The unit square can be on-line covered with any sequence of squares whose total area is not smaller than 4.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-1-6, author = {Janusz Januszewski}, title = {On-line Covering the Unit Square with Squares}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {57}, year = {2009}, pages = {57-62}, zbl = {1172.52011}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-1-6} }
Janusz Januszewski. On-line Covering the Unit Square with Squares. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) pp. 57-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba57-1-6/