Solving an elastic collision of equal masses in 1 dimension
In an elastic collision, both momentum and energy are conserved. The 1 dimensional case is mildly interesting: all collisions between points are 1 dimensional (an off-centre collision between balls is not 1 dimensional; see below).
So say we have 2 equal masses in an elastic collision on the line. All we know is that total momentum (m(v1+v2)) must be conserved, and so too must total kinetic energy (m*(v12/2+v22)). The velocities after the collision are unknowns, so we have 2 equations in 2 unknowns. We could solve them. We'd be likely to make a mistake, though.
So let's solve them without knowing anything! First, note we have a quadratic equation and a linear equation in our system, so we'll have 2 solutions (or a double solution, but we'll see that only happens if v1=v2). When this happens in Physics, it's invariably the case that one of the solutions is "unphysical" -- it cannot happen.
What's the infeasible solution (use a technical term, to sound more professional)? Easy! Suppose both velocities remained unchanged. Obviously total momentum and kinetic energy would be unchanged! The reason this solution is unphysical is that it involves the 2 objects passing right through each other, which is generally frowned upon by experimental physicists.
So now we have a hint how to get the other solution. The balls have equal mass, so if we just exchange v1 and v2 we're guaranteed to conserve kinetic energy and momentum. Hence this is the (only) other solution, therefore it describes what happens.
Here's another way of thinking about it: if the points were indistinguishable (say, both painted the same shade of pink, etc.), we'd expect not to be able to distinguish the result of the collision from no collision occurring. And this is indeed what we find.