This is the fifth problem in the book 200 Puzzling Physics Problems. The problem itself does not seem particularly unique, but the solution is remarkably ingenious:
Four snails are moving in uniform linear motion (each with a different velocity). Their trajectories are straight lines that intersect pairwise, but no three lines intersect at a single point; that is to say, their trajectories have six intersection points. Among them, five encounters have already occurred. The question is: will the sixth encounter necessarily occur? In other words, is it possible for only five encounters to take place?
Solution: The sixth encounter will definitely occur!
How do we solve this? One fairly simple approach is to draw a "spacetime diagram." That is, establish an OXYT Cartesian coordinate system and plot the motion of all four snails. Since they are moving in uniform linear motion, their spacetime paths must be four straight lines in three-dimensional space. The fact that they move in different directions implies that their spacetime lines are not parallel (i.e., they have different velocities). An encounter means that the spacetime lines intersect. One only needs to prove that if four lines exist such that no two are parallel, and they have five intersection points, then they must be coplanar and a sixth intersection point must exist. This argumentation process is also quite simple, as one only needs to prove that if three lines intersect at the same point, they can produce at most four intersection points.
However, there is another brilliant line of reasoning that demonstrates the concept of relative motion to its fullest! Since five encounters have already occurred, there must be one snail that has already met the other three snails. If we could shrink to the size of an ant and sit on that snail to observe, what would happen? We would see the other three snails still moving in uniform linear motion. Each of them would approach us at a certain moment, pass through us, and continue their journey. This means that, from our perspective, the trajectories of the other three snails are three lines passing through a common point (our position). At the same time, it can be proven that these three lines coincide (lie on the same line), because if they did not coincide, the fourth and fifth encounter events could not have occurred. Once they coincide, the sixth encounter must necessarily happen! Q.E.D.
What an exquisite proof, isn’t it? Changing one’s perspective to look at the world—this line of thought is truly worth our deep reflection...
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