many-body problem
Two people can plan a meeting easily; three friends already argue about where to eat; a hundred guests at a party split into ever-shifting clusters that no one can predict. The more participants who all influence each other, the harder it becomes to say what the whole crowd will do. That is the everyday shape of the many-body problem.
In physics it means this: write down the rules for how a huge number of particles — say, the electrons in a chunk of metal — push and pull on one another, then try to work out how they all move together. Each particle feels every other one, so you cannot solve for one at a time; the equations tangle them all together. Even with just a few dozen quantum particles, the exact answer outgrows any computer, because the difficulty explodes as you add particles.
It matters because nearly everything interesting in a material — magnetism, superconductivity, the very fact that some things are solid — comes from particles acting together, not alone. The honest reality: there is no general exact solution, so the field lives on clever approximations, simplified models, and powerful computers, each trusted only within limits.
Try to track the gravity between just the Sun, Earth, and Moon over millions of years and even that three-body case has no neat formula — astronomers must step it forward by computer. A speck of metal holds about 10^23 electrons, each tugging on all the rest. The many-body problem is that same impossibility, scaled up beyond imagination.
Even three bodies under gravity defy a tidy formula; a metal has 10^23 interacting electrons.
The trouble is not just 'many' particles but that they interact. A box of gas atoms that ignore each other is easy — you treat one and multiply. The problem becomes 'many-body' precisely when each particle's fate hinges on all the others.