三重内积(增加难度)的最小值

王守恩

11(行)×4(列)方格。第1行填1,2,3,4,5,6,7,8,9,10,11。第2行填1,1,2,3,4,5,6,7,8,9,10。第3行填1,1,1,2,3,4,5,6,7,8,9。

每列3个数的积填在第4行。若第4行11个数的和是最小时,  则这11个数有唯一答案。——唯一答案——这个可能最大。

northwolves

王守恩 发表于 2025-9-26 07:21
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 ...

w(7)=313

{{1,1,1,2,3,4,5,6,7},{6,7,8,2,4,1,1,3,5},{6,5,4,8,3,9,7,2,1},313}
{{1,1,1,2,3,4,5,6,7},{6,7,8,2,4,1,1,3,5},{6,5,4,9,3,8,7,2,1},313}


{36, 35, 32, 32, 36, 36, 35, 36, 35}
{36, 35, 32, 32, 36, 36, 35, 36, 35}

313

northwolves

w(9)=683
a={1,1,1,2,3,4,5,6,7,8,9};
b={10,9,8,6,2,5,3,1,1,4,7};
c={6,7,8,5,11,3,4,10,9,2,1};

{60,63,64,60,66,60,60,60,63,64,63}
683

王守恩

差不多还是这串数——Table[Ceiling[n*Power[(n - 2)! (n - 1)! n!, (n)^-1]], {n, 2, 20}]——三重内积的最小值—— 6# 。谢谢 mathe！！！

{3, 7, 17, 36, 68, 121, 199, 313, 471, 683, 961, 1318, 1767, 2324, 3005, 3826, 4807, 5967, 7327}

northwolves

王守恩 发表于 2025-9-27 04:39
差不多还是这串数——Table[Ceiling[n*Power[(n - 2)! (n - 1)! n!, (n)^-1]], {n, 2, 20}]——三重内积的 ...

按Stirling公式，$a(n) \approx n^4 \left(\frac{\left(\frac{2 \pi }{n-1}\right)^{\frac{1}{2 n}}}{e}\right)^3$

{4, 8, 17, 36, 69, 121, 200, 314, 472, 685, 963, 1320, 1770, 2327, 3008, 3830, 4811, 5971, 7332}

northwolves

w(8)=472?

a={1,1,1,2,3, 4,5,6,7,8};b={5,8,7,6,2,4,9,1,3,1};c={10,6,7,4,9,3,1,8,2,5}; {50,48,49,48,54,48,45,48,42,40}, 472

w(10)=963?

a={1,1,1,2,3, 4,5,6,7,8,9,10};b={10,9,8,6,5,7,3,1,11,1,2,4};c={8,9,10,7,6,3,5,12,1,11,4,2};{80,81,80,84,90,84,75,72,77,88,72,80},963

w(11)=1319?

a={1,1,1,2,3,4,5,6,7,8,9,10,11};b={8,9,10,7,12,6,4,3,2,1,11,5,1};c={13,11,10,8,3,4,5,6,7,12,1,2,9};{104,99,100,112,108,96,100,108,98,96,99,100,99},1319

王守恩

northwolves 发表于 2025-9-27 09:18
w(8)=472?

a={1,1,1,2,3, 4,5,6,7,8};b={5,8,7,6,2,4,9,1,3,1};c={10,6,7,4,9,3,1,8,2,5}; {50,48,49,48,5 ...

n = 10; a = {5, 7, 8, 2, 3, 1, 4, 1, 6, 1}; b = Join[{1}, Range[n - 1]]; Print[Total[a*b*(n + 1 - Ordering[Ordering[a*b]])]]
471。
n = 12; a = {7, 8, 10, 3, 4, 5, 2, 6, 1, 9, 1, 1}; b = Join[{1}, Range[n - 1]]; Print[Total[a*b*(n + 1 - Ordering[Ordering[a*b]])]]
962。——后面的还是来不了。——规律不好找。

王守恩

northwolves 发表于 2025-9-27 08:52
按Stirling公式，$a(n) \approx n^4 \left(\frac{\left(\frac{2 \pi }{n-1}\right)^{\frac{1}{2 n}}}{e}\ ...

{4, 8, 17, 36, 69, 121, 200, 314, 472, 685, 963, 1320, 1770, 2327, 3008, 3830, 4811, 5971, 7332}——有道理。

1. n = 17; a = {13, 14, 10, 7, 6, 11, 8, 4, 5, 9, 12, 3, 2, 1, 15, 1, 1}; b = Join[{1}, Range[n - 1]]; Print[Total[a*b*(n + 1 - Ordering[Ordering[a*b]])]]

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3832——大海捞针！——还有更小的吗？！

1. u = \[Infinity]; g = {}; Table[n = 17; a = {14, 13, A, B, 12, C, D, E, F, G, H, 11, 2, 1, 15, 1, 1}; W = DeleteCases[a, 1];
   
2. If[DuplicateFreeQ[W], b = Join[{1}, Range[n - 1]]; t = Total[a*b*(n + 1 - Ordering[Ordering[a*b]])];
   
3. If[t < u, u = t; g = {A, B, C, D, E, F, G, H};]], {A, 3, 10}, {B, 3,10}, {C, 3, 10}, {D, 3, 10}, {E, 3, 10}, {F, 3, 10}, {G, 3, 10}, {H, 3, 10}];
   
4. Print["最小结果: A=", g[[1]], ", B=", g[[2]], ", C=", g[[3]], ", D=", g[[4]], ", E=", g[[5]], ", F=", g[[6]], ", G=", g[[7]], ", H=", g[[8]], " \[RightArrow] ", u]

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最小结果: A=10, B=7, C=5, D=6, E=8, F=4, G=9, H=3 \[RightArrow] 3832——不懂半懂的学了一点——比大海捞针好一点。

王守恩

试试！走自己的路。

n(行)×4(列)方格。第1行填 1,2,3,...,n。第2行填1,1,2,3,...,n-1。第3行填1,1,1,..,(n-2)个1,2,3。

每列3个数的积填在第4行。则第4行n个数的和是最小时, 得到这样一串数: 2, 7, 13, 22, 36, 55, 80, 112, 152, 201, 260, 330, 412, 507, 616, 740, 880,

王守恩

试试！走自己的路。

1. Table[A = If[EvenQ[n], n/2, Floor[n/2]]; a = Join[ConstantArray[1, A], Range[2, n - A + 1]];
   
2.   B = If[EvenQ[n], n/2, Ceiling[n/2]]; b = Join[ConstantArray[1, B], Range[2, n - B + 1]];
   
3.   k = MinimalBy[DeleteDuplicates[Permutations[b]], Total[ReverseSort[a*#]*Range[n]] &] // First;
   
4.   m = Total[ReverseSort[a*k]*Range[n]];
   
5.   Print["n=", n, ": a=", a, " b=", k, " 和=", m], {n, 1, 16}];

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n=1: a={2} b={1} 和=2
n=2: a={1,2} b={1,2} 和=6
n=3: a={1,2,3} b={1,1,2} 和=13
n=4: a={1,1,2,3} b={1,2,3,1} 和=22
n=5: a={1,1,2,3,4} b={1,2,1,3,1} 和=36
n=6: a={1,1,1,2,3,4} b={1,2,3,1,4,1} 和=53
n=7: a={1,1,1,2,3,4,5} b={1,2,3,1,4,1,1} 和=75
n=8: a={1,1,1,1,2,3,4,5} b={1,2,3,4,1,5,1,1} 和=102
n=9: a={1,1,1,1,2,3,4,5,6} b={1,2,3,4,1,1,5,1,1} 和=135
n=10: a={1,1,1,1,1,2,3,4,5,6} b={1,2,3,4,6,1,1,5,1,1} 和=173
n=11: a={1,1,1,1,1,2,3,4,5,6,7} b={1,2,3,4,5,1,1,6,1,1,1} 和=219
n=12: a={1,1,1,1,1,1,2,3,4,5,6,7} b={1,2,3,4,5,7,1,1,1,6,1,1} 和=271
n=13: a={1,1,1,1,1,1,2,3,4,5,6,7,8} b={1,2,3,4,5,7,1,1,1,6,1,1,1} 和=331
n=14: a={1,1,1,1,1,1,1,2,3,4,5,6,7,8} b={1,2,3,4,5,6,8,1,1,1,7,1,1,1} 和=399
n=15: a={1,1,1,1,1,1,1,2,3,4,5,6,7,8,9} b={1,2,3,4,5,6,8,1,1,1,1,7,1,1,1} 和=476

SystemException["MemoryAllocationFailure"]——说来不了了。