Summer of Math Exposition

A video forced to obey one rule: no pixel may match its neighbor; left, right, up, down, or itself one frame ago.

Almost every real frame in videos already breaks that rule. This video does not. Each pixel is the original color or one 8 bit step away, so the picture stays the same and remains legal.

If you painted the same grid at random, the chance of a legal picture is F(k) / kⁿ, the count of proper colorings over every possible painting. That fraction is tiny. You are looking at one of the rare ones.

The mathematics is counting all of them. This is a small set of them

Many people assume that Greedy algorithms are a naive and inefficient approach to solving problems, but in fact they underpin many widely used algorithms and are an important concept of algorithm design.

This video aims to give a brief overview of some example Greedy algorithms on optimisation problems, how you prove their optimality using Matroid theory and the Rado-Edmonds theorem, and how you can apply these concepts to your algorithm design.

If you enjoyed the video and want to show some more support, then check out my music! Links are available here: https://linktr.ee/forethoughtmusic

A short animated introduction to the h-cobordism theorem, handle cancellation, and the high-dimensional Poincaré conjecture, imitating the look and feel of old rubber-hose cartoons.

The current version is unfinished and may be temporarily removed; I plan to revise it before voting begins.

My visual method for taking the derivative, plus the proofs of the rules and some extra techniques for finding the stationary points quickly.

Use Best Arm Identification algorithm to optimise shell-shards farming in Silksong!

When we have several tasks, each with its own random reward, and that we want to find the task with the best average reward as fast as possible, this is called a Best Arm Identification problem. In this video, we present two algorithms to deal with this problem: the simple Successive Rejects algorithm and the optimal Track and Stop algorithm.

This video introduces the concept of “approximate counting” algorithms.

Just for Maths Enthusiasm

This is a short video introducing the diffusive scaling property of brownian motion with an animation zooming in on one point. There is a voice-over math explanation for the first two minutes, and the last 4 minutes just consist of the zoom with an original ambient piece of background music.

Deterministic Pattern in Prime Distribution

This video presents a novel geometric and algebraic approach to unveiling the structural boundaries of prime numbers, shifting the paradigm from randomness to determinism. By partitioning the natural number continuum into distinct structural domains, we discover a crystalline architectural framework governed by “Probability Columns.”

Key Pillars of the Research: The Primorial Table System: Analyzing how the interaction of repetition joints creates a balanced, predictable framework.

Paper: https://www.researchgate.net/publication/403619862_Structural_Pattern_of_Primes

Note: The complete LaTeX paper, comprehensive research text, and interactive lab references are fully detailed in the YouTube video description.

In this video I describe how to construct a Turing machine using the Eragon magic system.

The film runs two minutes, states the halting problem, and stages the proof as a dispute between two people. No background is assumed: the film is built so that a viewer who has never met the problem can state it at the end, and say why running a program settles nothing.

The film opens and closes on AI, for the question presses on AI now. An AI is a procedure, and no design escapes what holds of procedure itself.

I met the halting problem writing my submission to the John Locke Institute’s Global Essay Prize, not in a computer science course: the essay turned on whether the reason behind an act can be decided from what a person does, and I turned to undecidability.

Everything was made by hand: no animation library, every motion keyframed in DaVinci Resolve, one property at a time.

The machine’s tape is a cable, on screen for almost the whole film. I photographed it coiled, then straightened it, fixed it to a sheet of white paper with double-sided tape, laid it on my bed, and moved it around the room for better light. Each move changed the white from shot to shot; I graded the shots until they matched.

The film starts from a photographed object, its metal, its wear, its place; the narration then says to ignore all of it, and the grade obeys. The physical goes, depth and tone with it, and everything on screen is one ink, the tape, the symbols written on it, the words. The picture makes the move the argument makes.

The sound took the longest. I recorded the effects on the microphone of a pair of earbuds, held in my hand, standing inside a wardrobe, for the clothes deaden the room; I recorded thirty seconds of silence in the same place, in case I needed it. The typing is mine, in two versions, loud and soft, so each moment could carry the weight it needed; a t-shirt over the keyboard gave the dead thud for the stops. When a line is struck through on screen, what you hear is a pencil dragged across paper.

Nothing was usable raw. The voice went through noise reduction, a leveler, a de-esser, equalization, compression, and a limiter; I recorded four takes and used one. I set the overall level wrong three times, too quiet, then distorting, then too quiet again, before it sat where it should.

I am seventeen and had no equipment. The style is a decision, not a consequence: white background, black text, nothing on screen that is not doing work. Nothing appears at once; every line is written in front of you.

In the proof, the program that breaks any predictor is built on its own text. In the film, the program is a person being petty: told what they will do, they do the other thing. That much anyone has already done, and it takes no notation.

Formal statements run under the narration in gray; nothing in the argument depends on reading them.

The machine halts because no rule matches the symbol under the head, not because it reaches a state marked accept or reject. The convention carries a trap: a machine that halts at the edge of the tape looks like one that ran out of tape, not one that ran out of rules. My first version was worse: I traced it by hand and found it looped forever. So this machine blanks a cell it has already visited, steps back onto it, and halts there, on a blank it wrote itself.

Before I called the film finished, I showed it to someone who had never met the problem and asked what the film said. The answer was right; that was the test I built the film to pass.

In this video, I create a new chess variant, real-valued chess, where the idea is that pieces aren’t constrained to the 64 squares offered by standard chess but are free to exist wherever they like on an 8x8 grid. I explore the construction of this variant and playtest it with a friend.

If you keep drawing triangles in a 1x1 grid, what is the average area. This seemingly easy problem is impossible to solve with standard calculus techniques.

I used a method that combines geometry probability and calculus to find a new simple solution to this problem.

If you’re curious about quantum computing or just want to explore something new. My video explains my first research paper on quantum computing. My paper explores how we can analyze and determine if a quantum algorithm is impossible to create. I hope you enjoy the video :)

Eternalism is the philosophy that time is a spacial dimension that shows itself as slices of a greater hypervolume that we see as the present. This video takes the philosophy to the next level, not only envisioning a world where one can step out of their temporal plane, but also how the laws of physics would operate in such a world.

In this video, we go through the development of Trigonometry from astronomical tool in ancient civilizations, to the modern state of the feild of mathematics.

This is a terrible first attempt at learning manim. I definitely ran out of time.

Club Zero Production

Hi everyone! This is my first video with math explanations, so I apologize in advance if anything was confusing or had slip-ups. I’ve tried to fix everything in the subtitles.

I’ve been passionate about math since childhood and was always a top student in school, though I was more about doing calculations and applying formulas. The real breakthrough came in 11th grade when I started preparing for the Unified State Exam (EGE) and got especially into parameter problems. I solved them on my own without peeking at solutions, and that significantly boosted my level. I scored 97 out of 100 on the EGE in 2025, though I still wasn’t completely satisfied – I’m something of a perfectionist.

After the EGE that summer, I kept studying math and started watching more English-language channels in the original, like 3Blue1Brown. This also improved my English – earlier it was hard to follow videos without subtitles, but now I often watch them without. I fell in love with intuitive explanations and realized how important it is to present material well. Once I got to university, I noticed how dry and fragmented math is taught there: they give formulas but don’t show the connections between them. I wanted to offer explanations that are intuitive, honest, and free from rote memorization.

At first, I started writing a detailed script for explaining the school curriculum, but then decided to focus on derivatives and make videos based on rough notes. Most of the work happened in June–July: I spent several days thinking through each idea and looking for the best explanation, cross-checked my hypotheses with AI, and came up with many explanations myself (e.g., for the derivative of power functions, composite functions, and the total differential). It was through tables that I figured out how to teach derivatives – and I haven’t seen that approach anywhere else.

During this time, I noticed especially how disconnected topics are in standard teaching: they give algorithms but don’t explain why they work or how they connect to what you’ve already learned. So in this video I tried to build a coherent picture of derivatives – not through examples, but through ideas.

I decided to join the Summer of Math Exposition (SoME) competition. For me, this is a chance to test my skills and show that understanding can be clear and vivid. I hope this video doesn’t go unnoticed. And yes – I finally published it!

Videos mentioned:

• Derivative of exponential function
• Derivative of sine and cosine
• Essence of calculus
• Linear algebra

Timestamps:

• 00:00 Introduction
• 01:23 The derivative – instantaneous rate of change
• 09:02 Derivative of e^x
• 11:50 Derivative of sine and cosine
• 14:47 Derivative of x^n
• 20:43 Constants factor out of the derivative
• 21:33 Derivative of a sum
• 23:52 Derivative of a product (+generalization)
• 27:10 Derivative of a quotient
• 31:36 Derivative of a composite function (+a^x)
• 35:16 Derivative of the inverse function
• 36:11 Deriving the remaining standard derivatives
• 41:38 Logarithmic derivative and bringing a function to base e before differentiating
• 45:21 L’Hôpital’s rule
• 50:06 Equation of the tangent line
• 53:18 Higher-order derivatives and differentials
• 57:46 Taylor’s formula
• 1:06:41 Critical points (extrema)
• 1:13:59 Functions of several variables. Partial derivatives and the total differential
• 1:23:28 Tangent plane / hyperplane
• 1:25:08 Directional derivative
• 1:32:16 Derivative of an implicit function
• 1:38:22 Gradient
• 1:43:35 Higher-order differentials for functions of several variables
• 1:47:36 Connection between higher-order differentials and binomial/polynomial coefficients
• 1:57:06 Taylor’s formula in a nutshell
• 1:58:39 Extrema of functions of several variables
• 2:11:40 Constrained extrema
• 2:25:28 Global extrema in a nutshell

Drawing app: DrawNote Subtitles made with riverside.fm

A general observation which makes you ponder about what is and how is Mathematics. Improve your Mathematics thinking.

This video addresses a foundational misconception in elementary mathematics education: the treatment of cardinality and ordinality as the sole defining properties of numbers. However, these properties derive specifically from Zermelo–Fraenkel set-theoretic formulations rather than representing the full utility of numbers.

By analyzing the mid-20th-century “New Math” curriculum reform and Piaget’s stages of cognitive development, the lesson demonstrates how introducing formal set abstractions to early to students winds up hiding other properties of numbers. The entry explores nominal numbers, collective numbers, multiplicative numbers, distributive numbers, disjunctive numbers, and more..

A beginner’s introduction to the exciting field of proof theory, outlining important milestones such as Hilbert’s Program, the Curry-Howard correspondence, and the world of proof assistants, all in under 1 minute.

Quantum teleportation