ICPC 2018 - J. Uncrossed Knight’s Tour
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/2018/J-uncrossed-knights-tour/solution.tex to update the written solution and
competitive_programming/icpc/2018/J-uncrossed-knights-tour/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
Uncrossed Knight’s Tour
Time limit: 2 seconds
A well-known puzzle is to “tour” all the squares of an 8 × 8 chessboard using a knight, which is a piece
that can move only by jumping one square in one direction and two squares in an orthogonal direction.
The knight must visit every square of the chessboard, without repeats, and then return to its starting
square. There are many ways to do this, and the chessboard size is manageable, so it is a reasonable
puzzle for a human to solve.
However, you have access to a computer, and some coding skills! So, we will give you a harder version
of this problem on a rectangular m × n chessboard with an additional constraint: the knight may never
cross its own path. If you imagine its path consisting of straight line segments connecting the centers of
squares it jumps between, these segments must form a simple polygon; that is, no two segments intersect
or touch, except that consecutive segments touch at their common end point. This constraint makes it
impossible to visit every square, so instead you must maximize the number of squares the knight visits.
We keep the constraint that the knight must return to its starting square. Figure J.1 shows an optimal
solution for the first sample input, a 6 × 6 board.
Figure J.1: An optimal solution for a 6 × 6 board.
Input
The input consists of a single line containing two integers m (1 ≤ m ≤ 8) and n (1 ≤ n ≤ 1015 ), giving
the dimensions of the rectangular chessboard.
Output
Display the largest number of squares that a knight can visit in a tour on an m × n chessboard that does
not cross its path. If no such tour exists, display 0.
Sample Input 1 Sample Output 1
6 6 12
Sample Input 2 Sample Output 2
8 3 6
Sample Input 3 Sample Output 3
7 20 80
Sample Input 4 Sample Output 4
2 6 0
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 2018\\J. Uncrossed Knight’s Tour}
\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;
}