Abstract Equilateral triangles of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into an equilateral triangle, provided that $$\ 1/2 1 / 2 < t ≤ 37 / 72 . Moreover, for t slightly greater than 1/2, squares of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into a square $$S_t$$ S t in such a way that some squares have a side parallel to a diagonal of $$S_t$$ S t and the remaining squares have a side parallel to a side of $$S_t$$ S t .

In λ-packing items are grouped in batches. Items arrive one by one (online) and they are stored in a buffer until either the total volume of stored items is greater than or equal to λ or all items have already arrived. Then items from the buffer are packed offline into a unit capacity bin and the buffer is emptied. We show that any sequence of cubes with total volume not greater than 1/4 can be 1/8-packed into a single bin (a unit cube).

In λ-packing items are grouped in batches. Items arrive one by one (online) and they are stored in a buffer until either the total volume of stored items is greater than or equal to λ or all items have already arrived. Then items from the buffer are packed offline into a unit capacity bin and the buffer is emptied. We show that any sequence of squares with total area not greater than 1/2 can be 1/4-packed into a single bin (a unit square).

Abstract A space bounded $$ O(d/\log d)$$ O ( d / log d ) -competitive hypercube packing algorithm with one active bin only is presented. As a starting point we give a simple 1-space bounded hypercube packing algorithm with competitive ratio $$ (3/2)^{d}+O((21/16)^d)$$ ( 3 / 2 ) d + O ( ( 21 / 16 ) d ) , for $$d\ge 3.$$ d ≥ 3 .

In this paper, we present a 1-space bounded cube packing algorithm with asymptotic competitive ratio 10.872.