Use this tag to describe mathematics questions dealing with the knight's tour problem. A knight's tour is a series of legal knight moves in chess that visits all of the squares on the board exactly once.
Questions tagged [knight-tours]
43 questions
28
votes
2 answers
A variant of the Knight's tour problem
The knight's tour problem is a famous problem in chess and computer science which asks the following question: can we move the knight on an $n \ \times \ n$ chessboard such that it visits every square exactly once? The answer is yes iff $n\geq5$.…
26
votes
3 answers
How many knight's tours are there?
The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from…
Charles Stewart
- 4,697
8
votes
1 answer
Tour of chess king
Consider lame chess king that can move only one cell left, down and diagonal upright. Consider square chess board.
Question: Can such a king visit all cells of a board (each cell only once) and end up in the cell to the left of the original?
I've…
Grigori
- 159
7
votes
1 answer
Is there a name for graphs with the following property?
The property of the graph is the following: For any vertex, there is a hamiltonian
path starting with this vertex, but the graph is not hamiltonian. The following graph is a small example:
Important examples are hypohamiltonian graphs (deleting…
Peter
- 86,576
7
votes
1 answer
Infinite Knight's Tour
Does there exists a one-to-one and onto map $f:\mathbb{N}\rightarrow \mathbb{Z}[i]$ such that $|f(n+1)-f(n)|=\sqrt{5}$?
That is to get from $f(n)$ to $f(n+1)$ you have to move like a knight in chess: move two vertically and then one horizontally or…
Mason
- 4,489
6
votes
1 answer
Proof that there is no closed knight tour on a $3\ \times\ 8$ - board
I want to prove that there is no closed knight tour on a $3\ \times\ 8$ - board by deleting $s$ vertices
of the corresponding knight graph to get a graph with more than $s$ connected
components (which would prove that it is not hamiltonian), but I…
Peter
- 86,576
6
votes
4 answers
Chess knight move in 8x8 chessboard
I've recently shown some interests in chess, and I wonder if there is a solution for the following problem:
In a 8x8 chessboard, labeling the cells with numbers from 1 to 8, is there any way to find the minimum moves that a knight should do to…
Dionis Beqiraj
- 195
5
votes
1 answer
Is it possible to start with a knight at some corner of a chess board and reach the opposite corner passing once through all the squares?
Is it possible to start with a knight at some corner of a chessboard and reach the opposite corner passing once through all the squares?
The knight can reach the other corner or any square for that matter. But if it were to pass through all the…
Tapi
- 1,763
5
votes
2 answers
chess board knight distance
Is there a formula to compute the "knight distance" on an infinite board? i.e. how many step a knight need to move from (0,0) to any point (i,j)?
athos
- 5,573
4
votes
1 answer
What does it mean to "unfold" a graph?
edit: more complicated graph
source (problem on pg.21): http://press.princeton.edu/chapters/s7714.pdf
I couldn't find any resources online explaining the unfolding process of a graph in layman's terms and the description from the pdf file doesn't…
iii
- 367
3
votes
1 answer
Knight's tours on the Sator square.
It is known that there are 1728 open knight's tours on a $5\times5$ chessboard.
Now I want to apply that to a very particular $5\times5$ square known as Sator square:
S A T O R
A R E P O
T E N E T
O P E R A
R O T A S
If we take letters from this…
Vosoni
- 271
3
votes
2 answers
Possible positions of the knight after moving $n$ steps in Chessboard.
Problem
There is a knight on an infinite chessboard. After moving one step, there are $8$ possible positions, and after moving two steps, there are $33$ possible positions. The possible position after moving n steps is $a_n$, find the formula for…
Aster
- 1,315
3
votes
2 answers
Does the existence of a knights tour depend on the initial position taken in any way?
Attempting to find solutions for a knights tour using basic bactracking algorithms and came across that most examples use one of the corners as the starting point, does the existence of the tour depend on the starting position.
raving_sage
- 31
3
votes
1 answer
Knightwise “Nearness” By Number Of Moves Required
Given an otherwise empty $n\times n$ chessboard with a knight on one of the squares, define the “knight-closedness” of this board as the maximum possible length of a minimal knight route from one square to another on that board. Determine a closed…
Lieutenant Zipp
- 1,837
3
votes
1 answer
Looking for a challenging task or variant related to the knight's tour problem
I recently took it upon myself to investigate the knight's tour problem for a math assessment.
I decided to investigate how the problem differs with a general knight $(m, n)$ that moves $m$ squares along one axis and $n$ squares along the other. I…
Perpenderp
- 31