This Seems Such A Simple Problem,

but still does not have a full answer,

even after 100 years! you have some holes in your jeans which you want to patch, they are all weird shapes, but never wider than an inch, taking some circular patches, each 1 inch in diameter should work, but will it? this is just a simple elementary geometry problem, but this problem, called the “universal cover” problem has not been solved in 100 years, You observe that no two points in the triangle are more than 1 inch apart, so a hole in your jeans might be this shape, but when you hold a circular patch against it, you notice that the circle covers two vertices of the triangle, but the third vertex sticks out,

the circle can’t cover the triangle, and the triangle can’t cover the circle either. Since a hole might be either shape, this means neither of these patches could cover every possible hole in your jeans, You could just use a really big patch to be safe, but you don’t want to waste precious material. So the question is: What’s the smallest patch needed to cover a hole at most 1 inch wide? An online search reveals that mathematicians have been thinking about this question too: They’ve been searching for a minimal universal cover for more than 100 years. They haven’t found it yet, but recent results are getting us closer to this ideal shape, so the question is: What’s the smallest patch needed to cover a hole at most 1 inch wide? for the fullest answer (and it is a long one), have a look here, photograph sewguide.