Drop a glass and it shatters. Wait as long as you like, and the shards never leap back together into a glass. Cream poured into coffee swirls into a uniform brown and never spontaneously separates back out. An egg breaks; it does not unbreak. You know, without being told, which way time runs in a video just by watching whether these things happen forwards or backwards. This lopsidedness — the fact that time seems to have a direction, a "before" that's fundamentally different from "after" — is what physicists call the arrow of time.

Here's the puzzle: nothing in the fundamental equations of physics points that arrow. Newton's laws of motion, Maxwell's equations of electromagnetism, Einstein's equations of general relativity, the equations of quantum mechanics — all of them share a property called time-reversal symmetry. That means if you take any process the laws allow and run the film backwards, the reversed process is also allowed by the same laws. Flip the sign of time, $t\to -t$, in the equations, and you get equations describing an equally valid physical process.

Think about two billiard balls colliding on a frictionless table. Ball A hits ball B, and they bounce off at some angles with some speeds. Now imagine the reverse film: balls moving in from the angles they ended at, colliding, and separating along the paths they originally came in on. Both films obey Newton's laws perfectly. There is nothing in the physics of that collision that says "this direction is allowed, the reverse is forbidden." A physicist watching only that one collision, with no other context, could not tell which film was playing forward and which was playing in reverse.

Processes like this — where the reversed sequence of events is just as physically valid as the original — are called reversible. A frictionless pendulum swinging back and forth is reversible: run the video backwards and it still looks like a pendulum obeying the laws of mechanics. But a shattering glass is irreversible: the forward process (intact glass to shards) obeys the laws of physics, and so, in principle, does the reverse (shards to intact glass) — yet we never see the reverse happen. That asymmetry between what's allowed and what actually happens is the whole mystery.