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style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>个</span> <sub><span lang=EN-US><!--[if gte vml 1]><v:shape
id="_x0000_i1030" type="#_x0000_t75" style='width:25.2pt;height:13.2pt'>
<v:imagedata src="./ex5.files/image011.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=34 height=17
src="./ex5.files/image012.gif" v:shapes="_x0000_i1030"><![endif]></span></sub></b><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>的两两正交的拉丁方。</span><span lang=EN-US><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解法一：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>利用书上第四节的定理，很容易构造出正交拉丁方。</span></p>
<p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>令</span> <span lang=EN-US
style='mso-text-raise:-8.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1031"
type="#_x0000_t75" style='width:54pt;height:19.8pt'>
<v:imagedata src="./ex5.files/image013.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=72 height=26
src="./ex5.files/image014.gif" v:shapes="_x0000_i1031"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>，</span><span lang=EN-US style='mso-text-raise:-8.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1032" type="#_x0000_t75" style='width:52.8pt;height:19.8pt'>
<v:imagedata src="./ex5.files/image015.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=70 height=26
src="./ex5.files/image016.gif" v:shapes="_x0000_i1032"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>，</span><span lang=EN-US style='mso-text-raise:-8.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1033" type="#_x0000_t75" style='width:54pt;height:19.8pt'>
<v:imagedata src="./ex5.files/image017.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=72 height=26
src="./ex5.files/image018.gif" v:shapes="_x0000_i1033"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>其中</span><span lang=EN-US style='mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1034" type="#_x0000_t75" style='width:85.8pt;height:19.2pt'>
<v:imagedata src="./ex5.files/image019.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=115 height=26
src="./ex5.files/image020.gif" v:shapes="_x0000_i1034"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>，</span><span lang=EN-US style='mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1035" type="#_x0000_t75" style='width:91.8pt;height:19.2pt'>
<v:imagedata src="./ex5.files/image021.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=123 height=26
src="./ex5.files/image022.gif" v:shapes="_x0000_i1035"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>，</span><span lang=EN-US style='mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1036" type="#_x0000_t75" style='width:91.2pt;height:19.2pt'>
<v:imagedata src="./ex5.files/image023.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=122 height=25
src="./ex5.files/image024.gif" v:shapes="_x0000_i1036"><![endif]></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>。</span></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解法二：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>如果你有空且有兴趣的话，也可以自己写出</span><span
lang=EN-US>3</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>个来：</span></p>
<p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>比如：</span><span lang=EN-US><b><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1037"
type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image025.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image026.gif" v:shapes="_x0000_i1037"><![endif]></span></b><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1037"
DrawAspect="Content" ObjectID="_1069593281">
</o:OLEObject>
</xml><![endif]--><b><span style='mso-tab-count:1'> </span><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1038"
type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image027.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image028.gif" v:shapes="_x0000_i1038"><![endif]></span><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1038"
DrawAspect="Content" ObjectID="_1069593282">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:1'> </span><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1039"
type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image029.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image030.gif" v:shapes="_x0000_i1039"><![endif]></span><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1039"
DrawAspect="Content" ObjectID="_1069593283">
</o:OLEObject>
</xml><![endif]--><o:p></o:p></b></span></p>
<p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>（其实是第</span><span lang=EN-US>5</span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>题的三个矩阵，</span><span lang=EN-US>hoho</span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>）</span></p>
<p class=MsoNormal><span lang=EN-US><span style="mso-spacerun:
yes"> </span></span></p>
<p class=MsoNormal style='margin-left:18.0pt;text-indent:-18.0pt;mso-list:l0 level1 lfo2;
tab-stops:list 18.0pt'><![if !supportLists]><b><span lang=EN-US>3．<span
style='font:7.0pt "Times New Roman"'> </span></span></b><![endif]><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>证明：不可能存在</span><span lang=EN-US>k</span></b><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>个</span><sub><span lang=EN-US><!--[if gte vml 1]><v:shape
id="_x0000_i1040" type="#_x0000_t75" style='width:27pt;height:13.8pt'>
<v:imagedata src="./ex5.files/image031.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=36 height=19
src="./ex5.files/image032.gif" v:shapes="_x0000_i1040"><![endif]></span></sub></b><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>两两正交的拉丁方。</span><span lang=EN-US><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>证明可以参见第一节</span><span
lang=EN-US>3.</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>正交拉丁方的基本性质。</span></p>
<p class=MsoNormal><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:18.0pt;text-indent:-18.0pt;mso-list:l0 level1 lfo2;
tab-stops:list 18.0pt'><![if !supportLists]><b><span lang=EN-US>4．<span
style='font:7.0pt "Times New Roman"'> </span></span></b><![endif]><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>判断下列两组拉丁方是否正交。</span><span lang=EN-US><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span lang=EN-US>(a) <span style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1041" type="#_x0000_t75" style='width:90pt;height:90pt'>
<v:imagedata src="./ex5.files/image025.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image033.gif" v:shapes="_x0000_i1041"><![endif]></span><span
style='mso-tab-count:1'> </span><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1042"
type="#_x0000_t75" style='width:90pt;height:90pt'>
<v:imagedata src="./ex5.files/image034.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image035.gif" v:shapes="_x0000_i1042"><![endif]></span><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>是正交拉丁方</span><span
lang=EN-US>.</span></p>
<p class=MsoNormal><b><span lang=EN-US>(b) <span style='mso-text-raise:-51.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1043" type="#_x0000_t75" style='width:106.8pt;height:108pt'>
<v:imagedata src="./ex5.files/image036.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=142 height=144
src="./ex5.files/image037.gif" v:shapes="_x0000_i1043"><![endif]></span><span
style='mso-tab-count:1'> </span><span
style='mso-text-raise:-51.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1044"
type="#_x0000_t75" style='width:106.8pt;height:108pt'>
<v:imagedata src="./ex5.files/image038.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=142 height=144
src="./ex5.files/image039.gif" v:shapes="_x0000_i1044"><![endif]></span><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>不是正交的拉丁方（没有</span><span
lang=EN-US>6</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>阶的正交拉丁方，</span><span lang=EN-US
style='font-family:Wingdings;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-font-family:
Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>J</span></span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>）</span></p>
<p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>实际上，（</span><span lang=EN-US>1</span><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>，</span><span lang=EN-US>3</span><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>）出现了两次。</span></p>
<p class=MsoNormal><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:18.0pt;text-indent:-18.0pt;mso-list:l0 level1 lfo2;
tab-stops:list 18.0pt'><![if !supportLists]><b><span lang=EN-US>5．<span
style='font:7.0pt "Times New Roman"'> </span></span></b><![endif]><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>判断下列的拉丁方是否两两正交。</span><span lang=EN-US><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span lang=EN-US style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape
id="_x0000_i1045" type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image025.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image040.gif" v:shapes="_x0000_i1045"><![endif]><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1045"
DrawAspect="Content" ObjectID="_1069593285">
</o:OLEObject>
</xml><![endif]--></span><span lang=EN-US><span style='mso-tab-count:1'> </span><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1046"
type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image027.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image041.gif" v:shapes="_x0000_i1046"><![endif]></span><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1046"
DrawAspect="Content" ObjectID="_1069593286">
</o:OLEObject>
</xml><![endif]--><span style='mso-tab-count:1'> </span><span
style='mso-text-raise:-42.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1047"
type="#_x0000_t75" style='width:90pt;height:90pt' o:ole="">
<v:imagedata src="./ex5.files/image029.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=120 height=120
src="./ex5.files/image042.gif" v:shapes="_x0000_i1047"><![endif]></span><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1047"
DrawAspect="Content" ObjectID="_1069593287">
</o:OLEObject>
</xml><![endif]--><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>是两两正交的拉丁方。</span></p>
<p class=MsoNormal><b><span lang=EN-US><span style="mso-spacerun:
yes"> </span><o:p></o:p></span></b></p>
<p class=MsoNormal style='margin-left:18.0pt;text-indent:-18.0pt;mso-list:l0 level1 lfo2;
tab-stops:list 18.0pt'><![if !supportLists]><b><span lang=EN-US>6．<span
style='font:7.0pt "Times New Roman"'> </span></span></b><![endif]><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>试构造一对</span><sub><span lang=EN-US><!--[if gte vml 1]><v:shape
id="_x0000_i1048" type="#_x0000_t75" style='width:36pt;height:13.2pt'>
<v:imagedata src="./ex5.files/image043.wmz" o:title=""/>
</v:shape><![endif]--><![if !vml]><img width=48 height=17
src="./ex5.files/image044.gif" v:shapes="_x0000_i1048"><![endif]></span></sub></b><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>的正交拉丁方。</span><span lang=EN-US><o:p></o:p></span></b></p>
<p class=MsoNormal><b><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>解：</span></b><span style='font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>利用书上</span><span
lang=EN-US>p250~251</span><span style='font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>中的结论构造。</span></p>
<p class=MsoNormal><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:18.0pt;text-indent:-18.0pt;mso-list:l0 level1 lfo2;
tab-stops:list 18.0pt'><![if !supportLists]><b><span lang=EN-US>7．<span
style='font:7.0pt "Times New Roman"'> </span></span></b><![endif]><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>试问下列区组是否为</span><span lang=EN-US>BIBD?</span></b><b><span
style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
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