This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.

中文翻译：

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将单位正方形严格包装成一个圆圈。

本文考虑了寻找最小的圆的任务，可以将固定数量的不旋转的正方形自由重叠成一个最小的圆。由于旋转角度的原因，将单位正方形填充到一个容器中要比圆形填充对应物更难解决。因此，到目前为止，最优布置仅对一个或两个单位正方形而言是最优的。通过基于间隔算术技术的计算机辅助方法，我们解决了三个正方形的情况，并针对此问题的每种最佳布置找到了严格的包围物。我们将正方形和圆形之间的关系建模为约束满足问题（CSP），并发现每个框可能在给定的半径上限内包含解。由于搜索域中的对称性，通用间隔方法太慢而无法直接解决CSP。为了克服这个困难，我们通过系统地将约束添加到每个正方形的中心，将问题分解为一组子问题。我们的证明要求解决分别具有1、2和3个单位平方的6、43和12个子问题。原则上，本文提出的方法可以推广到任意数量的平方。