Seminário para todos: "Milkboxes and their containers: a packing challenge"

Suppose you have a fixed number of milk boxes with a square base. These are transported in containers that also have a square base. In order to save space one wants to make containers small while still being able to transport all boxes. How large would the smallest such container be, and how would one need to pack the boxes into such a container? Questions such as these are called packing problems. In mathematical language, this particular question would be considered as square packing, an area that traces its origin to a paper by Erdös and Graham. For small numbers of packed squares, tight (and conjectured optimal) packings have been obtained by a variety of methods, including computerized search. In contrast, only a surprisingly small number of non-trivial cases have actually been proved to be optimal, all of which involving relatively simple packings. In this talk, which is accessible to undergraduate students, we give an overview on the subject and demonstrate the techniques used in proving square packings optimal.