# Problem 253: Tidying up
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. Every 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. As the caterpillar is built up in this
way, it forms distinct segments that gradually merge together. 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). For example: Piece
Placed Segments So Far121422936434554354…… Let M be the maximum number
of segments encountered during a random tidy-up of the caterpillar. For
a caterpillar of ten pieces, the number of possibilities for each M is M
Possibilities1512      2250912      31815264      41418112      5144000 
     so the most likely value of M is 3 and the average value is
385643⁄113400 = 3.400732, rounded to six decimal places. The most likely
value of M for a forty-piece caterpillar is 11; but what is the average
value of M? Give your answer rounded to six decimal places.