15 balls with the same spacing and speed move along a straight line, another 16 balls with the same spacing and speed move along the same straight line but at opposite direction. The balls are all the same size and mass and the spacing between the balls at both directions is the same. When two balls collide, they will reverse their moving directions, but maintain the same speed (frictionless moving and collision). How many collisions will occur?

解：

Let’s consider a general case where we have n balls on the left, and m balls on the right. Label these balls as L1, L2, …, Ln and R1, R2, …, Rm.

If a ball collides with another ball, both balls will bounce back with the same speed. Rather than seeing this as a collision, we may treat this as that both balls “go through” each other but exchange their labels. For each ball on the left, it has to “go through” m balls, so there are n*m “going-throughs”. Similarly, for each ball on the right, it has to “go through” n balls, so there are another n*m “going-throughs”. Each exchange corresponds to two “going-throughs”, so the total number of exchanges, or equivalently the total number of collisions, is n*m.

So we have 15 * 16 = 240 collisions.

Note: as far as I can see, equal spacing is not necessary.