Abstract
It is known that \({\sum_{i =1}^\infty {1/ i^2}={\pi^2/6}}\). We can ask what is the smallest \({\epsilon}\) such that all the squares of sides of length \({1, 1/2, 1/3, \ldots}\) can be packed into a rectangle of area \({{\pi^2/6}+\epsilon}\). A packing into a rectangle of the right area is called perfect packing. Chalcraft [4] packed the squares of sides of length \({1, 2^{-t}, 3^{-t}, \ldots}\) and he found perfect packings for \({1/2 < t \le 3/5}\). We generalize this problem and pack the 3-dimensional cubes of sides of length \({1, 2^{-t}, 3^{-t}, \ldots}\) into a right rectangular prism of the right volume. Moreover we show that there is a perfect packing for all t in the range \({0.36273 \le t \le 4/11}\).
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Joós, A. Perfect packing of cubes. Acta Math. Hungar. 156, 375–384 (2018). https://doi.org/10.1007/s10474-018-0858-z
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DOI: https://doi.org/10.1007/s10474-018-0858-z