## Learning Scheme, Euler problem 2

The second project Euler problem is:

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.

Brute force Fibonacci generator:

``````(define fibo
(lambda (n)
(cond
((= n 1)
1)
((= n 2)
2)
(else (+ (fibo (- n 1)) (fibo (- n 2)))))))``````

But that takes exponential time, and the problem requires generating all of the numbers anyway, so we can use something like:

``````(define fibo-sum
(lambda (a b lim)
(let* ((c (+ a b)))
(cond
((< lim c) 0)
(else (+ c (fibo-sum b c lim)))))))``````

to get the sum of all the fibonacci numbers less than lim (use `(fibo-sum 1 2 4000000)`). I could add a test for even numbers to solve the problem, but more elegantly we notice that the pattern of the fibonacci numbers has to be odd-odd-even-odd-odd-even, so we have:

``````(define euler2
(lambda (a b lim)
(let* ((c (+ a b))
(d (+ b c))
(e (+ c d)))
(cond
((< lim e) 0)
(else (+ e (euler2 d e lim)))))))
(euler2 1 0 4000000)``````

We start the fibonacci sequence with 1 0 so that the 2 is part of our sum. And we get the correct answer, 4613732, imperceptibly fast.

One bug that took me forever to find was a mistyped line:

``````(let* ((c (+ a b))
(d (+ b c))
(e (+ d e)))
``````

Where `e` was defined in terms of `d` and `e`, so it used the global value of `e`. If it hadn't been defined, it would have given me an error immediately, but unfortunately, DrScheme has it pre-defined as `#i2.718281828459045` so my code went happily along giving me the wrong answers.