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 September 7th, 2013, 12:12 PM #1 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 A binary problem Consider the partition of the positive integers defined as follows: Start with the digit 1. Now take its opposite in binary which is 0, and concatenate the two to create 10. Then take the opposite of 10 which is 01, and concatenate the two to create 1001. Repeat this process indefinitely. If the nth digit of this number is a 1, put n into partition A, and if it is a 0, put n into partition B. Now let $a_k$ represent the kth number (in order from smallest to largest) of partition A, and let $b_k$ represent the kth number of partition B. 1. For certain values of d, it is true that $|(a_k + a_{k+d} + a_{k+2d}) - (b_k + b_{k+d} + b_{k+2d})|$ is equal to 1 for every positive integer value of k. What are these values of d? 2. Determine, given arbitrary values of k and d, which of $a_k + a_{k+d} + a_{k+2d}$ and $b_k + b_{k+d} + b_{k+2d}$ is larger.

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