We show a new algorithm and improved bound for the online square-into-square packing problem using a hybrid shelf-packing approach. This 2-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. We seek the largest area α such that any set of squares with total area at most α can be packed. We show an algorithm that packs any online sequence of squares with total area α ≤ 2/5, improving upon recent results of 11/32 and 3/8 . Our approach allows all squares smeiller than a chosen maximum height h to be packed into the same fixed height shelf. We combine this with the introduction of variable height shelves for squares of height larger than h. Some of these techniques can be extended to the more general problems of rectangle packing with rotation and bin packing.
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