Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915658254516224 |
|---|---|
| author | Cashman, Caroline Cooper, Joseph Marquez, Raul Miller, Steven J. Shuffelton, Jenna |
| author_facet | Cashman, Caroline Cooper, Joseph Marquez, Raul Miller, Steven J. Shuffelton, Jenna |
| contents | The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks, proving the proportion of safe squares converges to $1/e^2$. We generalize and solve when randomly placing $n$ hyper-rooks and $n^{k-1}$ line-rooks on a $k$-dimensional board, using combinatorial and probabilistic methods, with the proportion of safe squares converging to $1/e^k$. We prove that the proportion of safe squares on an $n \times n$ board with bishops in 2 dimensions converges to $2/e^2$. This problem is significantly more interesting and difficult; while a rook attacks the same number of squares wherever it's placed, this is not so for bishops. We expand to the $k$-dimensional chessboard, defining line-bishops to attack along $2$-dimensional diagonals and hyper-bishops to attack in the $k-1$ dimensional subspace defined by its diagonals in the $k-2$ dimensional subspace. We then combine the movement of rooks and bishops to consider the movement of queens in 2 dimensions, as well as line-queens and hyper-queens in $k$ dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards Cashman, Caroline Cooper, Joseph Marquez, Raul Miller, Steven J. Shuffelton, Jenna Combinatorics Probability 60C05, 05A16 (primary) The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks, proving the proportion of safe squares converges to $1/e^2$. We generalize and solve when randomly placing $n$ hyper-rooks and $n^{k-1}$ line-rooks on a $k$-dimensional board, using combinatorial and probabilistic methods, with the proportion of safe squares converging to $1/e^k$. We prove that the proportion of safe squares on an $n \times n$ board with bishops in 2 dimensions converges to $2/e^2$. This problem is significantly more interesting and difficult; while a rook attacks the same number of squares wherever it's placed, this is not so for bishops. We expand to the $k$-dimensional chessboard, defining line-bishops to attack along $2$-dimensional diagonals and hyper-bishops to attack in the $k-1$ dimensional subspace defined by its diagonals in the $k-2$ dimensional subspace. We then combine the movement of rooks and bishops to consider the movement of queens in 2 dimensions, as well as line-queens and hyper-queens in $k$ dimensions. |
| title | Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards |
| topic | Combinatorics Probability 60C05, 05A16 (primary) |
| url | https://arxiv.org/abs/2409.04423 |