# Cover this disc

Can you place five discs of radius 1, such that they fully cover a disc of radius 2?

Here is proof that it is

not possible.

You can't even just cover the boundary of the $$R=2$$ disc with five unit discs. If the unit disc covers a section of the boundary, the two extremal points are at most two units apart (the diameter of the unit disc). A section of the boundary stretching for two units is exactly $$1/6$$ of the whole boundary (think of an inscribed regular hexagon, which will have side lengths equal to the radius), so it takes at least 6 discs to cover the boundary alone, and then you still haven't covered the centre of the disc. Therefore it takes at least $$7$$ unit discs to cover a disc of radius $$2$$.

• This is a better answer for this question +1 Commented Feb 27, 2023 at 14:22
• Agreed. +1 from me too Commented Feb 27, 2023 at 15:06
• Thank you, great answer. Commented Mar 1, 2023 at 13:08

No

The maximum radius that can be covered is 1.641+. See https://erich-friedman.github.io/packing/circovcir/

• Correct and thanks for the reference. I found this as one of Carnival games called "cover the spot": youtu.be/JZkXmf7sPGg Commented Feb 25, 2023 at 14:24