As a mathematician I love mathematical paradoxes because they are disturbing and thought-provoking. For example, Parrondo's Paradox tells us something counter-intuitive about probabilistic games; Simpson's paradox reminds us that we have to think carefully about statistics; and Curry's paradox is just mind-bending.
Paradoxes in the sciences are important because they make us think about our theories and where they don't quite match reality, driving new scientific ideas. My favourites include Olbers' Paradox (why is the sky dark at night?) and the EPR Paradox which shows us just how surprising the world is.
So I was delighted to come across, in an article by Oliver Southwick in the excellent magazine Chalkdust, a paradox that was new to me, the Mpemba Paradox. "If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?" (The background story is wonderful - read the article!) Not only is there no agreement on the answer, but it gives insights into the mathematical equations involved, and mathematical modelling may help us understand the effect.
Like all the best paradoxes, this is amusing but tells us something surprising about our world.