It is known that ∑i=1∞1/i2=π2/6\documentclass[12pt]{minimal}
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\begin{document}$${\sum_{i =1}^\infty {1/ i^2}={\pi^2/6}}$$\end{document}. We can ask what is the smallest ϵ\documentclass[12pt]{minimal}
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\begin{document}$${\epsilon}$$\end{document} such that all the squares of sides of length 1,1/2,1/3,…\documentclass[12pt]{minimal}
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\begin{document}$${1, 1/2, 1/3, \ldots}$$\end{document} can be packed into a rectangle of area π2/6+ϵ\documentclass[12pt]{minimal}
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\begin{document}$${{\pi^2/6}+\epsilon}$$\end{document}. 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,…\documentclass[12pt]{minimal}
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\begin{document}$${1, 2^{-t}, 3^{-t}, \ldots}$$\end{document} and he found perfect packings for 1/2<t≤3/5\documentclass[12pt]{minimal}
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\begin{document}$${1/2 < t \le 3/5}$$\end{document}. We generalize this problem and pack the 3-dimensional cubes of sides of length 1,2-t,3-t,…\documentclass[12pt]{minimal}
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\begin{document}$${1, 2^{-t}, 3^{-t}, \ldots}$$\end{document} 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≤t≤4/11\documentclass[12pt]{minimal}
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\begin{document}$${0.36273 \le t \le 4/11}$$\end{document}.
