Is Number of Balanced Integers in a Range easy or hard?

Number of Balanced Integers in a Range is categorized as a Hard problem. The difficulty comes from designing correct Digit DP states and handling constraints like digit position, prefix tightness, and balanced sum tracking.

Number of Balanced Integers in a Range Python/Java solution

Implement a Digit DP function that counts valid numbers up to a bound. The function processes digits from left to right, tracks the sum difference between halves, and respects the tight bound condition. The same logic translates cleanly to Python, Java, C++, Go, and TypeScript.

How to solve Number of Balanced Integers in a Range in O(d * states)?

Use Digit DP to count valid numbers up to a limit. Track the current digit position, tight constraint with the bound, and the difference between the first-half and second-half digit sums. Evaluate counts for high and subtract counts for low - 1 to get the final answer.

What is the best approach for Number of Balanced Integers in a Range?

Digit Dynamic Programming (Digit DP) is the most efficient approach. Instead of iterating through every number in the range, the algorithm builds numbers digit by digit while tracking the difference between the first-half and second-half digit sums. Memoization avoids recomputation of identical states, reducing the search space dramatically.

Is Number of Balanced Integers in a Range asked at Google/Amazon/Meta?

Digit DP counting problems similar to this frequently appear in interviews at companies like Google, Amazon, and Meta. Interviewers use them to test dynamic programming skills, state design, and the ability to optimize brute-force enumeration.

What data structure is used in Number of Balanced Integers in a Range?

The solution relies on dynamic programming with memoization, typically implemented using arrays or hash maps for DP states. Recursion or iterative DP is used to traverse digits while caching computed states.

What is the time complexity of Number of Balanced Integers in a Range?

The optimal Digit DP solution runs in roughly O(d * sum * states) time where d is the number of digits in the bound and sum represents possible digit sums. The brute force method takes O(N * d) time where N is the size of the range, which becomes impractical for large inputs.

Description Hints 5 Solution Video 23:56

3791. Number of Balanced Integers in a Range

Hard35.0% acceptance

You are given two integers low and high.

An integer is called balanced if it satisfies both of the following conditions:

It contains at least two digits.
The sum of digits at even positions is equal to the sum of digits at odd positions (the leftmost digit has position 1).

Return an integer representing the number of balanced integers in the range [low, high] (both inclusive).

Example 1:

Input: low = 1, high = 100

Output: 9

Explanation:

The 9 balanced numbers between 1 and 100 are 11, 22, 33, 44, 55, 66, 77, 88, and 99.

Example 2:

Input: low = 120, high = 129

Output: 1

Explanation:

Only 121 is balanced because the sum of digits at even and odd positions are both 2.

Example 3:

Input: low = 1234, high = 1234

Output: 0

Explanation:

1234 is not balanced because the sum of digits at odd positions (1 + 3 = 4) does not equal the sum at even positions (2 + 4 = 6).

Constraints:

1 <= low <= high <= 10¹⁵

Topics

Dynamic Programming

Hints open one at a time

Hint 1

Use digit dynamic programming.

Hint 2

Let f(x) be the number of balanced integers in [1, x]; the answer is f(high) - f(low - 1).

Hint 3

Track the difference sum(odd positions) - sum(even positions) while building from the most significant digit, and require it to be zero at the end.

Hint 4

Ignore leading zeros during transitions.

Hint 5

Enforce the length constraint by counting only when at least two digits have been placed.

23:56

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1class Solution:
2    def countBalanced(self, low: int, high: int) -> int:
3        

Starter codePython 3Spaces: 4

Examples

Case 1Case 2Case 3

Input

1
100

Input

120
129

Input

1234
1234

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