Packing of non-blocking squares into the unit square (Q6614507)

Let \(S_n\) be a square, for \(n = 1, 2,\ldots\), and let \(I\) be a square of sidelength 1. We say that the squares \(S_1\), \(S_2\), \(\ldots\) can be \textit{packed} into \(I\) if it is possible to apply translations and rotations to the sets \(S_n\) so that the resulting translated and rotated squares are contained in \(I\) and have mutually disjoint interiors. The packing is \textit{parallel} if every side of each packed square is parallel to a side of \(I\). Moon and Moser in 1967 showed that the squares \(S_1\), \(S_2\), \(\ldots\) can be packed parallel into \(I\) provided that the total area of the squares is not greater than \(1/2\). This upper bound is tight. Denote by an the sidelength of \(S_n\) for \(n = 1, 2, \ldots\). We say that the squares \(S_1\), \(S_2\), \(\ldots\) are \textit{non-blocking} if \(a_i + a_j \le 1\) for any \(i\ne j\).\N\NThe Main result of this paper is the following. Theorem. Any finite or infinite collection of non-blocking squares with total area no greater than 5/9 can be packed into the unit square. This upper bound is tight for parallel packing.