生日悖论：23人从何而来？
探明一个反直觉的概率结论：让任意两人同生日概率过半，所需人数远小于一年天数的一半。
14 min 7 scenes
Explore by topic
Math & Logic
Mathematics, probability, statistics, formal logic, and algorithmic principles.
Why Maps Distort the World
A flat map distorts the world because a round Earth cannot be flattened without stretching; every map projection trades off area, shape, distance, and direction.
8 min 4 scenes
Why the Tides Rise and Fall
The Moon’s gravity creates two ocean bulges—one facing it and one opposite—and Earth’s daily rotation carries each coast through both, producing two high tides a day.
8 min 4 scenes
镜中左右之谜
镜面只翻转与镜面垂直的深度轴；竖直镜的前-后翻转被我们“转过来面对自己”的想象读成了左右互换。
6 min 3 scenes
Why Market Prices Move
Market prices change as the balancing signal: when a shift leaves shortages or surpluses at the old price, buyers and sellers adjust their offers until quantity demanded equals quantity supplied.
8 min 4 scenes
Why Breakup Songs Rule Pop
Breakup songs rule pop when sad, specific lyrics are paired with an upbeat, danceable rhythm—letting listeners feel the heartbreak while moving through it.
6 min 3 scenes
梦与高维交流探秘
从大脑科学、空间维度和批判思维三个角度，重新看待'梦是高维交流'这个流行说法，形成自己的判断。
20 min 10 scenes
Slope of a Curve: Derivatives
Learners can explain what the derivative of a function at a point means and estimate it from a graph as the slope of the tangent line.
20 min 10 scenes
Slope of a Curve: Derivatives
14 min 7 scenes
Slope of a Curve: Derivatives
By the end of this course, you will be able to compute the slope of any polynomial curve at a given point using the limit definition of the derivative.
22 min 11 scenes
Slope: One Key Mistake to Avoid
You'll see that the numerator's sign tells you the direction of the slope, not just steepness.
10 min 5 scenes
Why Shadows Change Size
By the end of this course, learners can explain why shadow length changes with the Sun's position and predict how a shadow will change based on the angle of the light source.
22 min 11 scenes
Why Shadows Change Size
Learners can explain that a shadow's size changes because of the angle between the light source and the object, and the distance from the light source, using the model of straight-line light rays.
22 min 11 scenes
The Shadow Geometry
Learners will be able to explain how light source angle and object positioning dictate shadow length and shape.
18 min 9 scenes
Why Lightning Comes Before Thunder
Learners can explain why light from a storm arrives before sound, estimate how far away a storm is using the time gap, and identify the misconception that lightning happens 'long before' thunder.
24 min 12 scenes
Why Do Bubbles Pop?
Learners can explain, with evidence, the multiple physical processes that cause a soap bubble to pop, and can revise the common misconception that bubbles pop simply because air escapes through the film.
22 min 11 scenes
Why a Rubber Band Warms When Stretched
Learners can explain, with evidence, that stretching a rubber band warms it because polymer chains become more ordered, entropy drops, and the energy has to go somewhere measurable as heat.
26 min 13 scenes
盐为什么能融冰？影子计量验证
学习者能解释盐使冰融化的微观机理（凝固点降低），并设计一个用影子长度变化间接量化融冰速率的验证实验，记录数据并得出结论。
24 min 12 scenes
MathClue: Turning Mistakes into Mastery
Explain what MathClue is, the problem it solves, how its closed-loop works, and why its brand approach differs from typical AI homework tools.
8 min 4 scenes
If the Sun vanished, when would we know?
20 min 10 scenes