EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2018
ICPC 2018

I. Triangles

For your trip to Beijing, you have brought plenty of puzzle books, many of them containing challenges like the following: how many triangles can be found in Figure I.1? Figure I.1: Illustration of Sample Input 2. While these puzzles keep your interest for a...

Updated May 21, 2026
Track ICPC
Year 2018
Statement Text + PDF
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Problem statement

Problem Statement

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

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Time limit 6 seconds

For your trip to Beijing, you have brought plenty of puzzle books, many of them containing challenges like the following: how many triangles can be found in Figure I.1?

Figure I.1: Illustration of Sample Input 2.

While these puzzles keep your interest for a while, you quickly get bored with them and instead start thinking about how you might solve them algorithmically. Who knows, maybe a problem like that will actually be used in this year’s contest. Well, guess what? Today is your lucky day!

Input

The first line of input contains two integers r and c (1 ≤ r ≤ 3 000, 1 ≤ c ≤ 6 000), specifying the picture size, where r is the number of rows of vertices and c is the number of columns. Following this are 2r − 1 lines, each of them having at most 2c − 1 characters. Odd lines contain grid vertices (represented as lowercase x characters) and zero or more horizontal edges, while even lines contain zero or more diagonal edges. Specifically, picture lines with numbers 4k + 1 have vertices in positions 1, 5, 9, 13, . . . while lines with numbers 4k + 3 have vertices in positions 3, 7, 11, 15, . . . . All possible vertices are represented in the input (for example, see how Figure I.1 is represented in Sample Input 2). Horizontal edges connecting neighboring vertices are represented by three dashes. Diagonal edges are represented by a single forward slash (‘/’) or backslash (‘\’) character. The edge characters will be placed exactly between the corresponding vertices. All other characters will be space characters. Note that if any input line could contain trailing whitespace, that whitespace may be omitted.

Output

Display the number of triangles (of any size) formed by grid edges in the input picture.

Sample Tests

Sample 1
Sample Input 
 3 3
 x---x
  \ /
   x
  / \
 x    x
Sample Output 
1
Sample 2
Sample Input 
4 10
x    x---x---x     x
      \ /   / \
  x    x---x    x    x
      / \ / \    \
x    x---x---x---x
   /    / \   \ / \
  x---x---x---x---x
Sample Output 
12
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/2018/I-triangles/solution.texC++ implementationcompetitive_programming/icpc/2018/I-triangles/solution.cppStatement textcompetitive_programming/icpc/2018/I-triangles/statement.txtStatement PDFcompetitive_programming/icpc/2018/I-triangles/statement.pdfMetadatacompetitive_programming/icpc/2018/I-triangles/meta.jsonYear packetcompetitive_programming/icpc/2018/contest_problems.pdf