# Euler Solution 73

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# Solutions for Problem 73

**NOTE:** Another easter egg for the next person who solves it (no links/updates on the main page).

How many fractions are there of the form *n*/*d* (where *n* and *d* are positive integers with no common divisors) between 1/3 and 1/2 for *d* <= 12000?

Note: Limit was increased lately from 10,000 to 12,000. (Thomas's won't work now, or will at least need it's limit adjusted)

## Haskell by SanguineV

Runtime: unknown

{- Return all the fractions i in the range 1/3 < i < 1/2 for some divisor "n" - Note: results are sorted by default. -} fracs :: Double -> [Double] fracs n = map (\x -> x/n) (takeWhile (< n/2) (dropWhile (<= n/3) [1..])) {- Combine two sorted lists of doubles (can also use type: Eq a => [a] -> [a] -> [a]) - returning a sorted single list without duplicates. -} comb :: [Double] -> [Double] -> [Double] comb [] ys = ys comb xs [] = xs comb (x:xs) (y:ys) | x == y = [x] ++ (comb xs ys) comb (x:xs) (y:ys) | x > y = [y] ++ (comb (x:xs) ys) comb (x:xs) (y:ys) = [x] ++ (comb xs (y:ys)) {- Combine all the lists together for divisor 5..10000 and print the length -} main = print (length (foldl (\l n -> comb l (fracs n)) [] [5..10000]))

This is a slow brute force approach, can be done a lot faster!

## Python by Althalus

Runtime: 27.7 seconds. Using a (slightly modified) Stern-Brocot tree this time. Much faster, since we can instruct the tree to only give us the fractions between 1/2 and 1/3 by using those fractions as the starting point (instead of the original 0/1, 1/0)

import sys sys.setrecursionlimit(13001) def sternbrocot_count(depth,lower,upper,n1=1,d1=0,n2=0,d2=1,curdepth=1): n = n1+n2 d = d1+d2 if n > d or d > depth: return 0 if (float(n)/d < upper) and (float(n)/d > lower): res = 1 else: res = 0 if curdepth < depth: res += sternbrocot_count(depth,lower,upper,n,d,n2,d2,curdepth+1) res += sternbrocot_count(depth,lower,upper,n1,d1,n,d,curdepth+1) return res print sternbrocot_count(12000, 1.0/3, 1.0/2, 1,3, 1,2)

Original runtime: 496.6 seconds. Horrible.

third = 1.0/3 half = 1.0/2 fractions = {} for d in range(1,12001): if d % 100 == 0: print d for n in range(1,d): tmp = float(n)/d if tmp > third and tmp < half: f = gcd(n,d) fractions[(n/f,d/f)] = 1 print len(fractions.keys())

## Scheme by Althalus

Runtime: 1.03s

Essentially the same logic as my python code above, in a language I don't really know. I'm surprised I got it to work at all.

#lang scheme (define (stern-brocot depth n1 d1 n2 d2 curdepth) (define n (+ n1 n2)) (define d (+ d1 d2)) (cond [(or (> n d) (> d depth)) 0] [else (cond [(<= curdepth depth) (+ 1 (+ (stern-brocot depth n d n2 d2 (+ curdepth 1)) (stern-brocot depth n1 d1 n d (+ curdepth 1))))] [else 1]) ]) ) (stern-brocot 12000 1 3 1 2 1)