Sum of differences

[Susam Pal]

For a given even positive integer, we create two lists of
integers such that every positive integer less than or equal to
the given integer belongs to exactly one of the two lists and
both lists contain an equal number of integers. There is a third
list which is initially empty. Then, at every step, we remove
the smallest number from the first list, the largest number from
the second list and insert the positive difference between the
two numbers into the third list.

What do we get if we divide the square of the given even
positive integer by the sum of all the numbers in the third list
when the first list becomes empty?

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Originally posted at: http://cotpi.com/p/37/
Solutions to 'Splitting square into squares': http://cotpi.com/p/36/#36-1