NWERC11B - Bird tree

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Bird tree

The Bird tree1 is an infinite binary tree, whose first 5 levels look as follows:

                                   1                                    ∕1                     1∕2                                  2∕1          2∕3                1∕3                3∕1                3∕2    3        3         1        2        5         4        4        5    ∕5       ∕4       ∕4       ∕5        ∕2       ∕1       ∕3        ∕3 5   4    4    5   2    1    3   3    8    7   5    7   7    5    7   8 ∕8   ∕7   ∕5  ∕7   ∕7   ∕5  ∕7   ∕8  ∕3   ∕3   ∕1  ∕2   ∕5   ∕4  ∕4   ∕5

It can be defined as follows:

bird =             1∕1          1∕ (bird + 1)  (1∕bird) + 1

This is a co-recursive definition in which both occurrences of bird refer to the full (infinite) tree. The expression bird + 1 means that 1 is added to every fraction in the tree, and 1∕bird means that every fraction in the tree is inverted (so ab becomes ba).

Surprisingly, the tree contains every positive rational number exactly once, so every reduced fraction is at a unique place in the tree. Hence, we can also describe a rational number by giving directions (L for left subtree, R for right subtree) in the Bird tree. For example, 25 is represented by LRR. Given a reduced fraction, return a string consisting of L’s and R’s: the directions to locate this fraction from the top of the tree.

 

Input

On the first line a positive integer: the number of test cases, at most 100. After that per test case:

  • one line with two integers a and b (1 ≤ a,b ≤ 109), separated by a ’/’. These represent the numerator and denominator of a reduced fraction. The integers a and b are not both equal to 1, and they satisfy gcd(a,b) = 1.

For every test case the length of the string with directions will be at most 10 000.

 

Output

Per test case:

  • one line with the string representation of the location of this fraction in the Bird tree.

 

Sample in- and output

Input

Output

3
1/2
2/5
7/3
L
LRR
RLLR

1Hinze, R. (2009). The Bird tree. J. Funct. Program., 19:491–508.

Copyright notice

This problem text is copyright by the NWERC 2011 jury. It is licensed under the Creative Commons Attribution-Share Alike license version 3.0; The complete license text can be found at: http://creativecommons.org/licenses/by-sa/3.0/legalcode


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kelvin_chui: 2015-12-05 16:19:27

Is there any special case needed to check? I got correct results for all numbers in the given diagram, but wrong answer when I submitted it.

miodziu: 2015-07-24 10:05:05

hint: https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

soursugar: 2015-05-25 09:45:48

the figure doesn't seem to follow the recursive defn. e.g.: 1/((1/3)+1) is 3/4.
follow the figure or the recursive defn?

Bhavik: 2014-01-11 18:58:30

very good problem

Agnisnato Datta: 2013-06-07 05:43:23

nice prob...try to concentrate on fig

Paul Draper: 2012-11-20 02:11:43

Nice one, Jeroen! Thanks for the visual.

rohitjv: 2012-07-23 03:46:55

yeah...took a lot of time to understand

Ajey Golsangi: 2012-02-25 09:56:17

This one took a lot of time.

Santiago Palacio: 2012-01-12 18:56:45

Agree with bond, understanding it was the most difficult thing.

Devil D: 2011-12-31 11:44:57

At Last got it


Added by:Jeroen Bransen
Date:2011-11-02
Time limit:0.448s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ASM64
Resource:NWERC 2011 Jury