When I first heard about the solar neutrino puzzle, I had a little trouble taking it seriously. We know that the sun is powered by a chain of nuclear reactions and that in addition to emitting energy these reactions lead to the emission of neutrinos (uncharged fundamental particles that interact only via the weak nuclear force). The original solar neutrino puzzle was that when physicists made experiments to find these neutrinos, none of them were detected. But by the time I learned about the puzzle, physicists had in fact observed solar neutrinos — only the amount they found was only about 1/3 - 1/2 of the amount that other physicists had predicted. But I was skeptical that this deficit was really a problem — how could we make such an accurate prediction about the sun — an object 93 million miles away about which we can measure only so much? To give one example, the prediction for the neutrino flux was strongly temperature-dependent. Did we really know the temperature sufficiently accurately? Were we sure we understood heat transport inside the sun well enough to trust this prediction?
But I ended up changing my mind (along with many other initially skeptical physicists). The solar neutrino puzzle turned out to be a clue to some very interesting physics. It turns out that neutrinos mix. Every neutrino is labeled by the charged lepton with which it interacts via the weak nuclear force. (Charged leptons are particles like electrons — there are two heavier versions known as muons and taus.) It turns out the neutrinos have a bit of an identity crisis and can convert into each other as they travel through the sun and as they make their way to Earth. An electron neutrino can change into a tau neutrino. Since detectors were looking only for electron neutrinos, they missed the ones that had converted. And that was the very elegant solution to the solar neutrino puzzle. The predictions based on what we knew about the Standard Model of particle physics (that tells us what are the fundamental particles and forces) had been correct — hence change of mind #1. But the prediction had been inaccurate because no one had yet measured the masses and mixing angles of neutrinos. Subsequent experiments have searched for all types of neutrinos — not just electron neutrinos — and found the different neutrino types, thereby confirming the mixing.
And that leads me to a second thing I changed my mind about (along with much of the particle physics community). These neutrino mixing angles turned out to be big. That is, a significant fraction of electron neutrinos turn into muon neutrinos, and a big fraction of muon neutrinos turn into tau neutrinos (here it was neutrinos in the atmosphere that had gone missing). Few physicists had thought these mixing angles would be big. That is because similar angles in the quark sector (quarks are particles such as the up and down quarks inside protons and neutrons that interact via the strong nuclear force) are much smaller. Everyone based their guess on what was already known. These big neutrino mixing angles were a real surprise — perhaps the biggest surprise from particle physics measurements since I started studying the field.
Why are these angles important? First of all neutrino mixing does in fact explain the missing neutrinos from the sun and from the atmosphere. But these angles are also are an important clue as to the nature of the fundamental particles of which all known matter is made. One of the chief open questions about these particles is why there are three "copies" of the known particle types — that is heavier versions with identical charges? Another is why do these different versions have different masses? And a third question is why do these particles mix in the way they have been measured to do? When we understand the answers to these questions we will have a much greater insight into the fundamental nature of all known matter. We don't know yet if we'll get the right answers but these questions pose important challenges. And when we find the answer is is likely at this point that neutrinos will provide a clue.