Rational Number Tree

Time Limit: 2000/1000MS (Java/Others) Memory Limit: 128000/64000KB (Java/Others)

Problem Description

Consider an infinite complete binary tree where the root node is 1/1 and left and right childs of node p/q are p/(p+q) and (p+q)/q, respectively. This tree looks like:

         1/1
    ______|______
    |           |
   1/2         2/1
 ___|___     ___|___
 |     |     |     |
1/3   3/2   2/3   3/1
...

It is known that every positive rational number appears exactly once in this tree. A level-order traversal of the tree results in the following array:

1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, ...

Please solve the following two questions:

  1. Find the n-th element of the array, where n starts from 1. For example, for the input 2, the correct output is 1/2.
  2. Given p/q, find its position in the array. As an example, the input 1/2 results in the output 2.

Input

The first line of the input gives the number of test cases, T(1 ≤ T ≤ 100).

T test cases follow. Each test case consists of one line.

The line contains a problem id (1 or 2) and one or two additional integers:

  1. If the problem id is 1, then only one integer n is given, and you are expected to find the n-th element of the array.
  2. If the problem id is 2, then two integers p and q are given, and you are expected to find the position of p/q in the array.

p and q are relatively prime.

1 ≤ n, p, q ≤ 264-1

p/q is an element in a tree with level number ≤ 64.

Output

For each test case:

  1. If the problem id is 1, then output one line containing "Case #x: p q", where x is the case number (starting from 1), and p, q are numerator and denominator of the asked array element, respectively.
  2. If the problem id is 2, then output one line containing "Case #x: n", where x is the case number (starting from 1), and n is the position of the given number.

Sample Input

4
1 2
2 1 2
1 5
2 3 2

Sample Output

Case #1: 1 2
Case #2: 2
Case #3: 3 2
Case #4: 5

Source

codejam

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Information
Solved Number144
Submit Number574
Problem Tags
math
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