B—2。
(1)——A127180——有这么个公式。
- Table[Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]} & /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]] &][[1]]; Times @@ s + Times @@ t], {n, 3, 24}]
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{5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000}
(2)——可以短,速度还快了。
- Table[With[{t = n!}, Min[# + t/# &[Times @@ #] & /@ Subsets[Range[2, n], {1, Floor[n/2]}]]], {n, 3, 24}]
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{5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000}
(3)——这个可以把明细列出来。
- Table[Needs["Combinatorica`"]; m = Infinity; t = {}; Do[u = Total[Times @@@ v]; If[u < m, m = u; t = v];, {v, Combinatorica`KSetPartitions[Range[2, n], 2]}]; {m, t, Times @@@ t}, {n, 3, 24}]
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{5, {{2}, {3}}, {2, 3}},
{10, {{2, 3}, {4}}, {6, 4}},
{22, {{2, 5}, {3, 4}}, {10, 12}},
{54, {{2, 3, 4}, {5, 6}}, {24, 30}},
{142, {{2, 5, 7}, {3, 4, 6}}, {70, 72}},
{402, {{2, 3, 4, 8}, {5, 6, 7}}, {192, 210}},
{1206, {{2, 4, 8, 9}, {3, 5, 6, 7}}, {576, 630}},
{3810, {{2, 4, 5, 6, 8}, {3, 7, 9, 10}}, {1920, 1890}},
{12636, {{2, 5, 7, 9, 10}, {3, 4, 6, 8, 11}}, {6300, 6336}},
{43776, {{2, 3, 5, 8, 9, 10}, {4, 6, 7, 11, 12}}, {21600, 22176}},
{157824, {{2, 5, 8, 9, 10, 11}, {3, 4, 6, 7, 12, 13}}, {79200, 78624}},
{590520, {{2, 4, 6, 7, 8, 10, 11}, {3, 5, 9, 12, 13, 14}}, {295680, 294840}},
{2287080, {{2, 3, 4, 6, 8, 9, 10, 11}, {5, 7, 12, 13, 14 15}}, {1140480, 1146600}},
{9148320, {{2, 4, 6, 8, 9, 10, 11, 12}, {3, 5, 7, 13, 14, 15, 16}}, {4561920, 4586400}},
{37719360, {{2, 4, 5, 6, 7, 8, 9, 12, 13}, {3, 10, 11, 14, 15, 16, 17}}, {18869760, 18849600}},
{160029696, {{2, 3, 6, 8, 9, 11, 12, 13, 18}, {4, 5, 7, 10, 14, 15, 16, 17}}, {80061696, 79968000}},
{697553280, {{2, 5, 7, 8, 10, 13, 14, 18, 19}, {3, 4, 6, 9, 11, 12, 15, 16, 17}}, {348566400, 348986880}},
{3119552640, {{2, 3, 4, 8, 9, 11, 12, 18, 19, 20}, {5, 6, 7, 10, 13, 14, 15, 16, 17}}, {1560176640, 1559376000}},
{14295585696, {{2, 5, 7, 10, 11, 13, 14, 15, 17, 20}, {3, 4, 6, 8, 9, 12, 16, 18, 19, 21}}, {7147140000, 7148445696}},
{67052240640, {{2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15}, {3, 13, 16, 17, 18, 19, 20, 21, 22}}, {33530112000, 33522128640}},
{321571257120, {{2, 3, 4, 5, 8, 10, 12, 15, 16, 17, 18, 19}, {6, 7, 9, 11, 13, 14, 20, 21, 22, 23}}, {160745472000, 160825785120}},
{1575370944000, {{2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 18, 19}, {9, 10, 11, 12, 13, 20, 21, 22, 23, 24}}, {787652812800, 787718131200}}} |