三重内积(增加难度)的最小值

王守恩

试试！走自己的路。

w(1)=6,  1*2*1 + 1*2 + 3=7,
a={1},
b={2, 1},
c={1, 2, 3},

w(2)=17,  1*2*2 + 2*3*1 + 1*3 + 4=17,
a={1, 2},
b={2, 3, 1},
c={2, 1, 3, 4},

w(3)=36,  1*2*4 + 2*1*3 + 3*3*1 +4*2 + 5=36,
a={1, 2, 3},
b={2, 1, 3, 4},
c={4, 3, 1, 2, 5},

w(4)=69,  1*4*4 + 2*1*5 + 3*3*1 +4*3 + 5*2 + 2*6=69,
a={1, 2, 3, 4},
b={4, 1, 3,     5, 2},
c={4, 5, 1, 3, 2, 6},

w(5)=129,  1*2*5 + 2*5*2 + 3*7 +4*1*6 + 5*6*1 + 4*3 + 3*4=129,
a={1, 2, 3, 4, 5},
b={2, 5,     1, 6, 4, 3},——这个空格不能少？
c={5, 2, 7, 6, 1, 3, 4},

w(6)=215,  1*3*5 + 2*2*6 + 3*6*2 +4*8 + 5*1*7 + 6*7*1 + 5*3 + 4*4=215,
a={1, 2, 3, 4, 5, 6},
b={3, 2, 6,     1, 7, 5, 4},
c={5, 6, 2, 8, 7, 1, 3, 4},

northwolves

a,b可以用1补齐n个元素

northwolves

w(4)=68:

{24, {{{1, 1, 1, 2, 3, 4}, {2, 3, 5, 1, 4, 1}, {5, 4, 2, 6, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {2, 3, 5, 1, 4, 1}, {6, 4, 2, 5, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {2, 4, 5, 1, 1, 3}, {5, 3, 2, 6, 4, 1}, 68},
{{1, 1, 1, 2, 3, 4}, {2, 4, 5, 1, 1, 3}, {6, 3, 2, 5, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {2, 5, 3, 1, 4, 1}, {5, 2, 4, 6, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {2, 5, 3, 1, 4, 1}, {6, 2, 4, 5, 1, 3}, 68},
{{1, 1, 1, 2, 3, 4}, {2, 5, 4, 1, 1, 3}, {5, 2, 3, 6, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {2, 5, 4, 1, 1, 3}, {6, 2, 3, 5, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {3, 2, 5, 1, 4, 1}, {4, 5, 2, 6, 1, 3}, 68},
{{1, 1, 1, 2, 3, 4}, {3, 2, 5, 1, 4, 1}, {4, 6, 2, 5, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {3, 5, 2, 1, 4, 1}, {4, 2, 5, 6, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {3, 5, 2, 1, 4, 1}, {4, 2, 6, 5, 1, 3}, 68},
{{1, 1, 1, 2, 3, 4}, {4, 2, 5, 1, 1, 3}, {3, 5, 2, 6, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {4, 2, 5, 1, 1, 3}, {3, 6, 2, 5, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {4, 5, 2, 1, 1, 3}, {3, 2, 5, 6, 4, 1}, 68},
{{1, 1, 1, 2, 3, 4}, {4, 5, 2, 1, 1, 3}, {3, 2, 6, 5, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 2, 3, 1, 4, 1}, {2, 5, 4, 6, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 2, 3, 1, 4, 1}, {2, 6, 4, 5, 1, 3},68},
{{1, 1, 1, 2, 3, 4}, {5, 2, 4, 1, 1, 3}, {2, 5, 3, 6, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 2, 4, 1, 1, 3}, {2, 6, 3, 5, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 3, 2, 1, 4, 1}, {2, 4, 5, 6, 1, 3}, 68},
{{1, 1, 1, 2, 3, 4}, {5, 3, 2, 1, 4, 1}, {2, 4, 6, 5, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 4, 2, 1, 1, 3}, {2, 3, 5, 6, 4, 1}, 68}, {{1, 1, 1, 2, 3, 4}, {5, 4, 2, 1, 1, 3}, {2, 3, 6, 5, 4, 1}, 68}}}

northwolves

w(5)=121,有126996种组合：

{{1, 1, 1, 2, 3, 4, 5}, {1, 1, 2, 3, 4, 5, 6}, {1, 6, 4, 5, 3, 2, 7}, 121},
{{1, 1, 1, 2, 3, 4, 5}, {1, 1, 2, 3, 4, 5, 6}, {1, 6, 7, 4, 3, 2, 5}, 121},
{{1, 1, 1, 2, 3, 4, 5}, {1, 1, 2, 3, 4, 5, 6}, {2, 3, 6, 4, 5, 1, 7}, 121}...

王守恩

northwolves 发表于 2025-9-24 22:49
w(4)=68:

{24, {{{1, 1, 1, 2, 3, 4}, {2, 3, 5, 1, 4, 1}, {5, 4, 2, 6, 1, 3}, 68}, {{1, 1, 1, 2, 3, 4 ...

w(4)=68:——24个解。——因为a=1, 1, 1, 2, 3, 4,  前3位1,1,1相同。所以只有2个解。——这2个解,本质上还是一样的。
b=01=02=05=06=09=10=11=12=17=18=21=22——1个解。
b=03=04=07=08=13=14=15=16=19=20=23=24——另1个解。
01: {2, 3, 5, 1, 4, 1},
02: {2, 3, 5, 1, 4, 1},
03: {2, 4, 5, 1, 1, 3},
04: {2, 4, 5, 1, 1, 3},
05: {2, 5, 3, 1, 4, 1},
06: {2, 5, 3, 1, 4, 1},  
07: {2, 5, 4, 1, 1, 3},
08: {2, 5, 4, 1, 1, 3},
09: {3, 2, 5, 1, 4, 1},
10: {3, 2, 5, 1, 4, 1},  
11: {3, 5, 2, 1, 4, 1},
12: {3, 5, 2, 1, 4, 1},
13: {4, 2, 5, 1, 1, 3},
14: {4, 2, 5, 1, 1, 3},  
15: {4, 5, 2, 1, 1, 3},
16: {4, 5, 2, 1, 1, 3},
17: {5, 2, 3, 1, 4, 1},  
18: {5, 2, 3, 1, 4, 1},  
19: {5, 2, 4, 1, 1, 3},
20: {5, 2, 4, 1, 1, 3},
21: {5, 3, 2, 1, 4, 1},
22: {5, 3, 2, 1, 4, 1},  
23: {5, 4, 2, 1, 1, 3},
24: {5, 4, 2, 1, 1, 3},

王守恩

northwolves 发表于 2025-9-24 22:51
w(5)=121,有126996种组合：

{{1, 1, 1, 2, 3, 4, 5}, {1, 1, 2, 3, 4, 5, 6}, {1, 6, 4, 5, 3, 2, 7}, 121 ...

w(5)=121,  b有16种组合。121只有1种可能, 121=14+15+16+18+18+20+20。
1: b={2, 3, 4, 5, 6, 1, 1}
a*b={2, 3, 4,10,18,4, 5}
    c={7, 6, 5, 2, 1, 4, 3}

2: b={2, 3, 5, 4, 6, 1, 1}
a*b={2, 3, 5, 8,18, 4, 5}
    c={7, 6, 4, 2, 1, 5, 3}

3: b={2, 3, 6, 4, 1, 5, 1}
a*b={2, 3, 6, 8, 3,20, 5}
    c={7, 5, 3, 2, 6, 1, 4}

4: b={2, 3, 6, 4, 5, 1, 1}
a*b={2, 3, 6, 8,15, 4, 5}
    c={7, 6, 3, 2, 1, 5, 4}

5: b={2, 3, 6, 5, 1, 1, 4}
a*b={2, 3, 6,10, 3,4,20}
    c={7, 6, 3, 2, 5, 4, 1}

6: b={2, 3, 6, 5, 1, 4, 1}
a*b={2, 3, 6,10, 3,16,5}
    c={7, 6, 3, 2, 5, 1, 4}

7: b={2, 4, 6, 5, 1, 1, 3}
a*b={2, 4, 6,10, 3,4,15}
    c={7, 5, 3, 2, 6, 4, 1}

8: b={2, 5, 6, 4, 1, 1, 3}
a*b={2, 5, 6, 8, 3, 4,15}
    c={7, 4, 3, 2, 6, 5, 1}

9: b={3, 4, 5, 1, 6, 1, 2}
a*b={3, 4, 5, 2,18,4,10}
    c={6, 4, 3, 7, 1, 5, 2}

10: b={3, 4, 5, 1, 6, 2, 1}
  a*b={3, 4, 5, 2, 18,8, 5}
      c={6, 5, 4, 7, 1, 2, 3}

11: b={3, 4, 6, 1, 1, 5, 2}
  a*b={3, 4, 6, 2, 3,20,10}
      c={6, 4, 3, 7, 5, 1, 2}

12: b={3, 4, 6, 1, 5, 1, 2}
  a*b={3, 4, 6, 2,15,4,10}
      c={6, 5, 3, 7, 1, 4, 2}

13: b={3, 4, 6, 1, 5, 2, 1}
  a*b={3, 4, 6, 2,15, 8, 5}
      c={6, 5, 3, 7, 1, 2, 4}

14: b={3, 5, 6, 1, 1, 2, 4}
  a*b={2, 3, 4,10,18,4, 5}
      c={7, 6, 5, 2, 1, 4, 3}

15: b={3, 5, 6, 1, 1, 4, 2}
  a*b={3, 5, 6, 2, 3,16,10}
      c={6, 4, 3, 7, 5, 1, 2}

16: b={4, 5, 6, 1, 1, 2, 3}
  a*b={4, 5, 6, 2, 3, 8,15}
      c={5, 4, 3, 7, 6, 2, 1}

王守恩

你没搞懂。我也没搞懂。干脆搞得神秘一点。

7(行)×4(列)方格。第1行填1,2,3,4,5,6,7。第2行填1,1,2,3,4,5,6。第3行填1,1,1,2,3,4,5。

每列3个数的积填在第4行。若第4行7个数的和是最小时,  则这7个数有唯一答案。

northwolves

{7, 5, 6, 2, 1, 3, 4}, {6, 5, 4, 3, 2, 1, 1}, {1, 1, 1, 2, 3, 4, 5}

northwolves

{3,4,6,7,8,2,5,1},{7,6,5,2,1,3,1,4},{1,1,1,2,3,4,5,6}

王守恩

9(行)×4(列)方格。第1行填1,2,3,4,5,6,7,8,9。第2行填1,1,2,3,4,5,67,8。第3行填1,1,1,2,3,4,5,6,7。

每列3个数的积填在第4行。若第4行9个数的和是最小时,  则这9个数有唯一答案。