求相加之和为最小与最大的质数组

王守恩

王守恩 发表于 2016-9-13 09:37
解题思路

一、最小。
1、个位不能是0、4、6、8。即十位是0、4、6、8。
2、0在十位上，百位上肯定有数。
3、因为求最小，十个数最好分成1个三位数，3个二位数，1个一位数，
     且百位上的数码尽可能小，得基本竖式。
4、调整，个位上必须有1，百位数码只能换成2。
二、最大。
1、因为求最大，十个数码最好分成3个3位数，1个1位数，
     且百位上的数码尽可能大，得基本竖式。
2、调整，个位上必须有7，百位数码只能换成6。
三、说明
1、题目只要求给出405、2403就行。
2、题目没要求把一个一个一个一个素数都写出来。
3、交换同位上的数码，和不变。

奇数的世界

王守恩 发表于 2016-9-13 13:57
一、最小。
1、个位不能是0、4、6、8。即十位是0、4、6、8。
2、0在十位上，百位上肯定有数。

2 3 5 7 11 13 17 19 23 29

　　31 37 41 43 47 53 59 61 67 71

　　73 79 83 89 97 101 103 107 109 113

　　127 131 137 139 149 151 157 163 167 173

　　179 181 191 193 197 199 211 223 227 229

　　233 239 241 251 257 263 269 271 277 281

　　283 293 307 311 313 317 331 337 347 349
素数表中，我就没有看到203

angel_phoenix88

取最小值情形
1)0只能用在一个三位数质数的中间，三位数质数必须包含两个偶数(百位为2,4,6,8,十位为0)，由于201,203,207,209都是合数，最小三位数质素只能取401
2)最小和值必须包含质数5
3)4,6,8和1,3,7组成四组质数2，3,67,89

奇数的世界

天山草 发表于 2016-9-13 09:44
我完全同意你的这个观点。
丘成桐说过： 奥数就像是报纸上的娱乐版，看过之后也就扔到垃圾筒里了，根本 ...

要说编程解出答案，我可以做到，用穷举法就可以实现算法，只是很费时间，而且编出来意义不大。
这个题解答没有什么巧妙之处，只有靠排除法，一个个排除，总会有答案的。不过花的的时间实在很多，我问一下，这道题在所有的题目中能占多少分呢？

大傻8888888

大傻8888888 发表于 2016-9-13 11:45
还各有一组{2, 5, 7, 61, 83, 409}最小。{2, 607, 853, 941}最大。

因为大于两位数的素数的尾数只能是1,3,7,9。而307占用了3和7，同时还有4,6,8需要和1,9组成素数，这样其中必有一个大于400的素数，不符合题目相加之和为最小的素数组。

大傻8888888

解题思路
一、最小。
1、个位不能是0、4、6、8。即十位以上(含十位)是4、6、8。
2、0只能在十位上，百位上肯定有数。
3、因为求最小，十个数最好分成1个三位数，2个二位数，3个一位数，
     且百位上的数码尽可能小。
4、个位上必须有1，百位数码不能是2(因为201,203,207,209都不是素数),也不能是3(见25楼)，只能是4。
二、最大。
1、因为求最大，十个数码最好分成3个3位数，1个1位数，
     且百位上的数码尽可能大。
2、调整，个位上必须是1,3，7。十位数是0,4,5。百位数码是9,8，6。
三、
     交换同位上的数码，和不变。

大傻8888888

大傻8888888 发表于 2016-9-13 22:41
解题思路
一、最小。
1、个位不能是0、4、6、8。即十位以上(含十位)是4、6、8。

天山草先生
百位上为什么不能是 1 的道理同25楼。
奇数的世界
个位上必须有1是因为1本身不是素数，所以肯定有个位是1的素数。

在天山草老师面前我只能算小学生。

奇数的世界

大傻8888888 发表于 2016-9-14 11:46
天山草先生
百位上为什么不能是 1 的道理同25楼。
奇数的世界

“个位上必须有1是因为1本身不是素数，所以肯定有个位是1的素数。“
{2, 5, 89, 103, 467} 这组解存在，说明你的理由不充分。

天山草

下面这个程序，并不是本人所写，是数学研发网站的 zeroieme 先生的作品（本人目前的 mathematica 编程能力还差得很远哈，不下洪荒之力是提高无望啦）：

Module[{EligiblePrimes, PrimeLists},
PrimeLists = Composition[Gather, Select[#, MemberQ[IntegerDigits[#], 0] &] &, (EligiblePrimes =Select[#, Sort[#] ==
Union[#] &[IntegerDigits[#]] &]) &,   Array[Prime, #] &, PrimePi][999];
Do[PrimeLists = Join[Select[PrimeLists, MemberQ[Flatten[IntegerDigits /@ #], i] &],
     Function[{NowEligiblePrimes},
      Apply[Join, Table[DeleteCases[
         Table[If[Sort[#] == Union[#] &[Flatten[IntegerDigits /@ #]],
             Sort[#], {}] &[Append[\[Zeta], \[Omega]]], {\[Omega],
           NowEligiblePrimes}], {}], {\[Zeta], Select[PrimeLists, Not[MemberQ[Flatten[IntegerDigits /@ #], i]] &]}]]][
     Select[EligiblePrimes, MemberQ[IntegerDigits[#], i] &]]], {i, 9}];
SortBy[PrimeLists, Total]]

  运行结果：（每个集合各元素相加之和从小到大排序）

{
{2, 3, 5, 67, 89, 401}, {2, 5, 7, 61, 83, 409}, {5, 23, 67,  89, 401}, {5, 29, 67, 83, 401}, {2, 53, 67, 89, 401},
{2, 59, 67, 83, 401},  {2, 5, 83, 109, 467}, {2, 5, 83, 167, 409}, {2, 5, 89, 103, 467}, {2, 5, 89, 107, 463},
{2, 41, 67, 83, 509}, {2, 41, 67, 89, 503},  {2, 47, 61, 83, 509}, {2, 47, 61, 89, 503}, {2, 3, 5, 41,  89, 607},
{2, 3, 5, 47, 89, 601}, {2, 5, 7, 43, 89, 601}, {3, 5, 67, 281, 409}, {5, 7, 61, 283, 409}, {5, 7, 83, 269, 401},
{5, 7, 89, 263, 401}, {5, 23, 41, 89, 607}, {5, 23, 47, 89, 601}, {5, 29, 41, 83, 607},  {5, 29, 47, 83, 601},
{2, 41, 53, 89, 607}, {2, 41, 59, 83, 607}, {2, 47, 53, 89, 601}, {2, 47, 59, 83, 601}, {53, 67, 281, 409},
{59, 67, 283, 401},    {61, 83, 257, 409}, {67, 83, 251, 409}, {2, 5, 47, 109, 683}, {2, 5, 83, 109, 647},
{2, 5, 83, 149, 607},  {2, 5, 89, 103, 647}, {2, 5, 89, 107, 643}, {5, 103, 269, 487}, {5, 109, 263, 487},
{5, 109, 283, 467}, {5, 167, 283, 409}, {2, 5, 67, 389, 401}, {2, 5, 89, 307, 461}, {2, 5, 89, 367,  401},
{41, 67, 283, 509}, {43, 67, 281, 509}, {47, 61, 283,  509}, {67, 83, 241, 509}, {67, 89, 241, 503},
{2, 3, 5, 41, 67,  809}, {2, 3, 5, 47, 61, 809}, {2, 5, 7, 43, 61, 809}, {3, 5, 89, 241, 607}, {5, 23, 41, 67, 809},
{5, 23, 47, 61, 809}, {2, 41, 53,  67, 809}, {2, 47, 53, 61, 809}, {41, 59, 283, 607}, {43, 59, 281,  607},
{43, 89, 251, 607}, {43, 89, 257, 601}, {47, 59, 283,  601}, {53, 89, 241, 607}, {59, 83, 241, 607},
{2, 5, 43, 167,  809}, {2, 5, 47, 109, 863}, {2, 5, 47, 163, 809}, {5, 109, 283, 647}, {5, 149, 283, 607},
{2, 5, 41, 389, 607}, {2, 5, 47, 389,  601}, {2, 5, 89, 307, 641}, {2, 5, 89, 347, 601}, {5, 281, 367,  409},
{2, 3, 61, 409, 587}, {2, 3, 61, 487, 509}, {2, 7, 83, 401, 569}, {2, 7, 83, 461, 509}, {2, 7, 89, 401, 563},
{2, 7, 89, 461,  503}, {23, 61, 409, 587}, {23, 61, 487, 509}, {29, 61, 487,  503}, {67, 83, 409, 521},
{67, 83, 421, 509}, {67, 89, 401,  523}, {67, 89, 421, 503}, {2, 5, 17, 409, 683}, {2, 5, 83, 409,  617},
{2, 5, 83, 419, 607}, {3, 5, 29, 487, 601}, {3, 5, 67, 241,  809}, {3, 5, 89, 421, 607}, {5, 7, 29, 401, 683},
{5, 7, 41, 263,  809}, {2, 5, 89, 431, 607}, {2, 3, 59, 487, 601}, {2, 3, 89, 457,  601}, {2, 7, 59, 401, 683},
{2, 7, 83, 401, 659}, {2, 7, 89, 401,  653}, {2, 103, 487, 569}, {2, 109, 463, 587}, {2, 109, 487,  563},
{2, 163, 409, 587}, {2, 163, 487, 509}, {23, 59, 487,  601},  {23, 89, 457, 601}, {29, 53, 487, 601},
{29, 83, 457,  601}, {43, 61, 257, 809}, {43, 67, 251, 809}, {53, 67, 241,  809}, {53, 89, 421, 607},
{59, 83, 421, 607}, {2, 5, 71, 409,  683}, {2, 5, 79, 401, 683}, {2, 5, 83, 479, 601}, {2, 5, 89, 401,  673},
{2, 5, 83, 491, 607}, {2, 5, 97, 401, 683}, {5, 127, 409,  683}, {2, 5, 41, 367, 809}, {2, 5, 61, 347, 809},
{5, 241, 389,  607}, {5, 281, 349, 607}, {2, 3, 89, 541, 607}, {2, 3, 89, 547,  601}, {2, 7, 41, 509, 683},
{2, 7, 83, 509, 641}, {2, 7, 89, 503, 641}, {2, 103, 487, 659}, {2, 109, 457, 683}, {2, 109, 487, 653},
{2, 157, 409, 683}, {3, 269, 401, 587}, {3, 281, 467, 509},  {7, 281, 409, 563}, {7, 281, 463, 509},
{7, 283, 401, 569}, {7, 283, 461, 509}, {23, 89, 541, 607}, {23, 89, 547, 601}, {29, 43, 587, 601},
{29, 83, 541, 607}, {29, 83, 547, 601}, {41, 89, 523, 607}, {43, 89, 521, 607}, {47, 89, 523, 601},
{2, 5, 83, 401, 769}, {2, 5, 83, 409, 761}, {2, 5, 83, 461, 709}, {2, 5, 89, 463, 701}, {2, 3, 5, 7, 461, 809},
{2, 5, 13, 467, 809}, {2, 5, 17, 409, 863}, {2, 5, 17, 463, 809}, {3, 5, 61, 409, 827}, {3, 5, 67, 401, 829},
{3, 5, 67, 409, 821}, {3, 5, 67, 421, 809}, {5, 7, 23, 461, 809}, {5, 7, 29, 401, 863}, {5, 7, 61, 409, 823},
{5, 283, 409, 617}, {5, 283, 419, 607}, {2, 5, 31, 467, 809}, {2, 5, 37, 461, 809}, {2, 5, 67, 401, 839},
{2, 5, 67, 431, 809}, {5, 239, 487, 601}, {5, 281, 439, 607}, {2, 3, 61, 409, 857}, {2, 3, 61, 457, 809},
{2, 3, 67, 401, 859}, {2, 7, 53, 461, 809}, {2, 7, 59, 401, 863}, {2, 7, 61, 409, 853}, {2, 109, 547, 683},
{2, 109, 587, 643}, {7, 251, 409, 683}, {7, 281, 409, 653}, {7, 283, 401, 659}, {23, 61, 409, 857}, {23, 61, 457, 809},
{23, 67, 401, 859}, {29, 67, 401, 853}, {53, 61, 409, 827}, {53, 67, 401, 829}, {53, 67, 409, 821}, {53, 67, 421, 809},
{59, 67, 401, 823}, {2, 5, 71, 409, 863}, {2, 5, 71, 463, 809}, {2, 5, 73, 461, 809}, {2, 5, 79, 401, 863}, {5, 271, 409, 683},
{5, 281, 409, 673}, {5, 283, 479, 601}, {2, 5, 97, 401, 863}, {5, 283, 491, 607}, {5, 293, 487, 601}, {5, 103, 467, 829},
{5, 107, 463, 829}, {5, 109, 463, 827}, {5, 109, 467, 823}, {5, 127, 409, 863}, {5, 127, 463, 809}, {5, 163, 409, 827},
{5, 167, 409, 823}, {5, 241, 367, 809}, {5, 389, 421, 607}, {2, 3, 61, 547, 809}, {2, 3, 67, 541, 809}, {2, 7, 41, 509, 863},
{2, 7, 41, 563, 809}, {2, 103, 467, 859}, {2, 107, 463, 859}, {2, 109, 457, 863}, {2, 109, 463, 857}, {2, 109, 467, 853},
{2, 157, 409, 863}, {2, 157, 463, 809}, {2, 163, 409, 857}, {2, 163, 457, 809}, {2, 167, 409, 853}, {3, 281, 509, 647},
{7, 241, 509, 683}, {7, 281, 509, 643}, {7, 283, 509, 641}, {23, 61, 547, 809}, {23, 67, 541, 809}, {41, 67, 503, 829},
{41, 67, 509, 823}, {41, 67, 523, 809}, {43, 61, 509, 827}, {43, 67, 509, 821}, {43, 67, 521, 809}, {47, 61, 503, 829},
{47, 61, 509, 823}, {47, 61, 523, 809}, {2, 5, 41, 683, 709}, {2, 5, 83, 641, 709}, {2, 5, 89, 601, 743}, {2, 5, 89, 643, 701},
{2, 359, 487, 601}, {2, 389, 457, 601}, {5, 281, 463, 709}, {5, 283, 401, 769}, {5, 283, 409, 761}, {5, 283, 461, 709},
{2, 5, 67, 401, 983}, {2, 5, 83, 401, 967}, {2, 5, 83, 461, 907}, {2, 3, 5, 7, 641, 809}, {2, 5, 13, 647, 809}, {2, 5, 17, 643, 809},
{2, 5, 43, 617, 809}, {2, 5, 47, 613, 809}, {3, 5, 41, 607, 829}, {3, 5, 47, 601, 829}, {5, 7, 23, 641, 809}, {5, 7, 43, 601, 829},
{2, 5, 31, 647, 809}, {2, 5, 37, 641, 809}, {2, 5, 41, 607, 839}, {2, 5, 47, 601, 839}, {2, 5, 47, 631, 809}, {2, 3, 41, 607, 859},
{2, 3, 47, 601, 859}, {2, 7, 41, 653, 809}, {2, 7, 43, 601, 859}, {2, 7, 53, 641, 809}, {2, 109, 547, 863}, {2, 163, 547, 809},
{3, 251, 467, 809}, {3, 257, 461, 809}, {3, 269, 401, 857}, {7, 251, 409, 863}, {7, 251, 463, 809}, {7, 263, 401, 859},
{7, 269, 401, 853}, {23, 41, 607, 859}, {23, 47, 601, 859}, {29, 41, 607, 853}, {29, 43, 601, 857}, {29, 47, 601, 853},
{41, 53, 607, 829}, {41, 59, 607, 823}, {43, 59, 601, 827}, {43, 59, 607, 821}, {47, 53, 601, 829}, {47, 59, 601, 823},
{2, 5, 41, 673, 809}, {2, 5, 71, 643, 809}, {2, 5, 73, 641, 809}, {2, 349, 587, 601}, {2, 389, 541, 607}, {2, 389, 547, 601},
{5, 271, 409, 863}, {5, 271, 463, 809}, {5, 103, 647, 829}, {5, 107, 643, 829}, {5, 109, 643, 827}, {5, 109, 647, 823},
{5, 127, 643, 809}, {5, 149, 607, 823}, {5, 307, 461, 829}, {5, 367, 401, 829}, {5, 367, 409, 821}, {5, 367, 421, 809},
{2, 103, 647, 859}, {2, 107, 643, 859}, {2, 109, 643, 857}, {2, 109, 647, 853}, {2, 149, 607, 853}, {2, 157, 643, 809},
{2, 409, 587, 613}, {2, 487, 503, 619}, {2, 487, 509, 613}, {7, 241, 509, 863}, {7, 241, 563, 809}, {7, 263, 541, 809},
{7, 409, 521, 683}, {7, 421, 509, 683}, {2, 5, 41, 709, 863}, {2, 5, 43, 761, 809}, {2, 5, 61, 743, 809}, {2, 307, 461, 859},
{2, 367, 401, 859}, {2, 409, 587, 631}, {2, 439, 587, 601}, {2, 487, 509, 631}, {5, 241, 683, 709}, {5, 281, 643, 709},
{5, 283, 641, 709}, {2, 5, 41, 607, 983}, {2, 5, 41, 683, 907}, {2, 5, 47, 601, 983}, {2, 5, 83, 601, 947}, {2, 5, 83, 607, 941},
{2, 5, 83, 641, 907}, {5, 281, 463, 907}, {5, 283, 401, 967}, {5, 283, 461, 907}, {2, 409, 571, 683}, {2, 487, 503, 691},
{2, 487, 593, 601}, {5, 241, 607, 839}, {3, 241, 607, 859}, {3, 251, 647, 809}, {3, 257, 641, 809}, {7, 241, 653, 809},
{7, 251, 643, 809}, {2, 367, 541, 809}, {5, 241, 673, 809}, {5, 271, 643, 809}, {5, 307, 641, 829}, {5, 347, 601, 829},
{5, 349, 601, 827}, {5, 349, 607, 821}, {3, 401, 569, 827}, {3, 461, 509, 827}, {3, 467, 509, 821}, {3, 467, 521, 809},
{7, 401, 563, 829}, {7, 401, 569, 823}, {7, 409, 521, 863}, {7, 409, 563, 821}, {7, 421, 509, 863}, {7, 421, 563, 809},
{7, 461, 503, 829}, {7, 461, 509, 823}, {7, 461, 523, 809}, {7, 463, 509, 821}, {7, 463, 521, 809}, {2, 307, 641, 859},
{2, 347, 601, 859}, {2, 349, 601, 857}, {5, 241, 709, 863}, {5, 421, 683, 709}, {2, 5, 41, 863, 907}, {5, 241, 607, 983},
{5, 241, 683, 907}, {5, 281, 643, 907}, {5, 283, 601, 947}, {5, 283, 607, 941}, {5, 283, 641, 907}, {2, 409, 571,  863},
{2, 409, 683, 751}, {2, 463, 571, 809}, {5, 409, 613, 827}, {5, 409, 617, 823}, {5, 419, 607, 823}, {5, 409, 631,  827},
{5, 421, 607, 839}, {5, 431, 607, 829}, {5, 439, 601, 827}, {5, 439, 607, 821}, {2, 409, 613, 857}, {2, 409, 617,  853},
{2, 419, 607, 853}, {2, 457, 613, 809}, {3, 401, 659, 827}, {3, 421, 607, 859}, {3, 457, 601, 829}, {7, 401, 653,  829},
{7, 401, 659, 823}, {7, 409, 653, 821}, {7, 421, 653, 809}, {2, 409, 631, 857}, {2, 431, 607, 859}, {2, 439, 601,  857},
{2, 457, 601, 839}, {2, 457, 631, 809}, {5, 401, 673, 829}, {5, 409, 673, 821}, {5, 421, 673, 809}, {5, 479, 601,  823},
{5, 491, 607, 823}, {2, 401, 673, 859}, {2, 479, 601, 853}, {2, 541, 683, 709}, {2, 491, 607, 853}, {2, 547, 613,  809},
{3, 509, 641, 827}, {3, 509, 647, 821}, {3, 521, 647, 809}, {3, 541, 607, 829}, {3, 547, 601, 829}, {7, 503, 641,  829},
{7, 509, 641, 823}, {7, 509, 643, 821}, {7, 521, 643, 809}, {7, 523, 641, 809}, {2, 541, 607, 839}, {2, 547, 601,  839},
{2, 547, 631, 809}, {5, 401, 769, 823}, {5, 409, 761, 823}, {5, 421, 709, 863}, {5, 461, 709, 823}, {5, 463, 701,  829},
{5, 463, 709, 821}, {5, 241, 863, 907}, {5, 421, 607, 983}, {5, 421, 683, 907}, {2, 401, 769, 853}, {2, 409, 751,  863},
{2, 409, 761, 853}, {2, 461, 709, 853}, {2, 463, 701, 859}, {2, 463, 751, 809}, {2, 541, 673, 809}, {2, 571, 643,  809},
{2, 457, 601, 983}, {2, 487, 601, 953}, {2, 541, 709, 863}, {2, 541, 607, 983}, {2, 541, 683, 907}, {2, 547, 601,  983},
{5, 601, 743, 829}, {5, 641, 709, 823}, {5, 643, 701, 829}, {5, 643, 709, 821}, {5, 401, 823, 967}, {5, 421, 863,  907},
{5, 461, 823, 907}, {5, 463, 821, 907}, {2, 601, 743, 859}, {2, 641, 709, 853}, {2, 643, 701, 859}, {2, 643, 751,  809},
{2, 401, 853, 967}, {2, 461, 853, 907}, {2, 541, 863, 907}, {5, 601, 823, 947}, {5, 607, 823, 941}, {5, 641, 823,  907},
{5, 643, 821, 907}, {2, 601, 853, 947}, {2, 607, 853, 941}, {2, 641, 853, 907}
  }

王守恩

天山草 发表于 2016-9-14 19:29
下面这个程序，并不是本人所写，是数学研发网站的 zeroieme 先生的作品：

Module[{EligiblePrimes, Pri ...

大傻88888888是对的。我的答案是错了。正确答案是567.2403。这道题还是一道不错的题目。我们训练用的是这一题。用9，8，7，6，5，4，3，2，1，0十个数码组成若干个奇数。最大的数是三位数。问。1，若干个奇数的和最大是几。2，若干个奇数的和最小是几。