A comprehensive guide to the Hodge Conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, exploring its role as a bridge between algebraic geometry and topology and its deep mathematical background.
A detailed explanation of the full picture of the 'Poincaré Conjecture' that troubled mathematicians for over 100 years, the basics of topology, and the story of the dramatic proof by Grigori Perelman.
A detailed explanation of the Twin Prime Conjecture, an unsolved problem in mathematics, including its history, partial solutions, and the latest research trends.
Delve deeply into the history of the Continuum Hypothesis (CH) proposed by Georg Cantor, its mathematical background, and the proofs of its independence by Gödel and Cohen.
A deep explanation of the background and meaning of Euler's identity, considered the most beautiful equation in the history of mathematics.
We explain in detail the history of the Four Color Theorem, a difficult problem in the mathematical world, proof approaches, and the significance of computer-assisted proofs.
A detailed explanation of 'Goldbach's Conjecture', one of the most famous unsolved problems in mathematics, covering its history, meaning, and significance in modern mathematics.
An in-depth look at the overview of the ABC Conjecture and the groundbreaking approach by Professor Shinichi Mochizuki using Inter-Universal Teichmüller Theory (IUT Theory).
An explanation of the Law of Large Numbers, a crucial theorem in probability theory. We delve deep into casino profit structures, rigorous mathematical definitions (weak and strong laws), with diagrams and Python code.
The "Ancestor Paradox" occurs when the number of your ancestors going back in generations exceeds the world's population at the time. We explain this mystery, "Pedigree Collapse", in detail from mathematical and genetic perspectives.
Can we create an 'election system that is fair to everyone'? Through 'Arrow's Impossibility Theorem' proven by economist Kenneth Arrow, we explain the mathematical limits of democracy and how we should face elections and decision-making.
We explain Gödel's Incompleteness Theorems, a historic theorem that showed the limits of mathematics, including its meaning and how the proof works, in an easy-to-understand manner with concrete examples and diagrams.
We explain two representative randomized algorithms that utilize random numbers: the Monte Carlo and Las Vegas algorithms. We summarize specific algorithm implementations and their differences in an easy-to-understand manner.
Where are we in the history of humanity? We explain the 'Doomsday Argument,' which probabilistically predicts the timing of human extinction using the Copernican principle and Bayesian inference, with formulas and diagrams.
We explain the Halting Problem, one of the most famous (and proven) unsolved problems in computer science. It is explained easily using proof by contradiction, intuitive concrete examples, formulas, and diagrams.
The thought experiment 'Maxwell's Demon' that seems to break the second law of thermodynamics. We explain in detail how this paradox was unraveled by the 'physics of information,' using formulas and diagrams.
A detailed explanation of the famous 'Robert Floyd's cycle-finding algorithm (Tortoise and Hare Algorithm)' for detecting cycles in lists and graphs, featuring diagrams, mathematical proofs, and implementation in C++ and Rust.
We explain the 'Byzantine Generals Problem', a representative challenge in distributed systems, from its concept to mathematical proof, and its application in blockchain, featuring concrete examples and diagrams.
Why does the intuitively understandable 'Pigeonhole Principle' become the most important concept in the security of modern cryptography and hash functions? We explain it deeply using concrete examples, formulas, and diagrams.
Using the Greek philosophical paradox 'The Ship of Theseus' as a theme, we delve deeply into refactoring, system replacement in software development, and 'identity' in object-oriented programming.
