第14课完结。
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@@ -549,4 +549,170 @@ Fisher给出了另外一个存在$(v,k,\lambda)$-BIBD的必要条件。
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如果$A$和$B$是正交拉丁方，则它们的转置$A\trans$与$B\trans$也是一对正交拉丁方。
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\end{proposition}
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问题：给定$n$，在所有$n$阶拉丁方中，最多能有多少对两两互相正交的拉丁方组？
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\begin{definition}
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两两正交的一组拉丁方叫做正交拉丁方组。
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\end{definition}
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问题：给定$n$，在所有$n$阶拉丁方中，一个正交拉丁方组最多能有多少个？
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我们有
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\begin{theorem}
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如果$A_1, A_2, \dots, A_r$是$r$个两两正交的$n$阶正交拉丁方，则$r \leq n - 1$。
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\end{theorem}
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\begin{proof}
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根据正交拉丁方的性质，我们总可以对每个拉丁方$A_k$的元素进行适当的置换$p_k$，把$A_k$的第一行变为$1, 2, \dots, n$。置换后的$r$个拉丁方仍旧两两正交。因此我们不妨就直接设原来的$r$个拉丁方就是如下形式：
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\[A_k = \begin{bmatrix}
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1 & 2 & \cdots & n\\
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a_{21}^{(k)} & a_{22}^{(k)} & \cdots & a_{2n}^{(k)}\\
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\vdots & \vdots & \ddots & \vdots\\
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a_{n1}^{(k)} & a_{n2}^{(k)} & \cdots & a_{nn}^{(k)}
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\end{bmatrix},
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k = 1, 2, \dots, r\]
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下面我们考察$a_{21}^{(k)}$。每列不可能出现两个1，因此$a_{21}^{(k)} \neq 1, k = 1, 2, \dots, n$。当$h \neq k$时，$a_{21}^{(h)} \neq a_{21}^{(k)}$，否则$(a_{21}^{(h)}, a_{21}^{(k)})$与第一行的某一个是相同的，不符合正交拉丁方的定义。因此
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\[a_{21}^(1), a_{21}^{(2)}, \dots, a_{21}^{(r)}\]
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是两两不同且都不等于1的。这样的数最多有$n-1$个，因此$r \leq n - 1$。
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\end{proof}
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含有$n-1$个$n$阶正交拉丁方组称为$n$阶拉丁方的完备正交组。完备正交组的存在性问题是至今尚未解决的问题。
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通过元素置换将拉丁方的第一行化为自然顺序的过程称为规范化过程。如果对一个正交拉丁方组进行这个过程，也成为正交拉丁方规范化过程。这样的过程有利于比较拉丁方是不是``本质不同''的。
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1782年，欧拉提出了以下猜想：
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\begin{conjecture}
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如果$n = 4k + 2$，则不存在任何一对$n$阶正交拉丁方。
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\end{conjecture}
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在100多年后的1901年，法国人Tarry证明了$n = 6$时Euler猜想的正确性。然而，在1960年前后，Bose，Parker，Shrikhande相继证明了
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\begin{theorem}
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如果$n \neq 2, 6$，则必然存在一对$n$阶正交拉丁方。
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\end{theorem}
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\subsection{幻方}
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观察这样一组正交拉丁方：
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\begin{table}[H]
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\centering
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\begin{tabular}{|c|c|c|c|}
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\hline
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0 & 1 & 2 & 3\\
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\hline
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1 & 0 & 3 & 2\\
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\hline
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2 & 3 & 0 & 1\\
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\hline
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3 & 2 & 1 & 0\\
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\hline
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\end{tabular}
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\hspace{2cm}
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\begin{tabular}{|c|c|c|c|}
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\hline
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0 & 1 & 2 & 3\\
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\hline
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3 & 2 & 1 & 0\\
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\hline
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0 & 1 & 2 & 3\\
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\hline
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2 & 3 & 0 & 1\\
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\hline
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\end{tabular}
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\end{table}
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将其合在一起我们得到
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\begin{table}[H]
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\centering
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\begin{tabular}{|c|c|c|c|}
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\hline
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00 & 11 & 22 & 33\\
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\hline
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13 & 02 & 31 & 20\\
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\hline
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21 & 31 & 03 & 12\\
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\hline
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32 & 23 & 10 & 01\\
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\hline
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\end{tabular}
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\end{table}
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如果我们把表格中的每个数都看作一个两位的四进制数，再将它们转写回十进制，就有
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将其合在一起我们得到
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\begin{table}[H]
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\centering
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\begin{tabular}{|c|c|c|c|}
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\hline
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0 & 5 & 10 & 15\\
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\hline
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7 & 2 & 13 & 8\\
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\hline
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9 & 12 & 3 & 6\\
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\hline
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14 & 11 & 4 & 1\\
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\hline
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\end{tabular}
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\end{table}
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它每一行的和与每一列的和都是定值。这样的数表称为幻方（弱幻方）。如果每一列、每一行、对角线的和都相等，就是强幻方，例如下面这个幻方。
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\begin{table}[H]
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\centering
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\begin{tabular}{|c|c|c|}
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\hline
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4 & 9 & 2\\
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\hline
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3 & 5 & 7\\
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\hline
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8 & 1 & 6\\
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\hline
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\end{tabular}
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\end{table}
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有限几何与拉丁方也有着联系。例如：
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a1) at (0,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a2) at (2,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a3) at (4,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b1) at (0,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b2) at (2,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b3) at (4,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c1) at (0,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c2) at (2,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c3) at (4,4) {};
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\node at (0,4.5) {0};
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\node at (2,4.5) {1};
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\node at (4,4.5) {2};
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\draw (a1)--(a3) (b1)--(b3) (c1)--(c3) (a1)--(c1) (a2)--(c2) (a3)--(c3);
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\draw (c1)--(a3) (c2) ..controls (2.4,3.8) and (3.8,3)..(b3).. controls (3.8,1) and (2,0.6)..(a1);
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\draw (c3) ..controls (2,3.4) and (0.2,3).. (b1) .. controls(0.2,1) and (1.6,0.2) ..(a2);
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\draw[dashed] (c1)..controls (2,3.4) and (3.8,3) .. (b3) .. controls (3.8,1) and (2.4,0.2) .. (a2);
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\draw[dashed] (c2)..controls (1.6, 3.8) and (0.2,3) .. (b1) .. controls (0.2,1) and (2,0.6) ..(a3);
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\draw[dashed] (a1)--(c3);
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\end{tikzpicture}
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\end{figure}
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\begin{table}[H]
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\centering
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\begin{tabular}{|c|c|c|}
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\hline
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0 & 1 & 2\\
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\hline
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2 & 0 & 1\\
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\hline
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1 & 2 & 0\\
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\hline
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\end{tabular}
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\hspace{1cm}
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\begin{tabular}{|c|c|c|}
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\hline
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0 & 1 & 2\\
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\hline
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1 & 2 & 0\\
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\hline
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2 & 0 & 1\\
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\hline
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\end{tabular}
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\end{table}
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