论文标题
立方体的对称层式色彩色彩
Symmetric Layer-Rainbow Colorations of Cubes
论文作者
论文摘要
我们可以以$ n \ times n \ times n $ cube $ l $的$ n^3 $单元为$ n^2 $的颜色，以使每一层平行于每一脸的每一层都完全包含每种颜色一次，并且颜色是对称的，使得$ l_ {ij \ ell} = l_} = l_} \ {1，\ dots，n \} $和$ l_ {iij} = l_ {jj i}，l_ {iji} = l_ {jij}，l_ {ij j} = l_ {l_ {l_ {jii} $ 使用运输网络，我们表明，只有$ n \ equiv 0,2 \ mod 3 $（有两个例外，$ n = 1 $和$ n \ neq 3 $）才能表明这种着色是可能的。由实验的设计激励，对这些物体（无对称性）的研究是由Kishen和Fisher在1940年代启动的。这些对象也与已经进行了广泛研究的正交阵列密切相关，它们是对称拉丁正方形的天然三维类似物。
Can we color the $n^3$ cells of an $n\times n\times n$ cube $L$ with $n^2$ colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that $L_{ij\ell}=L_{j\ell i}=L_{\ell ij}$ for distinct $i,j,\ell \in \{1,\dots,n\}$, and $L_{iij}=L_{jj i}, L_{iji}=L_{jij}, L_{ij j}=L_{jii}$ for $i,j\in \{1,\dots,n\}$? Using transportation networks, we show that such a coloring is possible if and only if $n\equiv 0,2 \mod 3$ (with two exceptions, $n=1$ and $n\neq 3$). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940's. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.
