EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2009
ICPC 2009

C. The Return of Carl

Input file: carl.in Carl the ant is back! When we last left him (Problem A, 2004 World Finals), Carl was a little mixed-up, always taking strange, zigzag paths when traveling. But now, Carl has straightened out his life – literally. He now always takes the...

Updated May 21, 2026
Track ICPC
Year 2009
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

Statement text Statement PDF
Input file carl.in

Carl the ant is back! When we last left him (Problem A, 2004 World Finals), Carl was a little mixed-up, always taking strange, zigzag paths when traveling. But now, Carl has straightened out his life – literally. He now always takes the straightest, shortest path between any pair of points. This sounds simple, except for one small thing: Carl now spends most of his time on a paperweight in the shape of a regular octahedron. You may recall that an octahedron is one of the five Platonic solids and consists of eight equilateral triangles, as shown in Figure 3.

Figure 3: Regular octahedron

Carl has an innate (some say in-ant) ability to always take the shortest path when going from any starting point to any destination point on the paperweight. Your job is to verify this by determining the length of the shortest path when given two such points, not necessarily distinct.

Input

The input contains multiple test cases. Each test case consists of four integers θ1, φ1, θ2 and φ2, where 0 ≤ θi < 360 and 0 ≤ φi ≤ 180. The first two are the spherical coordinates of the start point, and the last two are the spherical coordinates of the destination point. As shown in Figure 3, θi is the azimuth and φi is the zenith angle, both of them given in degrees.

The input is terminated by a line containing four negative ones.

The paperweight is fixed for all test cases as follows: the octahedron is centered on the origin and each vertex lies on one of the axes. Every edge is exactly 10 cm long. You should suppose that Carl’s size is zero and ignore any supporting mechanisms that may be necessary to hold the paperweight in the correct position.

Output

For each test case, print the case number (starting with 1) and the length in centimeters of the shortest path from the start point to the destination point, rounded to the nearest thousandth. Follow the format of the sample output.

Sample Tests

Sample
Sample Input 
 0 90 90 90
 0 90 90 45
 0 0 0 180
 -1 -1 -1 -1
Sample Output 
Case 1: 10.000
Case 2: 8.660
Case 3: 17.321
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

Key Observations

  • Write the structural observations that make the problem tractable.

  • State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.

  • If the constraints matter, explain exactly which part of the solution they enable.

Algorithm

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. Mention corner cases explicitly if they affect the implementation.

Correctness Proof

We prove that the algorithm returns the correct answer.

Lemma 1.

State the first key claim.

Proof.

Provide a concise proof.

Lemma 2.

State the next claim if needed.

Proof.

Provide a concise proof.

Theorem.

The algorithm outputs the correct answer for every valid input.

Proof.

Combine the lemmas and finish the argument.

Complexity Analysis

State the running time and memory usage in terms of the input size.

Implementation Notes

  • Mention any non-obvious implementation detail that is easy to get wrong.

  • Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#include <bits/stdc++.h>
using namespace std;

namespace {

void solve() {
    // Fill in the full solution logic for the problem here.
}

}  // namespace

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    solve();
    return 0;
}
Resources

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TeX write-upcompetitive_programming/icpc/2009/C-the-return-of-carl/solution.texC++ implementationcompetitive_programming/icpc/2009/C-the-return-of-carl/solution.cppStatement textcompetitive_programming/icpc/2009/C-the-return-of-carl/statement.txtStatement PDFcompetitive_programming/icpc/2009/C-the-return-of-carl/statement.pdfMetadatacompetitive_programming/icpc/2009/C-the-return-of-carl/meta.jsonYear packetcompetitive_programming/icpc/2009/contest_problems.pdf