Russell's Fibonacci coin bags

Algebra Level 5

Russell has \( 2013 \) bags of coins to give away to two friends, Andrew and Brian. Bag \( i \) has \( F_i \) coins, where \( F_i \) is the \( i \)th term of the Fibonacci sequence, and each bag must be given to either Andrew or Brian. If there are \(N\) ways to distribute all of the bags so that Andrew and Brian get the same number of coins, what is \(\log_2 N\)?

This problem is shared by Russell F. from NIMO Summer Contest 2012.

Details and assumptions

The Fibonacci sequence is defined by \(F_1 = 1, F_2 = 1\) and \( F_{n+2} = F_{n+1} + F_{n}\) for \( n \geq 1 \).

Disregard physical impossibilities for the number of coins.


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