One of the most popular "paradoxes" in history, predicted trhough A. Einstein's special theory of relativity. Take two twins, born on the similar date on Earth. One, Albert, leaves home for trip around the Universe at extremely high speeds (extremely close to that of light), whereas the other, Henrik, stays at home at rests. Special relativity predicts that while Albert returns, he will determine himself much younger than Henrik.
Actually, that is not the paradox. The paradox stems from trying to naively analyze the condition to figure out why. From Henrik's view point (and from everyone else on Earth), Albert appear to speed off for a long time, linger around, and then return. Therefore he must be the younger one, which is what we view. But from Albert's view point, it's Henrik (and the whole of the Earth) that are travelling, not he. In according to special relativity, if Henrik is moving relative to Albert, then Albert must be measure his clock ticking slower and therefore Henrik is the one who must be younger. But it is not what happens.
Thus what's wrong with our analysis? Here the key point is that the symmetry was broken. Albert did something which Henrik did not did Albert accelerated in turning around. Henrik did no accelerating, as he & all the other people on the Earth can attest to (neglecting gravity). Thus Albert broke the symmetry, and while he returns, he is the younger one.