Prove whether or not any Fibonacci number, F(n), can be constructed by using
F(n) F(n-1)'s and the operators +,-,*,/
(Provided that F(n-1)>0 )
For example I can make a 5 using 5 3's: 3 + 3/3 + 3/3

The following solution assumes that brackets may be used with the given operators:

Where F(n-1) > 1, F(n-1) added to F(n-1)/F(n-1) iterated [F(n) - F(n-1)] times should produce the value F(n).
Remaining will be either an odd or even number of F(n-1)'s. If even, the number of F(n-1)'s remaining can be paired as either zeroes {[F(n-1) - F(n-1)]} to be added to the total, or ones
{F(n-1)/F(n-1)} to be multiplied to the total. If odd, the quanity of F(n-1)'s remaining will be >=5 [except for F(n-1)=5]. In these odd cases, one can multiply the odd number of the "fibs"
{[F(n-1)+F(n-1)]/F(n-1) - [F(n-1)/F(n-1)]} to the total. For the special case where F(n-1) = 5 {F(n) = 8}, i.e., 8 5's, there is the solution 8 = (5*5+5)/5 + 5/5 + 5/5.

Posted by Dej Mar on 2010-07-10 10:42:21