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Binary Circles

Problem 265

Published on Saturday, 21st November 2009, 09:00 am; Solved by 2851; Difficulty rating: 40%

2N binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct.

For N=3, two such circular arrangements are possible, ignoring rotations:


For the first arrangement, the 3-digit subsequences, in clockwise order, are:
000, 001, 010, 101, 011, 111, 110 and 100.

Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros as the most significant bits and proceeding clockwise. The two arrangements for N=3 are thus represented as 23 and 29:

00010111 2 = 23
00011101 2 = 29

Calling S(N) the sum of the unique numeric representations, we can see that S(3) = 23 + 29 = 52.

Find S(5).