Lebesgue's universal covering problem

Imagine you have a big piece of paper, and you want to cover it entirely with some little pieces of tape. But there are some rules you have to follow: the tape pieces can’t overlap or leave any gaps, and they have to be at least a certain size.

Lebesgue's universal covering problem is like this, but on a much, much larger scale! It's a question that mathematicians have been trying to solve for a long time. They want to know if you can cover every single point in a certain space using a certain kind of shape (called a "tile").

The shape they're using is called a unit square, which is like a little tile that's one unit wide and one unit tall. The space they're trying to cover is called the plane, which is like the entire surface of your big piece of paper.

But there are some rules they have to follow. The tiles can't overlap or leave any gaps, just like the tape pieces. And they have to be positioned in a certain way, so that every point on the plane is covered by one and only one tile.

Now, here's where it gets really tricky: mathematicians have actually already figured out how to cover some parts of the plane using these unit squares. But they don't know if it's possible to cover the entire plane using them! And that's what Lebesgue's universal covering problem is all about.

So imagine you have a ton of little unit squares, and you want to cover every single point on a giant piece of paper--without leaving any gaps or overlaps. That's what mathematicians are trying to figure out!