data/problems/314.yml
---
:id: 314
:name: The Mouse on the Moon
:url: https://projecteuler.net/problem=314
:content: "The moon has been opened up, and land can be obtained for free, but there
is a catch. You have to build a wall around the land that you stake out, and building
a wall on the moon is expensive. Every country has been allotted a 500 m by 500
m square area, but they will possess only that area which they wall in. 251001 posts
have been placed in a rectangular grid with 1 meter spacing. The wall must be a
closed series of straight lines, each line running from post to post.\n\nThe bigger
countries of course have built a 2000 m wall enclosing the entire 250 000 m<sup>2</sup>
area. The [Duchy of Grand Fenwick](http://en.wikipedia.org/wiki/Grand_Fenwick),
has a tighter budget, and has asked you (their Royal Programmer) to compute what
shape would get best maximum enclosed-area/wall-length ratio.\n\nYou have done some
preliminary calculations on a sheet of paper. For a 2000 meter wall enclosing the
250 000 m<sup>2</sup> area the enclosed-area/wall-length ratio is 125. \nAlthough
not allowed , but to get an idea if this is anything better: if you place a circle
inside the square area touching the four sides the area will be equal to π\\*250<sup>2</sup>
m<sup>2</sup> and the perimeter will be π\\*500 m, so the enclosed-area/wall-length
ratio will also be 125.\n\nHowever, if you cut off from the square four triangles
with sides 75 m, 75 m and 75√2 m the total area becomes 238750 m<sup>2</sup> and
the perimeter becomes 1400+300√2 m. So this gives an enclosed-area/wall-length ratio
of 130.87, which is significantly better.\n\n \n\nFind the maximum enclosed-area/wall-length ratio. \nGive
your answer rounded to 8 places behind the decimal point in the form abc.defghijk.\n\n"