The Stern-Brocot Number System

The Stern-Brocot tree is a beautiful way for constructing the set of all nonnegative fractions m where m 01 , 01 For example, the first step gives us one new entry between 01 and 10, and n are relatively prime. The idea is to start with two fractions operations as many times as desired: and then repeat the following ′′ Insert m+m between two adjacent fractions m and m . ′′ ()n n+n nn and the next gives two more: The next gives four more, 01 , 1 1 , 10 ; 01, 12, 1, 21, 10. 01, 31, 12, 23, 1, 32, 21, 31, 10; and then we will get 8, 16, and so on. The entire array can be regarded as an infinite binary tree structure whose top levels look like this: The construction preserves order, and we couldn’t possibly get the same fraction in two different places. We can, in fact, regard the Stern-Brocot tree as a number system for representing rational numbers, because each positive, reduced fraction occurs exactly once. Let’s use the letters ‘L’ and ‘R’ to stand for going down to the left or right branch as we proceed from the root of the tree to a particular fraction; then a string of L’s and R’s uniquely identifies a place in the tree. For example, LRRL means that we go left from 1 down to 12, then right to 32, then right to 34, then left to 57. We can consider LRRL to be a representation of 57. Every positive fraction gets represented in this way as a unique string of L’s and R’s.

2/2 Well, actually there’s a slight problem: The fraction 1 corresponds to the empty string, and we need a notation for that. Let’s agree to call it I, because that looks something like 1 and it stands for “identity”. In this problem, given a positive rational fraction, you are expected to represent it in Stern-Brocot number system. Input The input file contains multiple test cases. Each test case consists of a line contains two positive integers m and n where m and n are relatively prime. The input terminates with a test case containing two 1’s for m and n, and this case must not be processed. Output For each test case in the input file output a line containing the representation of the given fraction in the Stern-Brocot number system. Sample Input 57 878 323 11 Sample Output LRRL RRLRRLRLLLLRLRRR