Leetcode #1155: Number of Dice Rolls With Target Sum
In this guide, we solve Leetcode #1155 Number of Dice Rolls With Target Sum in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
You have n dice, and each dice has k faces numbered from 1 to k. Given three integers n, k, and target, return the number of possible ways (out of the kn total ways) to roll the dice, so the sum of the face-up numbers equals target.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Dynamic Programming
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: n = 1, k = 6, target = 3
Output: 1
Explanation: You throw one die with 6 faces.
There is only one way to get a sum of 3.
Python Solution
class Solution:
def numRollsToTarget(self, n: int, k: int, target: int) -> int:
f = [[0] * (target + 1) for _ in range(n + 1)]
f[0][0] = 1
mod = 10**9 + 7
for i in range(1, n + 1):
for j in range(1, min(i * k, target) + 1):
for h in range(1, min(j, k) + 1):
f[i][j] = (f[i][j] + f[i - 1][j - h]) % mod
return f[n][target]
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.