Introduction

In this blog post, we will explore how to solve the LeetCode problem "1223" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

A die simulator generates a random number from 1 to 6 for each roll. You introduced a constraint to the generator such that it cannot roll the number i more than rollMax[i] (1-indexed) consecutive times. Given an array of integers rollMax and an integer n, return the number of distinct sequences that can be obtained with exact n rolls. Since the answer may be too large, return it modulo 109 + 7. Two sequences are considered different if at least one element differs from each other. Example 1: Input: n = 2, rollMax = [1,1,2,2,2,3] Output: 34 Explanation: There will be 2 rolls of die, if there are no constraints on the die, there are 6 * 6 = 36 possible combinations. In this case, looking at rollMax array, the numbers 1 and 2 appear at most once consecutively, therefore sequences (1,1) and (2,2) cannot occur, so the final answer is 36-2 = 34. Example 2: Input: n = 2, rollMax = [1,1,1,1,1,1] Output: 30 Example 3: Input: n = 3, rollMax = [1,1,1,2,2,3] Output: 181 Constraints: 1 <= n <= 5000 rollMax.length == 6 1 <= rollMax[i] <= 15

Explanation

Here's the breakdown of the solution:

• Dynamic Programming: Use dynamic programming to build up the number of valid sequences. The state of the DP represents the number of rolls made so far, the last number rolled, and the number of consecutive times the last number has been rolled.
• State Transitions: For each state, consider all possible next rolls. If the next roll is different from the current last roll, reset the consecutive count to 1. If it's the same, increment the consecutive count, ensuring it doesn't exceed rollMax for that number.

• Base Case & Result: The base case is when the number of rolls is 0. The final result is the sum of all valid sequences of length n.

• Time & Space Complexity: O(n 6 15) time, O(n 6 15) space.

Code

    def dieSimulator(n: int, rollMax: list[int]) -> int:
    """
    Calculates the number of distinct sequences that can be obtained with exactly n rolls,
    given the constraints on consecutive rolls of the same number.

    Args:
        n: The number of rolls.
        rollMax: An array of integers representing the maximum consecutive rolls for each number (1-indexed).

    Returns:
        The number of distinct sequences modulo 10^9 + 7.
    """

    MOD = 10**9 + 7
    dp = [[[0] * 16 for _ in range(6)] for _ in range(n + 1)]

    # Base case: For 0 rolls, there are no sequences, but we initialize the first roll's states
    for i in range(6):
        dp[1][i][1] = 1

    # Iterate through the number of rolls
    for i in range(2, n + 1):
        # Iterate through the possible last rolls
        for j in range(6):
            # Iterate through the possible consecutive counts of the last roll
            for k in range(1, rollMax[j] + 1):
                # Iterate through the possible next rolls
                for l in range(6):
                    # If the next roll is different from the last roll
                    if l != j:
                        dp[i][l][1] = (dp[i][l][1] + dp[i - 1][j][k]) % MOD
                    # If the next roll is the same as the last roll, and we haven't exceeded the limit
                    elif k < rollMax[j]:
                        dp[i][j][k + 1] = (dp[i][j][k + 1] + dp[i - 1][j][k]) % MOD

    # Calculate the total number of distinct sequences
    total_sequences = 0
    for j in range(6):
        for k in range(1, rollMax[j] + 1):
            total_sequences = (total_sequences + dp[n][j][k]) % MOD

    return total_sequences