用n(n+1)/2个数码, 求最小的n位数

王守恩

用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, 求最小的n位数。

A={1,|2, 3,| 4, 5, 6,| 7, 8, 9, 1,| 0, 1, 1, 1, 2,| 1, 3, 1, 4, 1, 5,| 1, 6, 1, 7, 1, 8, 1,| 9, 2, 0, 2, 1, 2, 2, 2,| 3, 2, 4, 2, 5, 2, 6, 2, 7,| 2, 8, 2, 9, 3, 0, 3, 1, 3, 2,| 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3,| 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4,|4, 4, 5, 4, 6,
4, 7, 4, 8, 4, 9, 5, 0,| 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 7,| 5, 8, 5, 9, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6,| 5, 6, 6, 6, 7, 6, 8, 6, 9, 7, 0, 7, 1, 7, 2, 7,| 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1,| 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9,9, 0,|}

a(1)=1,
1,

a(2)=21,
3,
21,

a(3)=431,
6,
52,
431,

a(4)=6521,
9,
83,
741,
6521,
                  
a(5)=54110,
9,
82,
721,
6311,
54110,
                    
a(6)=541110,
9,
82,
731,
6311,
54211,
541110,
                     
a(7)=6521110,
9,
83,
831,
7411,
74111,
652111,
6521110,

a(8)=65211100,
9,
92,
832,
8321,
74211,
742111,
6521111,
65211100,

a(9)=653211110,
9,
93,
842,
8422,
75221,
752211,
7422111,
66321110,
653211110,

王守恩

最小的n位数。

{"1", "1"},
{"2", "21"},
{"3", "431"},
{"4", "6521"},
{"5", "54110"},
{"6", "541110"},
{"7", "6521110"},
{"8", "65211100"},
{"9", "653211110"},
{"10", "6632211100"},
{"11", "66432211100"},
{"12", "764322211100"},
{"13", "7654332211100"},
{"14", "76544322111000"},
{"15", "775543322111000"},
{"16", "7765443322111000"},
{"17", "87665433221111000"},
{"18", "887654433221111000"},
{"19", "9876654432221110000"},
{"20", "98765543322111100000"},
{"21", "987655432221111000000"},
{"22", "9876554322111111000000"},
{"23", "88776543322111111000000"},
{"24", "887665433221111111000000"},
{"25", "8876654332211111111000000"},
{"26", "88766544332211111111000000"},
{"27", "887665443322111111111000000"},
{"28", "8877655433221111111111000000"},
{"29", "88876554433221111111110000000"},
{"30", "987765544332211111111100000000"},
{"31", "9877765543322111111111110000000"},
{"32", "98776655433222111111111100000000"},
{"33", "987766544332221111111111000000000"},
{"34", "9877665443322221111111111000000000"},
{"35", "98776654433222221111111111000000000"},
{"36", "987766554433222221111111111000000000"},
{"37", "9877665544332222221111111111000000000"},
{"38", "98777665443322222221111111111000000000"},
{"39", "988776555443322222211111111111000000000"},
{"40", "9887766554333222222211111111110000000000"},
{"41", "98877665444332222222111111111110000000000"},
{"42", "988776654443332222222111111111100000000000"},
{"43", "9887766544433332222222111111111100000000000"},
{"44", "98877665544333322222222111111111100000000000"},
{"45", "988776655443333332222222111111111100000000000"},
{"46", "9888766655443333332222222111111111100000000000"},
{"47", "98887766554433333322222211111111111000000000000"},
{"48", "988877665544433333222222211111111110000000000000"},
{"49", "9888776655444333333222222211111111110000000000000"},
{"50", "98887766554444333333222222211111111110000000000000"},
{"51", "988877665544444333333222222211111111110000000000000"},
{"52", "9888776655544444333333222222211111111110000000000000"},
{"53", "98887766655444443333332222222111111111110000000000000"},
{"54", "988877666555444443333322222222111111111100000000000000"},
{"55", "9888776665554444433333322222222111111111100000000000000"},
{"56", "98887766655554444433333322222221111111111100000000000000"},
{"57", "988877766555554444433333322222221111111111100000000000000"},
{"58", "9988776665555544444333333222222221111111111000000000000000"},
{"59", "99887766665555444444333333222222211111111111000000000000000"},
{"60", "998877666655555444443333332222222211111111111000000000000000"},
{"61", "9988776666655554444443333332222222211111111111000000000000000"},
{"62", "99887776666555554444433333332222222111111111111000000000000000"},
{"63", "998877776666555554444433333322222222111111111110000000000000000"},
{"64", "9988777766665555544444433333322222222111111111110000000000000000"},
{"65", "99887777666665555444444333333322222222111111111110000000000000000"},
{"66", "998877777666655555444444333333222222221111111111100000000000000000"},
{"67", "9988877776666555555444444333333222222221111111111110000000000000000"},
{"68", "99888777776666555554444443333332222222211111111111100000000000000000"},
{"69", "998887777766666555554444433333332222222211111111111100000000000000000"},
{"70", "9988887777666665555544444433333322222222211111111111000000000000000000"},
{"71", "99888877777666665555544444433333322222222211111111111100000000000000000"},
{"72", "998888877777666655555544444333333322222222111111111111000000000000000000"},
{"73", "9998888777766666655555444444333333222222222111111111111000000000000000000"},
{"74", "99988887777766666555555444444333333222222222111111111111000000000000000000"},
{"75", "999888887777766666555554444443333333222222221111111111110000000000000000000"},
{"76", "9999888877777666665555544444433333332222222211111111111100000000000000000000"},
{"77", "99998888777766666655555444444333333322222222111111111111000000000000000000000"},
{"78", "999888888777766666555554444443333333222222221111111111110000000000000000000000"},
{"79", "9998888887777666665555544444433333332222222211111111111100000000000000000000000"},
{"80", "99988888877776666655555444444333333322222221111111111111000000000000000000000000"}},

代码——

1. g[m_] := Module[{i = {}, h = 1}, While[Length[i] < m, i = Join[i, IntegerDigits[h]]; h++]; Take[i, m]]
   
2. f[n_] := Module[{m = n (n + 1)/2, i = g[n (n + 1)/2], s, r, t, a, x, y, u, v, d, b, g, w, p, c, z}, s = Table[{k, {}}, {k, n}]; r = s; t = i; c = {};
   
3. While[Length[t] > 0, a = Max[t]; x = FirstPosition[t, a][[1]]; y = t[[x]]; t = Delete[t, x];  u[q_] := If[q == {}, 0, FromDigits[q]]; v = Min[u[#[[2]]] & /[url=home.php?mod=space&uid=6175]@[/url] r]; d = Select[r, u[#[[2]]] == v &];
   
4. If[Length[d] > 1, d = SortBy[d, First];]; b = First[d]; g = Append[b[[2]], y]; w = {b[[1]], g}; p = FirstPosition[r, b]; r = Delete[r, p];
   
5. If[Length[g] == w[[1]], AppendTo[c, w], AppendTo[r, w]];]; c = Join[c, r]; z = Select[c, #[[1]] == n &][[1]]; FromDigits[z[[2]]]]
   
6. results = Table[{n, f[n]}, {n, 80}]; e = {Automatic, {1 -> LightYellow, Table[i -> If[OddQ[i], Lighter[Yellow, 0.9], Lighter[Blue, 0.9]], {i, 2, 21}]}};
   
7. Grid[Prepend[results, {"n", "n位数"}], Frame -> All, Background -> e, Dividers -> {{Gray, {False, True}}, {Gray, True}}, Alignment -> {{Center, Left}}, ItemSize -> {Automatic, 1.5}]

复制代码

题目也是挺好的——用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, 求最小的n位数。——OEIS没有。

王守恩

用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, 求最小的n位数=R(n)。

A={1, 3, 6, 1, 0, 1, 5, 2, 1, 2, 8, 3, 6, 4, 5, 5, 5, 6, 6, 7, 8, 9, 1, 1, 0, 5, 1, 2, 0, 1, 3, 6, 1, 5, 3, 1, 7, 1, 1, 9, 0, 2, 1, 0, 2, 3, 1, 2, 5, 3, 2, 7, 6, 3, 0, 0, 3, 2,
5, 3, 5, 1, 3, 7, 8, 4, 0, 6, 4, 3, 5, 4, 6, 5, 4, 9, 6, 5, 2, 8, 5, 6, 1, 5, 9, 5, 6, 3, 0, 6, 6, 6, 7, 0, 3, 7, 4, 1, 7, 8, 0, 8, 2, 0, 8, 6, 1, 9, 0, 3, 9, 4, 6, 9, 9, 0, 1, 0,
3, 5, 1, 0, 8, 1, 1, 1, 2, 8, 1, 1, 7, 6, 1, 2, 2, 5, 1, 2, 7, 5, 1, 3, 2, 6, 1, 3, 7, 8, 1, 4, 3, 1, 1, 4, 8, 5, 1, 5, 4, 0, 1, 5, 9, 6, 1, 6, 5, 3, 1, 7, 1, 1, 1, 7, 7, 0, 1, 8, 3}

R(1)=1,
1,

R(2)=31,
6,
3, 1,

R(3)=110,
6,
3, 1,
1, 1, 0,

R(4)=2210,
6,
5, 1,
3, 1, 1,
2, 2, 1, 0,

R(5)=53210,
8,
6, 3,
6, 2, 1,
5, 4, 1, 1,
5, 3, 2, 1, 0,

R(6)=653210,
8,
8, 5,
7, 5, 2,
6, 6, 3, 1,
6, 5, 4, 1, 1,
6, 5, 3, 2, 1, 0,

R(7)=6532110,
9,
8, 5,
8, 5, 2,
7, 5, 2, 1,
6, 6, 3, 1, 1,
6, 5, 4, 1, 1, 0,
6, 5, 3, 2, 1, 1, 0,

R(8)=65321100,
9,
8, 5,
8, 5, 2,
7, 5, 3, 1,
6, 6, 3, 1, 1,
6, 5, 4, 1, 1, 1,
6, 5, 3, 2, 1, 1, 0,
6, 5, 3, 2, 1, 1, 0, 0,

R(9)=654211100,
9,
9, 5,
8, 5, 2,
8, 5, 2, 1，
7, 5, 3, 1, 1,
7, 5, 3, 1, 1, 1,
6, 6, 3, 2, 1, 1, 0,
6, 6, 3, 2, 1, 1, 0, 0,
6, 5, 4, 2, 1, 1, 1, 0, 0，

王守恩

用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, 求最小的n位数=R(n)。

A={2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0, 4, 5, 2, 3, 5, 3, 6, 0, 2, 8, 7, 4, 7, 1, 3, 5, 2, 6, 6, 2, 4, 9, 7, 7, 5, 7, 2, 4, 7, 0, 9, 3, 6, 9, 9, 9, 5, 9, 5, 7, 4, 9, 6, 6, 9, 6, 7, 6, 2, 7, 7, 2, 4, 0, 7, 6, 6, 3, 0, 3, 5, 3, 5, 4, 7, 5, 9, 4, 5, 7,
1, 3, 8, 2, 1, 7, 8, 5, 2, 5, 1, 6, 6, 4, 2, 7, 4, 2, 7, 4, 6, 6, 3, 9, 1, 9, 3, 2, 0, 0, 3, 0, 5, 9, 9, 2, 1, 8, 1, 7, 4, 1, 3, 5, 9, 6, 6, 2, 9, 0, 4, 3, 5, 7, 2, 9, 0, 0, 3, 3, 4, 2, 9, 5, 2, 6, 0, 5, 9, 5, 6, 3, 0, 7, 3, 8, 1, 3, 2, 3, 2, 8, 6, 2, 7, 9}

R (1) = 2,
2,

R (2) = 21,
7,
2, 1

R (3) = 721,
8,
8, 2,
7, 2, 1,

R (4) = 8211,
8,
8, 7,
8, 2, 2,
8, 2, 1, 1,

R (5) = 84210,
9,
8, 7,
8, 5, 2,
8, 4, 2, 1,
8, 4, 2, 1, 0,

R (6) =764210,
9,
8, 5,
8, 5, 3,
8, 5, 2, 2,
8, 4, 3, 2, 1,
7, 6, 4, 2, 1, 0,

R (7) = 7742210,
9,
8, 7,
8, 6, 3,
8, 5, 4, 2,
8, 5, 4, 2, 1,
8, 5, 3, 2, 1, 0,
7, 7, 4, 2, 2, 1, 0,

R (8) = 77532210,
9,
9, 5,
8, 7, 4,
8, 6, 4, 3,
8, 6, 4, 2, 2,
8, 6, 4, 2, 2, 1,
8, 5, 5, 3, 2, 1, 0,
7, 7, 5, 3, 2, 2, 1, 0,

R (9) = 776432100,
9,
9, 6,
9, 6, 4,
8, 7, 5, 4,
8, 7, 5, 4, 2,
8, 7, 5, 4, 2, 2,
8, 7, 5, 3, 2, 2, 1,
8, 7, 5, 3, 2, 2, 1, 0,
7, 7, 6, 4, 3, 2, 1, 0, 0,

手工往前走不了。各位！可有好方法？ 谢谢！！！

王守恩

用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, 求最小的n位数=R(n)。

A={3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8,
0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, 9, 8, 2, 1, 4, 8, 0, 8, 6, 5, 1, 3, 2, 8, 2, 3, 0, 6, 6, 4, 7, 0, 9, 3, 8, 4, 4, 6, 0, 9, 5, 5, 0, 5, 8, 2, 2, 3, 1, 7, 2, 5, 3, 5, 9, 4, 0, 8, 1, 2, 8, 4, 8, 1, 1, 1, 7, 4, 5, 0, 2, 8, 4, 1, 0, 2, 7, 0, 1, 9, 3}

R (1) = 3,
3,

R (2) = 31,
4,
3, 1,

R (3) = 431,
9,
5, 1,
4, 3, 1,

R (4) = 5321,
9,
6, 3,
5, 4, 2,
5, 3, 1, 1,

R (5) = 76421,  
9,
9, 5,
9, 5, 3,
8, 5, 3, 1,
7, 6, 4, 2, 1,

R (6) =764321,
9,
9, 5,
9, 5, 3,
8, 6, 3, 2,
8, 5, 3, 2, 1,
7, 6, 4, 3, 2, 1,

R (7) = 7643321,
9,
9, 5,
9, 5, 3,
8, 6, 3, 3,
8, 6, 3, 3, 2,
8, 5, 4, 3, 2, 1,
7, 6, 4, 3, 3, 2, 1,

R (8) = 86533210,
9,
9, 6,
9, 6, 3,
9, 5, 4, 3,
8, 8, 4, 3, 2,
8, 7, 5, 3, 3, 2,
8, 7, 5, 3, 2, 2, 1,
8, 6, 5, 3, 3, 2, 1, 0,

R (9) = 886432110,
9,
9, 8,
9, 7, 5,
9, 7, 5, 3,
9, 7, 4, 3, 3,
9, 6, 5, 3, 3, 2,
9, 6, 5, 3, 3, 2, 1,
8, 8, 6, 4, 3, 2, 2, 1,
8, 8, 6, 4, 3, 2, 1, 1, 0,

R (10) = 8864332100,
9,
9, 8,
9, 8, 5,
9, 7, 5, 4,
9, 7, 5, 3, 3,
9, 7, 5, 3, 3, 2,
9, 7, 5, 3, 3, 2, 1,
9, 6, 6, 4, 3, 2, 1, 1,
8, 8, 6, 5, 3, 2, 2, 1, 0,
8, 8, 6, 4, 3, 3, 2, 1, 0, 0,

R (11) = 97653322100,
9,
9, 8,
9, 8, 6,
9, 8, 5, 4,
9, 8, 5, 4, 3,
9, 8, 5, 4, 3, 3,
9, 8, 5, 4, 3, 2, 2,
9, 7, 7, 4, 3, 3, 2, 1,
9, 7, 6, 5, 4, 3, 2, 1, 0,
9, 7, 6, 5, 4, 3, 2, 2, 1, 0,
9, 7, 6, 5, 3, 3, 2, 1, 1, 0, 0,

R (12) = 887543221000,
9,
9, 8,
9, 8, 6,
9, 8, 6, 4,
9, 8, 6, 4, 3,
9, 8, 6, 4, 3, 3,
9, 8, 5, 5, 3, 3, 2,
9, 7, 7, 5, 4, 3, 2, 1,
9, 7, 6, 5, 4, 3, 2, 1, 1,
9, 7, 6, 5, 4, 3, 2, 2, 1, 0,
9, 7, 6, 5, 4, 3, 2, 2, 1, 0, 0,
8, 8, 7, 5, 4, 3, 2, 2, 1, 0, 0, 0,

R (13) = 9865443221000,
9,
9, 8,
9, 8, 7,
9, 8, 7, 5,
9, 8, 7, 5, 4,
9, 8, 7, 5, 4, 3,
9, 8, 7, 5, 4, 3, 2,
9, 8, 7, 5, 4, 3, 2, 2,
9, 8, 6, 6, 4, 3, 2, 2, 1,
9, 8, 6, 6, 4, 3, 3, 2, 1, 0,
9, 8, 6, 6, 4, 3, 3, 2, 1, 1, 0,
9, 8, 6, 5, 5, 3, 3, 2, 2, 1, 0, 0,
9, 8, 6, 5, 4, 4, 3, 2, 2, 1, 0, 0, 0,

R (14) = 98664432211000,
9,
9, 8,
9, 8, 7,
9, 8, 7, 6,
9, 8, 7, 5, 4,
9, 8, 7, 5, 4, 3,
9, 8, 7, 5, 4, 3, 3,
9, 8, 7, 5, 4, 3, 3, 2,
9, 8, 7, 5, 4, 3, 2, 2, 2,
9, 8, 7, 5, 4, 3, 2, 2, 1, 1,
9, 8, 6, 6, 5, 4, 3, 2, 1, 1, 0,
9, 8, 6, 6, 5, 3, 3, 2, 2, 1, 0, 0,
9, 8, 6, 6, 4, 4, 3, 2, 2, 1, 1, 0, 0,
9, 8, 6, 6, 4, 4, 3, 2, 2, 1, 1, 0, 0, 0,

R (15) = 888654332210000,
9,
9, 8,
9, 8, 7,
9, 8, 7, 6,
9, 8, 7, 6, 4,
9, 8, 7, 6, 4, 3,
9, 8, 7, 5, 4, 4, 3,
9, 8, 7, 5, 4, 3, 3, 2,
9, 8, 7, 5, 4, 3, 3, 2, 2,
9, 8, 7, 5, 4, 3, 3, 2, 2, 1,
9, 8, 6, 6, 5, 4, 3, 2, 2, 1, 1,
9, 8, 6, 6, 5, 4, 3, 2, 2, 1, 1, 0,
9, 8, 6, 6, 5, 4, 3, 2, 2, 1, 1, 0, 0,
9, 8, 6, 6, 5, 4, 3, 2, 2, 1, 1, 0, 0, 0,
8, 8, 8, 6, 5, 4, 3, 3, 2, 2, 1, 0, 0, 0, 0,

人脑的方法我已成熟！！！各位！这电脑的方法应该怎么走？ 谢谢！！！

王守恩

乘积最小与乘积最大。一并解解决。两者还是有区别的。

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

乘积最小。用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最小时, n位数=R(n)。譬如。
n=1, {1}, R(1)=1,
n=2, {1,23}, R(2)=23,
n=3, {1,24,356}, R(3)=356,
n=4, {1,14,257,3689}, R(4)=3689,
n=5, {1,13,245,2467,35689}, R(5)=35689,
n=6, {1,13,135,2357,24678,246899}, R(6)=246899,
n=7, {1,13,135,1356,24578,246789,2467899}, R(7)=2467899,
n=8, {1,13,134,1356,23567,245678,2457889,24678999}, R(8)=24678999,
n=9, {1,12,134,1345,13467,235678,2356788,24567899,245678999}, R(9)=245678999,

乘积最大。用A前n(n+1)/2个数码, 恰好可以组成n个数, 其中有1个1位数,1个2位数,1个3位数,1个4位数,...,1个n位数,  当n个数的乘积最大时, n位数=R(n)。譬如。
n=1, {1}, R(1)=1,
n=2, {3,21}, R(2)=21,
n=3, {6,52,431}, R(3)=431,
n=4, {9,83,741,6521}, R(4)=6521,
n=5, {9,84,742,6532,65311}, R(5)=65311,
n=6, {9,94,853,8532,76421,763211}, R(6)=763211,
n=7, {9,96,954,8742,86531,864321,7753211}, R(7)=7753211,
n=8, {9,96,965,9643,87632,875432,8754311,87542211}, R(8)=87542211,
n=9, {9,97,975,9654,96543,885432,8765321,87643221,876432111}, R(9)=876432111,