Given two integers N, M denoting N×M chessboard, the task is to count the number of ways a knight can reach (N, M) starting from (0, 0). Since the answer can be very large, print the answer modulo 109+7.
Input: N =3, M= 3
Two ways to reach (3, 3) form (0, 0) are as follows:
(0, 0) → (1, 2) → (3, 3)
(0, 0) → (2, 1) → (3, 3)
Input: N=4, M=3
Explanation: No possible way exists to reach (4, 3) form (0, 0).
Approach: Idea here is to observe the pattern that each move increments the value of the x-coordinate + value of y-coordinate by 3. Follow the steps below to solve the problem.
- If (N + M) is not divisible by 3 then no possible path exists.
- If (N + M) % 3==0 then count the number of moves of type (+1, +2) i.e, X and count the number of moves of type (+2, +1) i.e, Y.
- Find the equation of the type (+1, +2) i.e. X + 2Y = N
- Find the equation of the type (+2, +1) i.e. 2X + Y = M
- Find the calculated values of X and Y, if X < 0 or Y < 0, then no possible path exists.
- Otherwise, calculate (X+Y)CY.
Below is the implementation of the above approach:
Time Complexity: O(X + Y + log(mod)).
Auxiliary Space: O(1)
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