This is an exercise from the Kürschák Mathematics competition from 1947:
The radius of each small disc is half of the large disc. How many small disks are needed to cover the large disc completely?
I assume they ask for the minimal number of small disks.
So, to cover the plane with disks with "minimal overlapping" we have to do something like:
The question is how to place the large disk on this picture so that it overlaps with the least amount of small disks.
If we place the large disk so that its middlepoint is in the middle as so:
then it overlaps with all the $12$ small circles on the picture but I recon we can do better. If we shift the large disk so its centerpoint coincides with one of the small disks' then it will overlap with $9$ small disks.
this looks minimal, but I have a hard time finding a rigorous way showing this. I tried looking at the lattice the small disks' middlepoints define and reason just using the middlepoints but without any good conclusion.
I have solutions on the back of the book (where I found the exercise) but it would be better if I myself (with maybe some help from MSE) could cook up something reasonable.
Is my approach good at all? All the above pictures look really symmetrical, maybe it is better to approach the problem in a "less symmentric way".
Could anyone give me some HINTS how to think here?