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


2 Answers 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 $2$.

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


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

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

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