The first 2n positive integers are arbitrarily divided into two groups of n numbers each. The numbers in the first group are sorted in ascending order: a1 < a2 < ... < an; the numbers in the second group are sorted in descending order: b1 > b2 > ... > bn.
Find, with proof, the value of the sum |a1 − b1| + |a2 − b2| + ... + |an − bn|.
The table below is generated at random, in two stages. Firstly, a value of n in the range 2..20 is chosen at random. Then, n numbers in the range 1..2n are chosen at random. These numbers are sorted to form the sequence ai; the remaining numbers are sorted to form the sequence bi. For each i, the value |ai − bi| is calculated, and the sum is given.
Here are some generated random table . Use the tables to formulate a conjecture about each pair, (ai, bi.) Prove your conjecture, and hence find the value of the sum.
Edited by BMAD, 09 April 2013 - 05:45 PM.