Determine which two fractions are adjacent to 34/87 in this listing.
Is it possible to generalize this to finding the fractions adjacent to a/b?
Source: Modified from Problem 2/3/12 at the USA Mathematical Talent Search.
First the solutions:
45/86 < 11/21 < 43/82
25/64 < 34/87 < 9/23
Stephanie Higueret's solution is most straightforward:
The irreducible fractions between 0 and 1, with denominators at most 99, are listed in ascending order is the Farey series (generally Stern Brocot tree). Three adjacent fractions c/d, a/b, e/f followed the properties : ad - bc = 1 be - af = 1 ((c+e)/(d+f)=a/b, a/b is the median fraction between c/d and e/f) So with a/b = 11/21 11d - 21c = 1 gives c/d = 45/86 21e - 11f = 1 gives e/f = 43/82 and a/b = 34/87 34d - 87c = 1 gives c/d = 25/64 87e - 34f = 1 gives e/f = 9/23 I forget to precise that they could exist many solutions for ad-bc = 1. It was obvious to select the one's with the highest denominator. In case of a/b = 34/87 there's only one solution, but for a/b = 11/21 if we search c/d we found 5 solutions 1/2, 12/23, 23/44, 34/65 and 45/86 of course only the last one matched the criteria.