It’s been a couple of days since I finished Problem 1 of Project Euler and I did in fact finish Problem 2 the very next day. Unfortuantely, I must admit that I have been making some very basic maths quite complicated in the hopes of understanding how some people have solved Problem 2.
Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Find the sum of all the even-valued terms in the sequence which do not exceed four million.
At first this seems like an insurmountable task because four million is quite a large number and common sense leads us to believe that there will be hundreds of numbers in the Fibonacci sequence below four million. Fortunately this is a paradox as there are only a mear 33 numbers below four million in the Fibonacci sequence.
I spent quite some time looking for a formulae that would let me work out all the even numbers in the fibonacci sequence (i.e. even numbers are always three terms apart, so F(3n) is always an even number). Not being a mathematician and not having done any kind of maths for several years I gave up (I also do not have that much time on my hands) and found a list of Fibonacci numbers and used a calculator to add up all the numbers below four million.
After getting the correct answer I did however look in to other peoples solutions, and there are some quite facinating solutions indeed. It’s made me learn about the Golden Ratio and how it links to the Fibonacci sequence, though I have not delved too far, see previous comment about not being a mathematician.
I’m going to very briefly go over one of the properties of the Fibonacci sequence and how it can be used to find the answer to this problem.
Something can be said to have the Golden Ratio if the ratio between two things is the same as the ratio between the larger of the two things and the sumation of the two things. Given that B is greater than A, B and A are in the Golden Ratio when:
A:B = B:(A+B)
This links to the Fibonacci sequence because the ratio between two consecutive terms in the Fibonacci sequence is close to the Golden Ratio (it is more complicated than this, but for our purposes this is all we need to know). I’m going to express Fibonacci numbers as Fib(n) where n is the term, so F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2… F(10) = 55.
The nth term of the Fibonacci sequence can be expressed as so:
Fib(n) = Fib(n-1) + Fib(n-2)
And so the Ratio between two consecutive terms n and n-1 is as phi.
Fib(n)/Fib(n-1) = Φ
Eventually we want the ratio between three terms, because we know that even numbers are three terms apart.
Fib(n)/Fib(n-3) = ?
We can immediately infer that the ratio is going to be 3Φ because we know that the ratio between any Fibonacci number is going to be Φ, but for sake of argument I’ll show you what took me so long to work my head around. The ratio between three Fibonacci numbers is given by the following:
- ( Fib(n) / (Fib(n-1) ) / ( Fib(n-2) / Fib(n-1) )
Now this is the point where I got confused, how can the first part be transformed into the second part? I tried many different ways, thinking it was a ratio of ratios, or an inverse ratio of the ratio or some crazy things like that and finally someone on the irc channel #maths helped me out with the simple answer of “that’s how fractions work”. If you Divide top and bottom by Fib(n-1) then you get part 2. The use of this is because we know that Fib(n)/Fib(n-1) is Φ, and because Fib(n-2)/Fib(n-1) is basically the inverse of Fib(n)/Fib(n-1) it is 1/Φ. This gives us the last two steps which are a replacement
- (Φ)/(1/Φ) : Replace Fib(n)/Fib(n-1) and Fib(n-2/Fib(n-1) with Φ and 1/Φ respectively
- 2Φ : x/(1/Φ) can also be written as x * Φ.
Finally, we want to know what Fib(n)/Fib(n-3) is, which can also be written like this:
( Fib(n) / Fib(n-2) )/ ( Fib(n-3) / Fib(n-2) )
- 2Φ / (1 / Φ) : Fib(n) / Fib(n-2) = 2Φ (From earlier) and Fib(n-3) / Fib(n-2) is 1/Φ (From earlier).
- 3Φ : As earlier, x/(1/Φ) = x*Φ.
And we’re done, we now know that even Fibonacci numbers are a ratio of 3Φ apart and use a calculator to work out the sum of all the even terms under four million.
There is one other solution that I don’t really understand that I might put some work into understanding, but I will have to see how much time I have, before that…
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?