yaworsw/euler-manager

---
:id: 253
:name: Tidying up
:url: https://projecteuler.net/problem=253
:content: "A small child has a “number caterpillar” consisting of forty jigsaw pieces,
  each with one number on it, which, when connected together in a line, reveal the
  numbers 1 to 40 in order.\n\nEvery night, the child's father has to pick up the
  pieces of the caterpillar that have been scattered across the play room. He picks
  up the pieces at random and places them in the correct order.  \n As the caterpillar
  is built up in this way, it forms distinct segments that gradually merge together.
  \ \n The number of segments starts at zero (no pieces placed), generally increases
  up to about eleven or twelve, then tends to drop again before finishing at a single
  segment (all pieces placed).\n\nFor example:\n\n| **Piece Placed** | **Segments
  So Far** |\n| 12 | 1 |\n| 4 | 2 |\n| 29 | 3 |\n| 6 | 4 |\n| 34 | 5 |\n| 5 | 4 |\n|
  35 | 4 |\n| … | … |\n\nLet <var>M</var> be the maximum number of segments encountered
  during a random tidy-up of the caterpillar.  \nFor a caterpillar of ten pieces,
  the number of possibilities for each <var>M</var> is\n\n| **<var>M</var>** | **Possibilities**
  |\n| 1 | 512&nbsp; &nbsp;&nbsp; &nbsp; |\n| 2 | 250912&nbsp; &nbsp;&nbsp; &nbsp;
  |\n| 3 | 1815264&nbsp; &nbsp;&nbsp; &nbsp; |\n| 4 | 1418112&nbsp; &nbsp;&nbsp; &nbsp;
  |\n| 5 | 144000&nbsp; &nbsp;&nbsp; &nbsp; |\n\nso the most likely value of <var>M</var>
  is 3 and the average value is <sup>385643</sup>⁄<sub>113400</sub> = 3.400732, rounded
  to six decimal places.\n\nThe most likely value of <var>M</var> for a forty-piece
  caterpillar is 11; but what is the average value of <var>M</var>?\n\nGive your answer
  rounded to six decimal places.\n\n"