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Must a mathematical proof find its answer step by step? The mathematician Erdős offered a marvelous idea: if we can prove that “a random choice has some chance of success,” then we have proved that a solution must exist. Make some coffee and join us in appreciating these artful proofs. Author: silverxz Reviewed by: phy东西 Paul Erdős Image source: zbMATH   Paul Erdős is not exactly a household name, but within mathematics, almost everyone knows him. He was a legendary mathematician: prolific, wide-ranging, and responsible for remarkable results in number theory, set theory, analysis, geometry, and many other fields. His standing in combinatorics is especially unassailable.   There are plenty of stories to tell about Erdős—his unusual way of living and doing mathematics, the many ways in ...

This summer, the annual Perseid meteor shower has arrived right on schedule. Why do meteor showers return at the same time every year? And why are they named after constellations? Let us begin with a speck of space dust and trace the cosmic paths behind meteors. Author: emmm Reviewed by: Situ Mo   A meteor may be the briefest and most brilliant “firework” in the night sky. A streak of light tears through the darkness without warning, vanishes an instant later, and leaves only cries of surprise behind. Right now, the Perseid meteor shower is approaching its peak, bringing the best time of the year to “run into” a meteor. Before you marvel at the sky, however, have you ever wondered what lies behind that flash of light? What Is a Meteor?   Before introducing meteors formally, let us dist ...

⚠️ This science article comes with a certain aroma! Why is poop usually yellowish brown no matter what we eat? Does the color come from food, or from an epic journey through the body? By the end of this article, you will know where the familiar color comes from—and what other stool colors may be trying to tell you. Authors: 阿氯 and 时光 Reviewer: 司徒陌   You probably do not study your poop closely after using the bathroom. Take an occasional look, though, and you will notice that it is usually yellowish brown. Braised pork and leafy greens generally end up producing the same color, although a few foods—notably dragon fruit—can tint the stool. The normal yellow-brown color comes mainly from stercobilin, a product of the reactions that bilirubin undergoes in the body.   So bilirubin appears to ...

After rain, mysterious rings of mushrooms can appear in grass—and even in the game Sky: Children of the Light. Some have been growing for decades or even centuries. How do these strange rings form? Why do some make the grass greener while others are classified as turf diseases? Let us uncover the secrets behind mushroom rings. Authors: 阿氯、时光 Reviewed by: 星海中的蓝玫瑰   After the rain clears, have you ever seen clusters of mushrooms arranged in a circle on a lawn, as though someone had deliberately planted a ring of tiny umbrellas? In folklore, the phenomenon is often called a fairy ring or mushroom ring; in scientific Chinese, it is usually known as a xùn ring. What you may not know is that some large mushroom rings have been growing underground for more than a century.   What we call a “mus ...

Can the digit string 1145141919810 be turned into an equation? Astonishingly, every integer with at least eight digits can become a true equation if we insert addition, subtraction, multiplication, division, parentheses, and an equals sign. From counterexamples and constructive arguments to a computer search, this article proves a mathematical claim that sounds absurd at first. Author: 硅基·飙尘葆光 A Brief Introduction to the Silicon–Silence Balance Integer Theorem and Its Proof   In daily life, sophon interference from the Trisolarans often disrupts the numbers we try to transmit. Humanity has therefore developed many ways to disguise digit strings and thwart this evil plan. One method inserts arithmetic operators into a string until it becomes a true equation. For example, “114514” can be ...

Can electricity be generated without “boiling water,” using nothing more than a temperature difference? Why should that difference make electric charges move? And how can such a seemingly weak effect power a spacecraft in deep space? This article introduces the Seebeck effect and explains how “heat” can become electricity. Author: ST Reviewer: 时光   When people talk about generating electricity, many describe it as “boiling water”: heat the water, let the steam turn a turbine, and convert mechanical energy into electrical energy. The most obvious example is fossil-fuel power generation, where coal, natural gas, oil, or another fuel is burned to heat water. Nuclear, geothermal, biomass, and even waste-to-energy plants often use the same basic route. Other technologies—wind turbines, photo ...

The number of comparisons made by a deterministic comparison-based sorting algorithm on its worst-case input has a lower bound of Ω(nlg⁡n)\Omega(n\lg n)Ω(nlgn). But what about the expected number of comparisons made by a randomized algorithm on its worst-case input? This article introduces Yao’s principle and uses it to analyze the lower bound on comparisons in randomized sorting algorithms. Author: silverxz Reviewed by: phy东西 Introduction   The lower bound of Ω(nlg⁡n)\Omega(n\lg n)Ω(nlgn) on the number of comparisons made by a deterministic comparison-based sorting algorithm on its worst-case input is a famous result. It appears in textbooks such as Introduction to Algorithms and in most algorithms courses, so readers are probably familiar with it. (This article assumes only an introd ...

When you put a bottle of mineral water in the refrigerator, will it chill faster lying down or standing upright? A horizontal bottle seems to expose more surface area to the cold air, but does that really make it cool more quickly? Read on to find out. Author: 锅炉-251 Reviewed by: 薛定谔的猫 Introduction   Suppose you put a room-temperature bottle of mineral water in the refrigerator and want it to cool as quickly as possible. How would you position it? Many people might answer intuitively, “Lay it on its side—it seems to expose more area to the cold air.” Others might argue, “Standing it upright is better because cold air moves downward.”   This simple question from everyday life attracts conflicting answers, but it can be explained through an important concept in heat transfer: natural con ...

Mathematics has not always been rigorous from the outset. Throughout its history, both Bertrand’s paradox and disputes over infinitesimals have exposed the uncertainty created by vague definitions. In response, mathematics gradually moved toward formal expression. Author: silverxz Reviewer: ΔδDelta\Delta\delta DeltaΔδDelta Introduction: Ask “What Is It?” First   Whether we are learning mathematics or solving a mathematical problem, asking “what is it?” first is a natural instinct.   A high-school student encountering sets for the first time must begin by knowing what a set is, even if the answer is only the informal statement that “a set is a whole formed by putting some objects together.” Only after knowing what it is can the student answer other questions about sets or move on to con ...