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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1$\begingroup$ Where is this exercise from? I don't see how this is possible as well. $\endgroup$– Toby MakOct 14, 2020 at 9:44
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3$\begingroup$ They can't be disjoint. They can be distinct, though. $\endgroup$– ArthurOct 14, 2020 at 9:47
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$\begingroup$ Ok, I just got an email, it was a typo, the disks are supposed to be distinct, not disjoint. Thanks for your efforts. $\endgroup$– 3nondaturOct 14, 2020 at 9:49
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1 Answer
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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 $$d\approx0.02031$$