Mid/SeniorDynamic Programming
Knight Probability in Chessboard
On an n x n chessboard, a knight starts at the cell (row, column) and attempts to make exactly k moves. The rows and columns are 0-indexed, so the top-left cell is (0, 0), and the bottom-right cell is (n - 1, n - 1).
A chess knight has eight possible moves it can make. Each move is two cells in a cardinal direction, then one cell in an orthogonal direction.
Each time the knight is to move, it chooses one of eight possible moves uniformly at random, even if the piece would go off the chessboard, and moves there.
The knight continues moving until it has made exactly k moves or has moved off the chessboard.
Return the probability that the knight remains on the board after it has stopped moving.
Example 1
Input
n = 3, k = 2, row = 0, column = 0Output
0.0625There are two first moves that keep the knight on the board, and from each of those positions there are two more valid moves, so the total probability is 0.0625.
Example 2
Input
n = 1, k = 0, row = 0, column = 0Output
1The knight makes no moves and remains on the only board cell.
Constraints
- 1 <= n <= 25
- 0 <= k <= 100
- 0 <= row, column <= n - 1