[组合] 填数字游戏(四年级奥数)

本帖最后由 realnumber 于 2016-11-26 08:39 编辑

如图，1，2，3，...，填入以下12个交点，使得6条边上的四个数和相等.已用程序算出结果.

free pascal2.4.0程序如下，依次沿外面边界的点，取变量a1,a2,...,a11,以及常量12,其中a1是某正三角形顶点.
var
a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11 : integer;
b:integer;
begin
for a1:=1 to 11 do
begin
for a2:=1 to 11 do
      begin
      if a2=a1 then continue;
      for a3:=1 to 11 do
         begin
         if a3=a1 then continue;
         if a3=a2 then continue;
         for a4:=1 to 11 do
            begin
            if a4=a1 then continue;
            if a4=a2 then continue;
            if a4=a3 then continue;
            for a5:=1 to 11 do
            begin
               if a5=a1 then continue;
               if a5=a2 then continue;
               if a5=a3 then continue;
               if a5=a4 then continue;
               if a1+a2+a4+a5<>26 then continue;
               for a6:=1 to 11 do
                  begin
                  if a6=a1 then continue;
                  if a6=a2 then continue;
                  if a6=a3 then continue;
                  if a6=a4 then continue;
                  if a6=a5 then continue;
                  for a7:=1 to 11 do
                  begin
                  if a7=a1 then continue;
                  if a7=a2 then continue;
                  if a7=a3 then continue;
                  if a7=a4 then continue;
                  if a7=a5 then continue;
                  if a7=a6 then continue;
                  if a3+a4+a6+a7<>26 then continue;
                  for a8:=1 to 11 do
                     begin
                     if a8=a1 then continue;
                     if a8=a2 then continue;
                     if a8=a3 then continue;
                     if a8=a4 then continue;
                     if a8=a5 then continue;
                     if a8=a6 then continue;
                     if a8=a7 then continue;
                     for a9:=1 to 11 do
                        begin
                        if a9=a1 then continue;
                        if a9=a2 then continue;
                        if a9=a3 then continue;
                        if a9=a4 then continue;
                        if a9=a5 then continue;
                        if a9=a6 then continue;
                        if a9=a7 then continue;
                        if a9=a8 then continue;
                        if a5+a6+a8+a9<>26 then continue;
                        for a10:=1 to 11 do
                           begin
                           if a10=a1 then continue;
                           if a10=a2 then continue;
                           if a10=a3 then continue;
                           if a10=a4 then continue;
                           if a10=a5 then continue;
                           if a10=a6 then continue;
                           if a10=a7 then continue;
                           if a10=a8 then continue;
                           if a10=a9 then continue;
                           a11:=66-a1-a2-a3-a4-a5-a6-a7-a8-a9-a10;
                           if a7+a8+a10+a11<>26 then continue;
                           if a3+a2+a11+12<>26 then continue;
                           if a1+a9+a10+12<>26 then continue;
                           write(a1,' ',a2,' ',a3,' ',a4,' ',a5,' ',a6,' ');
                           write(a7,' ',a8, ' ',a9,' ',a10,' ',a11,' ');
                           writeln;
                           end;
                        end;
                     end;
                  end;
                  end;
            end;
            end;
         end;
      end;
end;

end.

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realnumber

2^#

发表于 2016-11-26 08:59 | 只看该作者

1 4 2 10 11 5 9 3 7 6 8
1 4 7 11 10 2 6 9 5 8 3
1 5 7 9 11 4 6 8 3 10 2
1 6 3 11 8 2 10 7 9 4 5
1 6 5 10 9 7 4 8 2 11 3
1 7 2 10 8 3 11 6 9 4 5
1 7 3 8 10 9 6 5 2 11 4
1 8 4 6 11 7 9 5 3 10 2
1 9 2 6 10 7 11 4 5 8 3
1 9 3 11 5 4 8 10 7 6 2
2 3 5 11 10 1 9 7 8 4 6
2 5 6 9 10 7 4 8 1 11 3
2 6 1 10 8 4 11 5 9 3 7
2 6 5 11 7 1 9 10 8 4 3
2 6 7 10 8 4 5 11 3 9 1
2 7 3 8 9 10 5 6 1 11 4
2 8 1 10 6 4 11 7 9 3 5
2 10 3 5 9 7 11 6 4 8 1
2 10 3 8 6 4 11 9 7 5 1
3 2 8 10 11 1 7 9 5 6 4
3 4 1 8 11 7 10 2 6 5 9
3 4 8 10 9 1 7 11 5 6 2
3 5 1 11 7 4 10 6 9 2 8
3 7 1 11 5 4 10 8 9 2 6
3 8 2 6 9 11 7 5 1 10 4
3 8 5 4 11 7 10 6 2 9 1
3 8 5 9 6 2 10 11 7 4 1
3 11 1 8 4 7 10 9 6 5 2
3 11 2 8 4 7 9 10 5 6 1
4 1 5 10 11 2 9 6 7 3 8
4 1 7 11 10 5 3 9 2 8 6
4 3 5 9 10 1 11 7 8 2 6
4 3 6 11 8 7 2 10 1 9 5
4 5 1 6 11 10 9 2 3 7 8
4 5 2 11 6 3 10 8 9 1 7
4 5 3 10 7 2 11 8 9 1 6
4 6 5 9 7 1 11 10 8 2 3
4 7 1 5 10 11 9 3 2 8 6
4 10 1 5 7 11 9 6 2 8 3
4 11 1 6 5 10 9 8 3 7 2
4 11 2 5 6 9 10 8 3 7 1
5 1 3 11 9 4 8 6 7 2 10
5 2 1 10 9 7 8 4 6 3 11
5 3 9 7 11 4 6 10 1 8 2
5 4 3 11 6 10 2 9 1 8 7
5 4 9 6 11 3 8 10 2 7 1
5 4 9 7 10 2 8 11 3 6 1
5 9 1 10 2 7 8 11 6 3 4
6 1 2 9 10 7 8 4 5 3 11
6 1 9 8 11 2 7 10 3 5 4
6 2 3 8 10 4 11 5 7 1 9
6 2 8 7 11 1 10 9 5 3 4
6 2 9 8 10 4 5 11 1 7 3
6 4 8 7 9 1 10 11 5 3 2
6 8 2 3 9 11 10 5 1 7 4
6 8 2 9 3 5 10 11 7 1 4
7 1 2 10 8 9 5 6 3 4 11
7 1 9 8 10 3 6 11 2 5 4
7 2 1 9 8 10 6 5 3 4 11
7 3 9 5 11 4 8 10 1 6 2
7 4 2 10 5 11 3 9 1 6 8
7 8 1 9 2 10 6 11 3 4 5
8 1 3 11 6 7 5 9 4 2 10
8 2 9 6 10 4 7 11 1 5 3
8 3 1 9 6 11 5 7 2 4 10
8 3 2 11 4 7 6 10 5 1 9
8 3 7 6 9 2 11 10 5 1 4
8 6 1 9 3 11 5 10 2 4 7
8 6 7 3 9 5 11 10 2 4 1
8 7 3 2 9 10 11 6 1 5 4
9 1 5 10 6 7 4 11 2 3 8
9 1 7 5 11 4 10 8 3 2 6
9 5 6 2 10 7 11 8 1 4 3
10 2 4 9 5 7 6 11 3 1 8
10 2 7 6 8 4 9 11 3 1 5
10 4 2 7 5 11 6 9 1 3 8
11 3 5 4 8 7 10 9 2 1 6
11 3 6 4 8 7 9 10 1 2 5
11 4 3 5 6 10 8 9 1 2 7
11 4 3 6 5 9 8 10 2 1 7

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isee

3^#

发表于 2016-11-26 23:39 | 只看该作者

1 4 2 10 11 5 9 3 7 6 8
1 4 7 11 10 2 6 9 5 8 3
1 5 7 9 11 4 6 8 3 10 2
1 6 3 11 8 2 10 7 9 4 5
...
realnumber 发表于 2016-11-26 08:59

12个点应该是12个数，这里为啥只11个数？

这让我想起了类似某道点阵图。。。。

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realnumber

4^#

发表于 2016-11-27 07:55 | 只看该作者

回复 3# isee

数字12位置已经固定，
a11=66-a1-a2-...-a10所以是10个循环。

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isee

5^#

发表于 2016-11-27 08:07 | 只看该作者

回复 isee

数字12位置已经固定，
a11=66-a1-a2-...-a10所以是10个循环。
realnumber 发表于 2016-11-27 07:55

sorry 没看首楼的程序语言。。

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realnumber

6^#

发表于 2016-11-27 12:28 | 只看该作者

本帖最后由 realnumber 于 2016-11-27 12:29 编辑

小小修改了下，去掉关于数字12所在轴对称，有44个解。
var
a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11 : integer;
begin
for a1:=1 to 11 do
begin
for a2:=1 to 11 do
      begin
      if a2=a1 then continue;
      for a3:=1 to 11 do
         begin
         if (a3=a1) or (a3=a2) then continue;
         for a4:=1 to 11 do
            begin
            if (a4=a1) or (a4=a2) or (a4=a3) then continue;
            for a5:=1 to 11 do
            begin
               if (a5=a1) or (a5=a2) or (a5=a3) or (a5=a4)  then continue;
               if a1+a2+a4+a5<>26 then continue;
               for a6:=1 to 11 do
                  begin
                  if (a6=a1) or (a6=a2) or (a6=a3) then continue;
                  if (a6=a4) or (a6=a5)  then continue;
                  for a7:=2 to 11 do
                  begin
                  if (a7=a1) or (a7=a2) or (a7=a3)  then continue;
                  if (a7=a4) or (a7=a5) or (a7=a6)  then continue;
                  if a3+a4+a6+a7<>26 then continue;
                  for a8:=2 to 11 do
                     begin
                     if (a8=a1) or (a8=a2) or (a8=a3) or (a8=a4) then continue;
                     if (a8=a5) or (a8=a6) or (a8=a7)  then continue;
                     for a9:=2 to 11 do
                        begin
                        if (a9=a1) or (a9=a2) or (a9=a3) or (a9=a4) then continue;
                        if (a9=a5) or (a9=a6) or (a9=a7) or (a9=a8) then continue;
                        if a5+a6+a8+a9<>26 then continue;
                        for a10:=2 to 11 do
                           begin
                           if (a10=a1) or (a10=a2) or (a10=a3) then continue;
                           if (a10=a4) or (a10=a5) or (a10=a6) then continue;
                           if (a10=a7) or (a10=a8) or (a10=a9) then continue;
                           a11:=66-a1-a2-a3-a4-a5-a6-a7-a8-a9-a10;
                           if a11=1 then  continue;
                           if a7+a8+a10+a11<>26 then continue;
                           if a3+a2+a11+12<>26 then continue;
                           if a1+a9+a10+12<>26 then continue;
                           write(a1,' ',a2,' ',a3,' ',a4,' ',a5,' ',a6,' ');
                           write(a7,' ',a8, ' ',a9,' ',a10,' ',a11,' ');
                           writeln;
                           end;
                        end;
                     end;
                  end;
                  end;
            end;
            end;
         end;
      end;
end;
end.

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realnumber

7^#

发表于 2016-11-27 12:31 | 只看该作者

本帖最后由 realnumber 于 2016-11-27 12:48 编辑

数字12放在顶点位置，同之前猜测的有一样多的44个解，估计可以构造一个一一映射，把6楼和7楼的解对应起来.
var
a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11 : integer;
begin
for a1:=1 to 11 do
begin
for a2:=1 to 11 do
      begin
      if a2=a1 then continue;
      for a3:=1 to 11 do
         begin
         if (a3=a1) or (a3=a2) then continue;
         for a4:=1 to 11 do
            begin
            if (a4=a1) or (a4=a2) or (a4=a3) then continue;
            if 12+a1+a3+a4<>26 then continue;
            for a5:=1 to 11 do
            begin
               if (a5=a1) or (a5=a2) or (a5=a3) or (a5=a4)  then continue;
               for a6:=1 to 11 do
                  begin
                  if (a6=a1) or (a6=a2) or (a6=a3) then continue;
                  if (a6=a4) or (a6=a5)  then continue;
                  if a2+a3+a5+a6<>26 then continue;
                  for a7:=2 to 11 do
                  begin
                  if (a7=a1) or (a7=a2) or (a7=a3)  then continue;
                  if (a7=a4) or (a7=a5) or (a7=a6)  then continue;
                  for a8:=2 to 11 do
                     begin
                     if (a8=a1) or (a8=a2) or (a8=a3) or (a8=a4) then continue;
                     if (a8=a5) or (a8=a6) or (a8=a7)  then continue;
                     if a4+a5+a7+a8<>26 then continue;
                     for a9:=2 to 11 do
                        begin
                        if (a9=a1) or (a9=a2) or (a9=a3) or (a9=a4) then continue;
                        if (a9=a5) or (a9=a6) or (a9=a7) or (a9=a8) then continue;
                        for a10:=2 to 11 do
                           begin
                           if (a10=a1) or (a10=a2) or (a10=a3) then continue;
                           if (a10=a4) or (a10=a5) or (a10=a6) then continue;
                           if (a10=a7) or (a10=a8) or (a10=a9) then continue;
                           if a6+a7+a9+a10<>26 then continue;
                           a11:=66-a1-a2-a3-a4-a5-a6-a7-a8-a9-a10;
                           if a11=1 then  continue;
                           if a1+a2+a10+a11<>26 then continue;
                           if a8+a9+a11+12<>26 then continue;
                           write(a1,' ',a2,' ',a3,' ',a4,' ',a5,' ',a6,' ');
                           write(a7,' ',a8, ' ',a9,' ',a10,' ',a11,' ');
                           writeln;
                           end;
                        end;
                     end;
                  end;
                  end;
            end;
            end;
         end;
      end;
end;
end.

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realnumber

8^#

发表于 2016-11-29 23:30 | 只看该作者

.....写了个随笔，见附件

爸爸还没做出来.doc (55.5 KB)

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