The square covering problem asks for the largest square that can be completely covered by 
congruent unit squares, allowing
the unit squares to overlap and to have arbitrary orientations.
Pegg (2026) reported a construction by R. Chi in which 12 unit squares cover a square of side length
 |
(1)
|
and hence area
 |
(2)
|
where 
is the third real root of
 |
(3)
|
Optimality is not known.
Pegg, E. Jr. Mathematical Games. Episode
45: "Filling Space with Similar Parts." Sep. 24, 2026.