Brauer–Manin obstruction

So, imagine you have a really cool toy car that you want to put together. But you don't have all the pieces! You're missing the wheels and the engine. You look for them everywhere, but you just can't find them. So even though you have the body of the car, you can't actually use it because it's missing important parts.

Now, let's talk about numbers, specifically whole numbers that are called "rational numbers." People who love math (like mathematicians) like to study these rational numbers and see if they have patterns. They want to know if there are any missing pieces in a special set of these numbers.

This special set of numbers is called the set of "solutions to equations," and it's made up of all the sets of numbers that could solve a particular math problem. Just like your toy car was missing pieces, sometimes these sets of numbers are missing important pieces too.

The Brauer-Manin obstruction is like a detective who tries to figure out if there are any missing pieces in these sets of numbers. It's a fancy way of saying there might be something stopping us from seeing the complete solution to the math problem.

The Brauer-Manin obstruction looks at a type of math called "algebraic geometry" to help it solve the mystery of the missing pieces. Basically, it uses a bunch of fancy math tools to see if there are any invisible parts missing from the solutions to these math problems.

So, to sum it up: the Brauer-Manin obstruction is like a detective that helps mathematicians figure out if there are any invisible missing pieces in a special set of numbers that could solve a math problem. Sort of like how your toy car had invisible missing pieces so you couldn't play with it properly!