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Same (Sum of) Differences - Problem-A-Day

joshua_greenSep. 20th, 2009 12:39 pm Same (Sum of) Differences

For nN, suppose that the set S ≡ {1, 2, 3, ..., 2n} is partitioned into two sets A and B of size n each.  (That is, AB = S and |A| = n = |B|.)  Say A = {ai}in= 1 and B = {bi}in= 1 with a1 < a2 < a3 < ... < an and b1 < b2 < b3 < ... < bn.  Prove that Σin= 1 |ai - bn-i+1| = n2.

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From:the_s3ntinel
Date:December 5th, 2010 05:10 pm (UTC)
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Nice puzzle - pity this community seems to be dead now. Can't resist answering (I'm like in that xkcd cartoon - someone who can be immediately derailed by a mathematical puzzle, regardless of circumstances.)

The sum in question equals n2 when A = {1,...,n}. And the sum is unchanged whenever two adjacent numbers, one in A and the other in B, are 'flipped over'.