A note on perfect packing of squares and cubes
A packing of squares into a rectangle is said to be perfect if its area equals to the sum of areas of the squares. Analogously, using volume, a perfect packing of cubes into a rectangular parallelotope is defined. The authors prove that the sequence of squares of sides of length \(1, 2^{-t}, 3^{-t}, \dots\) can be perfectly packed into a square provided \(2/3 < t \leq 4/11\). In order to show this, they modify a lemma of \textit{A. Joós} [``Perfect square packings'', Math. Rep., Buchar. (to appear)]. Moreover, the authors prove that the sequence of cubes of length \(1, 2^{-t}, 3^{-t}, \dots\) can be packed perfectly into a rectangular parallelotope if \(1/3 < t \leq 4/11\). This statement is obtained by a slight modification of the proof by \textit{A. Joós} [Acta Math. Hungar. 156, 375--384 (2018; Zbl 1413.05047)], who showed a bit weaker property.
scientific article; zbMATH DE number 7364764