We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

我们解决了用六个相同的圆覆盖矩形的问题，这六个圆的半径要最小化。我们专注于Melissen和Schuur的未结案件（Discrete Appl Math 99：149–156，2000）。根据矩形的边长，圆的不同配置（对应于其放置方式的不同）会产生最佳的覆盖率。我们证明了与开放案例相对应的两种配置的最优性。对于第一个，我们提出了一种数学混合整数非线性优化公式，该公式可以计算全局最优解。对于第二个，我们提供了最佳半径的解析表达式，该表达式是矩形边长之一的函数。因此，关闭所有打开的盒子，以最佳覆盖六个圆的矩形。