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I need to show that one can cover the disk of radius $1.01$ with three distinct unit disks. But after trying around a bit I am not sure that this is even possible. Could you please give me a hint how to do this?

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    $\begingroup$ Where is this exercise from? I don't see how this is possible as well. $\endgroup$
    – Toby Mak
    Commented Oct 14, 2020 at 9:44
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    $\begingroup$ They can't be disjoint. They can be distinct, though. $\endgroup$
    – Arthur
    Commented Oct 14, 2020 at 9:47
  • $\begingroup$ Ok, I just got an email, it was a typo, the disks are supposed to be distinct, not disjoint. Thanks for your efforts. $\endgroup$
    – 3nondatur
    Commented Oct 14, 2020 at 9:49

1 Answer 1

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If you offset the unit disk centers by $d$ in symmetric directions, the circles intersect in pairs at a distance $r$ of the origin such that

$$\left(r-\frac d2\right)^2+\frac{3d^2}4=1.$$

The relevant root is

$$r=\frac{\sqrt{4-3d^2}+d}2$$ and it achieves a maximum when $$d=\frac1{\sqrt3},$$ corresponding to

$$r=\frac2{\sqrt3}>1.01$$


The minimum decentering is obtained with

$$\left(1.01-\frac d2\right)^2+\frac{3d^2}4=1,$$

or $$d\approx0.02031$$

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