ICPC 2015 - J. Tile Cutting
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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competitive_programming/icpc/2015/J-tile-cutting/solution.tex to update the written solution and
competitive_programming/icpc/2015/J-tile-cutting/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem J
Tile Cutting
Time limit: 15 seconds
Youssef is a Moroccan tile installer who specializes in mosaics like the one
shown on the right. He has rectangular tiles of many dimensions at his
disposal, and the dimensions of all his tiles are integer numbers of centime-
ters. When Youssef needs parallelogram-shaped tiles, he cuts them from
his supply on hand. To make this work easier, he invented a tile cutting
machine that superimposes a centimeter grid on the cutting surface to guide
the cuts on the tiles. Due to machine limitations, aesthetic sensibilities, and
Youssef’s dislike of wasted tiles, the following rules determine the possible
cuts.
1. The rectangular tile to be cut must be positioned in the bottom left corner of the cutting surface
and the edges must be aligned with the grid lines.
2. The cutting blade can cut along any line connecting two different grid points on the tile boundary
as long as the points are on adjacent boundary edges.
3. The four corners of the resulting parallelogram tile must lie on the four sides of the original
rectangular tile.
4. No edge of the parallelogram tile can lie along an edge of the rectangular tile.
Figure J.1 shows the eight different ways in which a parallelogram tile of area 4 square centimeters can
be cut out of a rectangular tile, subject to these restrictions.
Figure J.1: The eight different ways for cutting a parallelogram of area 4.
Youssef needs to cut tiles of every area between alo and ahi . Now he wonders, for which area a in this
range can he cut the maximum number of different tiles?
Input
The input consists of multiple test cases. The first line of input contains an integer n (1 ≤ n ≤ 500), the
number of test cases. The next n lines each contain two integers alo , ahi (1 ≤ alo ≤ ahi ≤ 500 000), the
range of areas of the tiles.
Output
For each test case alo , ahi , display the value a between alo and ahi such that the number of possible ways
to cut a parallelogram of area a is maximized as well as the number of different ways w in which such
a parallelogram can be cut. If there are multiple possible values of a display the smallest one.
Sample Input 1 Sample Output 1
2 4 8
4 4 6 20
2 6
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
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
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
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.
Code
Exact copied C++ implementation from solution.cpp.
#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;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2015\\J. Tile Cutting}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#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;
}