yaworsw/euler-manager

---
:id: 316
:name: Numbers in decimal expansions
:url: https://projecteuler.net/problem=316
:content: "Let <var>p</var> = <var>p<sub><font size=\"-2\">1</font></sub> p<sub><font
  size=\"-2\">2</font></sub> p<sub><font size=\"-2\">3</font></sub></var> ... be an
  infinite sequence of random digits, selected from {0,1,2,3,4,5,6,7,8,9} with equal
  probability.  \nIt can be seen that <var>p</var> corresponds to the real number
  0.<var>p<sub><font size=\"-2\">1</font></sub> p<sub><font size=\"-2\">2</font></sub>
  p<sub><font size=\"-2\">3</font></sub></var> ....   \nIt can also be seen that choosing
  a random real number from the interval [0,1) is equivalent to choosing an infinite
  sequence of random digits selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.\n\nFor
  any positive integer <var>n</var> with <var>d</var> decimal digits, let <var>k</var>
  be the smallest index such that   \n<var>p<sub><small>k</small>, </sub></var><var>p<sub><small>k+1</small></sub></var>,
  ...<var>p<sub><small>k+d-1</small></sub></var> are the decimal digits of <var>n</var>,
  in the same order.  \nAlso, let <var>g</var>(<var>n</var>) be the expected value
  of <var>k</var>; it can be proven that <var>g</var>(<var>n</var>) is always finite
  and, interestingly, always an integer number.\n\nFor example, if <var>n</var> =
  535, then  \nfor <var>p</var> = 31415926 **535** 897...., we get <var>k</var> =
  9  \nfor <var>p</var> = 35528714365004956000049084876408468 **535** 4..., we get
  <var>k</var> = 36  \netc and we find that <var>g</var>(535) = 1008.\n\nGiven that
  ![p316_decexp1.gif]({{ images_dir }}/p316_decexp1.gif), find ![p316_decexp2.gif]({{
  images_dir }}/p316_decexp2.gif)\n\n<u><i>Note</i></u>: ![p316_decexp3.gif]({{ images_dir
  }}/p316_decexp3.gif) represents the floor function.\n"