Unmeasurable with Takagi’s Monster There are functions that refuse to be smooth. They wiggle, they bend, they squirm, and the closer you look, the worse they behave. In real analysis, such creatures live everywhere — but a few have become infamous for how innocently they present themselves before revealing their pathological depths. One such monster is the Takagi function , sometimes called the “blancmange function.” On first sight, it looks like a child’s zig-zag doodle. Defined by a simple infinite sum of tent functions, it has the unnerving property of being continuous everywhere, but differentiable nowhere. When I first met it, I thought: “Fine, I know Weierstrass did this already. This is just another example of pathological analysis.” But then I asked a simple question: I ( α ) = ∫ 0 1 f α ( x )   d x , I(\alpha) = \int_0^1 f_\alpha(x) \, dx, where f α ( x ) = ∑ n = 0 ∞ α n   ϕ ( 2 n x ) , ϕ ( x ) = min ⁡ k ∈ Z ∣ x − k ...

you can find the full paper here First Attempts, and the Limits of Energy When I first began thinking about angular occupancy, I wasn’t looking for a brand-new problem. I was chasing a theme that has long fascinated me in discrete geometry: how do we measure the geometric richness of a finite set of points? This fascination is not unique to me. Erdős, way back in 1946, posed the distinct distances problem : given n points in the plane, how many distinct distances can they determine? That question alone gave birth to entire decades of work in combinatorial geometry. Later, Fishburn and Füredi asked about distinct angles , while Pach and Sharir studied repeated angles . Each time, the same underlying itch was being scratched: when you place points in the plane, what kinds of diversity can they generate? Why angles, why bins? It was natural to think of angles — after all, distances and directions had already been well studied. But if you go straight for “distinct angles,” you run in...