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Program to find the percentage of places where knight can move and not left the board in Python
Suppose we have four values n, x, y, and k. Here n indicates an n x n chessboard and x, y coordinate represents a knight is placed at (x, y). The knight has to take exactly k steps, where at each step it can move any of the 8 directions uniformly at random. We have to find the percentage chance (nearest integer) that the knight remains in the chessboard after taking k moves. We have to follow a condition that it cannot enter the board again once it leaves it.
So, if the input is like n = 8, (x = 0, y = 0), k = 1, then the output will be 50, as here we have 8x8 chessboard and the initial position of the knight is (1, 1). It can take k = 1 step. Taking one step it will lie inside the board only at 4 out of 8 positions, and will lie outside at other positions. So 50%.
To solve this, we will follow these steps −
- Make a list of moves [(1, 2) ,(1, -2) ,(-1, 2) ,(-1, -2) ,(2, 1) ,(2, -1) ,(-2, 1) ,(-2, -1) ]
- Define a function dfs() . This will take x, y, k
- if (x, y) are not in range of the board, then
- return 0
- if k is same as 0, then
- return 1
- s = empty list
- for all moves (dx, dy) in moves −
- x = dfs(x + dx, y + dy, k - 1) / 8
- insert x into s
- return sum of the elements in s
- From the main method do the following −
- return rounded result of (dfs(x, y, k) * 100) to nearest integer
Let us see the following implementation to get better understanding −
Example
moves = [(1, 2), (1, -2), (-1, 2), (-1, -2), (2, 1), (2, -1), (-2, 1), (-2, -1)] class Solution: def solve(self, n, x, y, k): def dfs(x, y, k): if x < 0 or y < 0 or x >= n or y >= n: return 0 if k == 0: return 1 return sum(dfs(x + dx, y + dy, k - 1) / 8 for dx, dy in moves) return int(dfs(x, y, k) * 100) ob = Solution() n = 8 x = 1 y = 1 k = 1 print(ob.solve(n, x, y, k))
Input
8, 1, 1, 1
Output
0
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