project Euler309巷道交叉梯子问题(要求全部整数解)

mathematica

问题来源:http://projecteuler.net/index.php?section=problems&id=309 问题描述(请参考下面的图): 我还是稍微翻译一下,凭借我个人的理解, 在一个巷道里,放置了两个梯子. 这两个梯子的长度都是正整数, 短的梯子的长度是x, 长的梯子的长度是y,(x小于等于y) 两个梯子的交点到地面的距离是h,h也是整数, 此时存在在巷道的宽度w也是整数的情况,因此 要求所有的(x,y,h,w)(这四个量都是整数). 当然,这两个梯子所在的平面垂直于巷道的地面. 我写了一个mathematica的求解的子函数,但是只是求解巷道的宽度等的子函数, 可以使用三重循环(长梯子的宽度,短梯子的宽度,交叉点高度)来穷举, 但是我觉得太慢了,不知道谁有高招!

1. Clear["Global`*"];(*Clearallvariables*)
2. (*定义子函数求解方程*)
3. (*子函数的输出量依次为a,b,w,c,d,其意义见图所示*)
4. mySolu[x0_,y0_,h0_]:=
5. Module[{x=x0,y=y0,h=h0,(*已知的初始值*)
6. a,b,c,d,w, (*待求解的变量的值*)
7. eq1,eq2,eq3}, (*中间变量*)
8. eq1=NSolve[{c/w==h/b,
9. d/w==h/a,
10. a+b==w,
11. c^2+w^2==x^2,
12. d^2+w^2==y^2
13. },{a,b,w,c,d},10];
14. eq2={a,b,w,c,d}/.eq1; (*只保留数据,去掉变量符号*)
15. eq3=Select[eq2,Arg[#]=={0,0,0,0,0}&](*提取结果都大于等于零的结果*)
16. ]
17. mySolu[70, 119, 30](*用来验证子函数是否正确*)
18. {{16.0000000, 40.00000000, 56.0000000, 42.00000000, 105.0000000}}
19. (*上面的结果告诉我们,巷道的宽度是56*)

复制代码

mathematica

上面的代码!

mathematica

mySolu[70, 119, 30] mySolu[74, 182, 21] mySolu[87, 105, 35] mySolu[100, 116, 35] mySolu[119, 175, 40] {{16.0000000, 40.00000000, 56.0000000, 42.00000000, 105.0000000}} {{8.7500000, 61.2500000, 70.0000000, 24.00000000, 168.0000000}} {{26.2500000, 36.75000000, 63.0000000, 60.0000000, 84.0000000}} {{33.3333333, 46.66666667, 80.0000000, 60.0000000, 84.0000000}} {{30.0000000, 75.0000000, 105.0000000, 56.0000000, 140.0000000}} 由上面的求解结果直到,巷道的宽度依次是56,70,63,80,105

gxqcn

2# mathematica 没必要，总是发代码的截图，因为它不会给阅读者带来更多的信息。

mathematica

2# mathematica 没必要，总是发代码的截图，因为它不会给阅读者带来更多的信息。 gxqcn 发表于 2011-4-10 11:36

有必要截图的,如果直接把代码粘贴上去,在论坛上看还是很费劲的, 如果截图了,看起来更好看的,可是如果好看却又不能复制, 所以两者兼考虑一下,我就把代码和图全部传上来了, 当然图还有防止代码被转换错误的功能.

wayne

{40, {{401, 58, 38}}} {56, {{119, 70, 30}}} {63, {{663, 87, 55}, {105, 87, 35}}} {70, {{182, 74, 21}}} {80, {{116, 100, 35}}} {96, {{296, 104, 35}}} {105, {{273, 175, 90}, {175, 119, 40}}} {112, {{238, 113, 14}}} {117, {{533, 195, 120}}} {119, {{7081, 169, 118}}} {140, {{364, 175, 80}}} {144, {{219, 156, 44}}} {165, {{915, 429, 275}, {429, 187, 72}}} {168, {{210, 182, 45}}} {224, {{1799, 476, 340}}} {225, {{1695, 255, 112}}} {252, {{1148, 273, 96}, {420, 273, 80}}} {260, {{1700, 676, 455}}} {264, {{814, 275, 70}, {561, 286, 90}}} {275, {{715, 365, 176}}} {300, {{500, 375, 144}}} {315, {{1435, 357, 150}}} {364, {{1300, 455, 224}}} {385, {{1375, 407, 120}}} {400, {{1625, 500, 252}}} {408, {{442, 425, 70}}} {448, {{3152, 848, 585}}} {455, {{7969, 481, 153}, {2137, 679, 406}}} {460, {{1196, 667, 336}}} {462, {{2562, 770, 495}}} {483, {{1725, 667, 360}}} {495, {{3729, 825, 560}, {825, 583, 210}}} {496, {{2015, 1054, 630}}} {520, {{1105, 754, 350}, {650, 533, 90}}} {525, {{1875, 875, 504}}} {528, {{1628, 660, 315}, {1122, 822, 385}}} {552, {{1702, 1173, 630}}} {555, {{4181, 1443, 1008}}} {576, {{1224, 776, 351}}} {585, {{4407, 689, 336}, {689, 663, 168}}} {595, {{3637, 757, 414}, {2125, 1547, 840}, {1547, 1099, 561}}} {600, {{870, 845, 306}, {750, 625, 126}}} {612, {{1887, 1020, 560}}} {616, {{4334, 1309, 910}, {1166, 770, 315}}} {630, {{6630, 1306, 975}}} {637, {{28987, 763, 414}}} {651, {{6851, 1085, 770}}} {660, {{4972, 1199, 832}}} {663, {{1937, 1105, 595}}} {672, {{4728, 840, 455}, {2353, 1022, 574}}} {675, {{9125, 1125, 819}}} {680, {{986, 697, 126}}} {693, {{3843, 707, 135}}} {728, {{2422, 1547, 858}, {1547, 1378, 630}}} {735, {{2625, 959, 495}}} {741, {{4845, 1235, 819}}} {759, {{1495, 809, 230}}} {760, {{3838, 950, 495}, {1976, 1615, 800}}} {765, {{1989, 901, 378}}} {775, {{9703, 2015, 1560}}} {792, {{6558, 2442, 1705}, {2442, 1067, 546}}} {800, {{4040, 1160, 693}}} {819, {{15981, 3731, 2964}, {15981, 1869, 1520}, {5355, 981, 490}}} {825, {{1625, 1375, 616}}} {880, {{1276, 884, 77}, {1100, 979, 260}}} {918, {{23418, 2682, 2275}}} {920, {{4646, 943, 198}}} {931, {{22819, 3325, 2800}}} {936, {{2886, 2210, 1155}, {1560, 975, 224}}} {945, {{21273, 2639, 2208}, {1233, 1071, 308}}} {952, {{1802, 1073, 374}}} {975, {{1455, 1105, 351}}} {980, {{4949, 2548, 1584}}} {992, {{7967, 1240, 680}}}

mathematica

{40, {{401, 58, 38}}} {56, {{119, 70, 30}}} {63, {{663, 87, 55}, {105, 87, 35}}} {70, {{182, 74, 21}}} {80, {{116, 100, 35}}} {96, {{296, 104, 35}}} {105, {{273, 175, 90}, {175, 119, 40}}} {112 ... [size=2000]wayne 发表于 2011-4-10 14:15

只有结果的回复没有太大的意义,过程很重要

wayne

http://bbs.emath.ac.cn/thread-2048-1-1.html

mathematica

http://bbs.emath.ac.cn/thread-2048-1-1.html wayne 发表于 2011-4-11 09:22

这个问题我在论坛上也研究过,具体请看 http://bbs.emath.ac.cn/viewthread.php?tid=2987&page=1#pid35308

mathematica

http://bbs.emath.ac.cn/viewthrea ... romuid=865#pid35309 看这个回复吧