  We provide a tight result for a fundamental problem arising from packing
squares into a circular container: The critical density of packing squares into
a disk is $\delta=\frac{8}{5\pi}\approx 0.509$. This implies that any set of
(not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always
be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there
are sets of squares of total area $\frac{8}{5}+\varepsilon$ that cannot be
packed, even if squares may be rotated. This settles the last (and arguably,
most elusive) case of packing circular or square objects into a circular or
square container: The critical densities for squares in a square
$\left(\frac{1}{2}\right)$, circles in a square
$\left(\frac{\pi}{(3+2\sqrt{2})}\approx 0.539\right)$ and circles in a circle
$\left(\frac{1}{2}\right)$ have already been established, making use of
recursive subdivisions of a square container into pieces bounded by straight
lines, or the ability to use recursive arguments based on similarity of objects
and container; neither of these approaches can be applied when packing squares
into a circular container. Our proof uses a careful manual analysis,
complemented by a computer-assisted part that is based on interval arithmetic.
Beyond the basic mathematical importance, our result is also useful as a
blackbox lemma for the analysis of recursive packing algorithms. At the same
time, our approach showcases the power of a general framework for
computer-assisted proofs, based on interval arithmetic.
