How Many Rotations Does the Coin Make?
The Key to Success in the Age of AI: Clear Thinking
In today's era where the wave of artificial intelligence is sweeping through all industries, many people worry about being replaced by AI. However, instead of fearing it, it's better to actively cultivate core competencies that are even more important in the AI age: clear thinking, precise problem-solving skills, and the practical ability to turn ideas into reality. The author will periodically share thought-provoking questions and methods to practice these skills, hoping to help readers stand out in the AI era.
Today, let's start with an interesting geometry problem:
Problem: Two coins of the same size, coin A, are placed against coin B and rolled along its edge. When coin A goes around coin B once and returns to its original position, how many times has coin A rotated?
Please choose: (a) 2/3 rotation (b) one rotation (c) two rotations (d) six rotations
You might solve it like this: Coin A travels along the circumference of coin B, so the total length is the circumference. Since the two coins are the same size, going around B means traveling one circumference, which is one rotation.
Let me tell you the answer is not (b), not one rotation. Think about it again.
Or take out two coins and try rolling them.
This seemingly simple problem often leads people to intuitively choose "one rotation." However, actual operation or deeper thinking will reveal that the answer is not so straightforward. Let's analyze it carefully:
When coin A rolls around coin B, its rotation involves not only moving along the circumference of coin B but also rotation around its own axis. You can imagine a point on coin A that initially touches a point on coin B. When coin A rolls halfway around, this point will move to the very top of coin A. And when coin A rolls a full circle back to the starting position, this point has actually rotated around its own axis twice!
The correct answer is (c) two rotations.
Why is this the case? We can understand it this way:
Relative Motion: Relative to the center of coin B, the center of coin A moves one full circle. This corresponds to one rotation of coin A itself.
Self-Rotation: At the same time, because coin A is rolling on the circumference of coin B, it also needs to rotate once on its own to fully contact the entire circumference of coin B.
Therefore, the superposition of these two rotations means that after coin A goes around coin B once, it has rotated a total of two times.
This simple coin problem actually contains a profound truth. In the age of AI, the answers to many questions are not as obvious as they seem on the surface. AI can quickly process large amounts of data and provide seemingly reasonable answers, but true insight often comes from our clear thinking and in-depth exploration.
Clear thinking helps us identify the essence of a problem and avoid being misled by superficial information. Just like this coin problem, the intuitive first answer is "one rotation," but only through carefully analyzing the components of the motion can we arrive at the correct answer. Precise problem-solving skills allow us to ask the right questions, guiding us to the crux of the issue. If we only stick to the surface understanding that "rolling around once is one rotation," we won't be able to discover the hidden relative motion and self-rotation. Practical ability encourages us to get our hands dirty and verify our thinking through actual operation. Take out two coins and try rolling them; you will more intuitively feel that coin A does indeed rotate twice.
In the future of rapidly advancing AI technology, having the ability to think independently, ask precise questions, and dare to practice and verify will be the key to not being eliminated by the times and even mastering the AI wave. Let us work together to cultivate these core competencies and embrace the challenges and opportunities of the AI era.