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Mathematics, probability, statistics, formal logic, and algorithmic principles.
The distance formula (d = √((x₂−x₁)² + (y₂−y₁)²)) and midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2) operate purely on endpoint coordinates, so axis crossings and quadrant changes affect only the signs and magnitudes inside the formulas, not the formulas themselves.
The length of a segment is the Pythagorean distance between its endpoints, and the midpoint is the average of the x-coordinates and the average of the y-coordinates.
Voting rules translate a profile of rankings into a single winner by emphasizing different things — frequency of being first, average rank, or head-to-head victories — and these emphases are mathematically incompatible whenever a Condorcet loser exists.
A voting rule defines what 'most preferred' means, so changing the rule can change the winner without any voter changing their mind.
有限域上的代数曲线本质上是满足多项式方程的有限点集；当 p 增大时这些点在视觉上趋近于抛物线，但其精确结构由模 p 算术的离散性决定，而非实数几何。
This investigation explains why rainbows need both raindrops and sunlight, and why they always appear as a colored arc opposite the Sun.
试除法判断素数的最坏步数约为 √n；n 有 d 位时，这约等于 10^(d/2) 次候选除法，所以位数线性增加、工作量却指数爆炸——30 位还能勉强算，60 位就超过宇宙年龄。
When the price cannot adjust, the market stays out of balance: a price stuck below equilibrium creates persistent shortages, and a price stuck above equilibrium creates persistent surpluses.
理解 RSA、有限域离散对数与椭圆曲线离散对数各自依赖什么数学难题，以及这些差异如何导致密钥长度和运算方式不同。
Why majority rule can fail to produce a single clear winner, and why voting procedures matter.
生日悖论不是趣味谜题，它把哈希函数的抗碰撞能力从 n 位压缩到 n/2 位，成为密码学中必须正视的现实威胁。
A parabolic mirror gathers parallel waves into one focus, which is why it powers satellite dishes, solar concentrators, and car headlights.
One formula turns any normal curve into the standard normal curve, so z-scores provide a common scale for comparing values across different distributions.
A parabola's width at the focus is exactly twice its focus–directrix distance, so one measured distance reveals the scale of the whole curve.
The 68-95-99.7 rule works because the normal curve has a fixed relative shape: when distance from the mean is measured in standard deviations, fixed areas under the curve—68%, 95%, and 99.7%—always contain the data.
A parabola's width is set by the leading coefficient a: larger |a| makes it narrower, smaller |a| makes it wider.
How the normal distribution's bell-shaped curve is completely described by two numbers—its center and spread—and why that pattern explains many real-world measurements.
一个简洁的公式（a = m² - n², b = 2mn, c = m² + n²）就能生成无穷多个整数边直角三角形。
理解碰撞概率的核心思想，并用生日悖论和泊松近似解释现实中的巧合与风险。
用成对因子、√n 边界和质数候选来组织试除，把大合数快速分解成质因数乘积。
A concrete construction showing that the unit group of the binary Leavitt algebra is not sofic, thereby disproving the soficity conjecture.
从几何定义到密码学基础：理解椭圆曲线的形状、点加法与标量乘法，以及它们如何支撑现代加密。
通过质因数分解，判断并证明‘每个大于2的整数要么被4整除，要么含某个奇素因子’是否成立。
费马大定理的证明不是一条初等捷径，而是一条由椭圆曲线、模形式和伽罗瓦表示铺成的现代数学高速公路，理解这条路线比记忆细节更重要。