We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper.
翻译:我们研究在平面上完全铺设砖块的问题,并探索使用分立的正方形铺设矩形的可能性。 给定了一系列可区分的正方形( 或等同一组不同的自然数字 ), 并且必须决定它能否把平面叠成砖块。 以前, 人们已经证明, 使用一组奇数或无限的自然数字序列( 包括两个奇数 ) 铺设平面是不可行的。 问题存在于奇数是任意性的各种不同情况中。 除了为这个特殊情况提供解决办法外, 我们讨论一些公开的问题, 将平面和矩形在本文中打乱 。