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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › 百度之星的一道与斐波那契数列有关的题 ...
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[提问] 百度之星的一道与斐波那契数列有关的题

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chyanog
发表于 2012-12-13 18:38:55 | 显示全部楼层 |阅读模式

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题目来源：http://astar.baidu.com/index.php?r=home/detail&id=2

du熊对数学一直都非常感兴趣。最近在学习斐波那契数列的它，向你展示了一个数字串，它称之为“斐波那契”串：

11235813471123581347112358........

聪明的你当然一眼就看出了这个串是这么构造的：
1.先写下两位在0~9范围内的数字a, b，构成串ab；
2.取串最后的两位数字相加，将和写在串的最后面。
上面du熊向你展示的串就是取a = b = 1构造出来的串。
显然，步骤1之后不停地进行步骤2，数字串可以无限扩展。现在，du熊希望知道串的第n位是什么数字。
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2012-12-13 21:09:49 | 显示全部楼层
猜想,不管a,b是什么开始的,最终都构成一个周期重复性的数字串.
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2012-12-13 21:17:37 | 显示全部楼层
不是猜想,是必然的.
总共也就100种情况.可以用程序穷举验证.
  1. Flatten[Table[{{i, j}, xx = Table[0, {i, 10}, {j, 10}];
  2.    NestWhile[(xx[[#[[-2]] + 1, #[[-1]] + 1]] = 1; Join[#, IntegerDigits[Total[Take[#, -2]]]]) &, {i, j}, xx[[#[[-2]] + 1, #[[-1]] + 1]] == 0 &]}, {i, 0, 9}, {j, 0, 9}], 1]
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2012-12-13 22:22:15 | 显示全部楼层

  2. {0,0}        {0,0,0}
  3. {0,1}        {0,1,1,2,3,5,8,1,3,4,7,1,1}
  4. {0,2}        {0,2,2,4,6,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  5. {0,3}        {0,3,3,6,9,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  6. {0,4}        {0,4,4,8,1,2,3,5,8,1,3,4,7,1,1,2}
  7. {0,5}        {0,5,5,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  8. {0,6}        {0,6,6,1,2,3,5,8,1,3,4,7,1,1,2}
  9. {0,7}        {0,7,7,1,4,5,9,1,4}
  10. {0,8}        {0,8,8,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  11. {0,9}        {0,9,9,1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  12. {1,0}        {1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  13. {1,1}        {1,1,2,3,5,8,1,3,4,7,1,1}
  14. {1,2}        {1,2,3,5,8,1,3,4,7,1,1,2}
  15. {1,3}        {1,3,4,7,1,1,2,3,5,8,1,3}
  16. {1,4}        {1,4,5,9,1,4}
  17. {1,5}        {1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  18. {1,6}        {1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  19. {1,7}        {1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  20. {1,8}        {1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  21. {1,9}        {1,9,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  22. {2,0}        {2,0,2,2,4,6,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  23. {2,1}        {2,1,3,4,7,1,1,2,3,5,8,1,3}
  24. {2,2}        {2,2,4,6,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  25. {2,3}        {2,3,5,8,1,3,4,7,1,1,2,3}
  26. {2,4}        {2,4,6,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  27. {2,5}        {2,5,7,1,2,3,5,8,1,3,4,7,1,1,2}
  28. {2,6}        {2,6,8,1,4,5,9,1,4}
  29. {2,7}        {2,7,9,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  30. {2,8}        {2,8,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  31. {2,9}        {2,9,1,1,2,3,5,8,1,3,4,7,1,1}
  32. {3,0}        {3,0,3,3,6,9,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  33. {3,1}        {3,1,4,5,9,1,4}
  34. {3,2}        {3,2,5,7,1,2,3,5,8,1,3,4,7,1,1,2}
  35. {3,3}        {3,3,6,9,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  36. {3,4}        {3,4,7,1,1,2,3,5,8,1,3,4}
  37. {3,5}        {3,5,8,1,3,4,7,1,1,2,3,5}
  38. {3,6}        {3,6,9,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  39. {3,7}        {3,7,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  40. {3,8}        {3,8,1,1,2,3,5,8,1,3,4,7,1,1}
  41. {3,9}        {3,9,1,2,3,5,8,1,3,4,7,1,1,2}
  42. {4,0}        {4,0,4,4,8,1,2,3,5,8,1,3,4,7,1,1,2}
  43. {4,1}        {4,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  44. {4,2}        {4,2,6,8,1,4,5,9,1,4}
  45. {4,3}        {4,3,7,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  46. {4,4}        {4,4,8,1,2,3,5,8,1,3,4,7,1,1,2}
  47. {4,5}        {4,5,9,1,4,5}
  48. {4,6}        {4,6,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  49. {4,7}        {4,7,1,1,2,3,5,8,1,3,4,7}
  50. {4,8}        {4,8,1,2,3,5,8,1,3,4,7,1,1,2}
  51. {4,9}        {4,9,1,3,4,7,1,1,2,3,5,8,1,3}
  52. {5,0}        {5,0,5,5,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  53. {5,1}        {5,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  54. {5,2}        {5,2,7,9,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  55. {5,3}        {5,3,8,1,1,2,3,5,8,1,3,4,7,1,1}
  56. {5,4}        {5,4,9,1,3,4,7,1,1,2,3,5,8,1,3}
  57. {5,5}        {5,5,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  58. {5,6}        {5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  59. {5,7}        {5,7,1,2,3,5,8,1,3,4,7,1,1,2}
  60. {5,8}        {5,8,1,3,4,7,1,1,2,3,5,8}
  61. {5,9}        {5,9,1,4,5,9}
  62. {6,0}        {6,0,6,6,1,2,3,5,8,1,3,4,7,1,1,2}
  63. {6,1}        {6,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  64. {6,2}        {6,2,8,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  65. {6,3}        {6,3,9,1,2,3,5,8,1,3,4,7,1,1,2}
  66. {6,4}        {6,4,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  67. {6,5}        {6,5,1,1,2,3,5,8,1,3,4,7,1,1}
  68. {6,6}        {6,6,1,2,3,5,8,1,3,4,7,1,1,2}
  69. {6,7}        {6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  70. {6,8}        {6,8,1,4,5,9,1,4}
  71. {6,9}        {6,9,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  72. {7,0}        {7,0,7,7,1,4,5,9,1,4}
  73. {7,1}        {7,1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  74. {7,2}        {7,2,9,1,1,2,3,5,8,1,3,4,7,1,1}
  75. {7,3}        {7,3,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  76. {7,4}        {7,4,1,1,2,3,5,8,1,3,4,7,1,1}
  77. {7,5}        {7,5,1,2,3,5,8,1,3,4,7,1,1,2}
  78. {7,6}        {7,6,1,3,4,7,1,1,2,3,5,8,1,3}
  79. {7,7}        {7,7,1,4,5,9,1,4}
  80. {7,8}        {7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  81. {7,9}        {7,9,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  82. {8,0}        {8,0,8,8,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  83. {8,1}        {8,1,9,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  84. {8,2}        {8,2,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  85. {8,3}        {8,3,1,1,2,3,5,8,1,3,4,7,1,1}
  86. {8,4}        {8,4,1,2,3,5,8,1,3,4,7,1,1,2}
  87. {8,5}        {8,5,1,3,4,7,1,1,2,3,5,8,1,3}
  88. {8,6}        {8,6,1,4,5,9,1,4}
  89. {8,7}        {8,7,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  90. {8,8}        {8,8,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  91. {8,9}        {8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  92. {9,0}        {9,0,9,9,1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  93. {9,1}        {9,1,1,0,1,1,2,3,5,8,1,3,4,7,1,1}
  94. {9,2}        {9,2,1,1,2,3,5,8,1,3,4,7,1,1}
  95. {9,3}        {9,3,1,2,3,5,8,1,3,4,7,1,1,2}
  96. {9,4}        {9,4,1,3,4,7,1,1,2,3,5,8,1,3}
  97. {9,5}        {9,5,1,4,5,9,1,4}
  98. {9,6}        {9,6,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  99. {9,7}        {9,7,1,6,7,1,3,4,7,1,1,2,3,5,8,1,3}
  100. {9,8}        {9,8,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
  101. {9,9}        {9,9,1,8,9,1,7,8,1,5,6,1,1,2,3,5,8,1,3,4,7,1,1}
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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chyanog
 楼主| 发表于 2012-12-13 23:07:58 | 显示全部楼层
不是猜想,是必然的.
总共也就100种情况.可以用程序穷举验证.Flatten;
   NestWhile[(xx[[#[[-2]] + 1, #[[-1]] + 1]] = 1; Join[#, IntegerDigits[Total[Take[#, -2]]]]) &, {i, j}, xx[[#[[-2]] + 1, #[[-1]] +  ...
wayne 发表于 2012-12-13 21:17


厉害！再问个问题，用Join是不是比AppendTo更好呢
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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chyanog
 楼主| 发表于 2012-12-13 23:40:14 | 显示全部楼层
python用reduce(更像Fold)模仿Mathematica的Nest

  2. intDigits=lambda n: map(int, str(n))
  3. print reduce(lambda x,y: x+intDigits(sum(x[-2:])), range(10), [1,1])
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2012-12-14 10:54:54 | 显示全部楼层
AppendTo和Join的差别可以根据在文档中的路径看得出来
Mathematica > Core Language > Defining Variables and Functions > Assignments > AppendTo
Mathematica > Core Language > Expressions > Structural Operations on Expressions > Join
一个是属于赋值性质的,是要修改原值的, 一个是对结构的操作.
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2012-12-14 11:09:13 | 显示全部楼层
6# chyanog

我一定要抽个时间深入学一下Python,这语言挺不错的.
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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郭先抢
发表于 2012-12-14 13:23:05 | 显示全部楼层
这个题目还真有些意思,matlab应该也能干这个活
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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郭先抢
发表于 2012-12-14 13:23:55 | 显示全部楼层
其实也就是双重循环,然后再加上一个子函数
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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