LeetCode #264 — MEDIUM

Ugly Number II

Move from brute-force thinking to an efficient approach using hash map strategy.

The Problem

Problem Statement

An ugly number is a positive integer whose prime factors are limited to 2, 3, and 5.

Given an integer n, return the n^th ugly number.

Example 1:

Input: n = 10
Output: 12
Explanation: [1, 2, 3, 4, 5, 6, 8, 9, 10, 12] is the sequence of the first 10 ugly numbers.

Example 2:

Input: n = 1
Output: 1
Explanation: 1 has no prime factors, therefore all of its prime factors are limited to 2, 3, and 5.

Constraints:

1 <= n <= 1690

Step 01

Brute Force Baseline

Problem summary: An ugly number is a positive integer whose prime factors are limited to 2, 3, and 5. Given an integer n, return the nth ugly number.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Hash Map · Math · Dynamic Programming

Example 1

Example 2

Core Insight

What unlocks the optimal approach

The naive approach is to call <code>isUgly</code> for every number until you reach the nth one. Most numbers are not ugly. Try to focus your effort on generating only the ugly ones.
An ugly number must be multiplied by either 2, 3, or 5 from a smaller ugly number.
The key is how to maintain the order of the ugly numbers. Try a similar approach of merging from three sorted lists: L1, L2, and L3.
Assume you have Uk, the kth ugly number. Then Uk+1 must be Min(L1 * 2, L2 * 3, L3 * 5).

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #264: Ugly Number II
class Solution {
    public int nthUglyNumber(int n) {
        Set<Long> vis = new HashSet<>();
        PriorityQueue<Long> q = new PriorityQueue<>();
        int[] f = new int[] {2, 3, 5};
        q.offer(1L);
        vis.add(1L);
        long ans = 0;
        while (n-- > 0) {
            ans = q.poll();
            for (int v : f) {
                long next = ans * v;
                if (vis.add(next)) {
                    q.offer(next);
                }
            }
        }
        return (int) ans;
    }
}

// Accepted solution for LeetCode #264: Ugly Number II
func nthUglyNumber(n int) int {
	h := IntHeap([]int{1})
	heap.Init(&h)
	ans := 1
	vis := map[int]bool{1: true}
	for n > 0 {
		ans = heap.Pop(&h).(int)
		for _, v := range []int{2, 3, 5} {
			nxt := ans * v
			if !vis[nxt] {
				vis[nxt] = true
				heap.Push(&h, nxt)
			}
		}
		n--
	}
	return ans
}

type IntHeap []int

func (h IntHeap) Len() int           { return len(h) }
func (h IntHeap) Less(i, j int) bool { return h[i] < h[j] }
func (h IntHeap) Swap(i, j int)      { h[i], h[j] = h[j], h[i] }
func (h *IntHeap) Push(x any) {
	*h = append(*h, x.(int))
}
func (h *IntHeap) Pop() any {
	old := *h
	n := len(old)
	x := old[n-1]
	*h = old[0 : n-1]
	return x
}

# Accepted solution for LeetCode #264: Ugly Number II
class Solution:
    def nthUglyNumber(self, n: int) -> int:
        h = [1]
        vis = {1}
        ans = 1
        for _ in range(n):
            ans = heappop(h)
            for v in [2, 3, 5]:
                nxt = ans * v
                if nxt not in vis:
                    vis.add(nxt)
                    heappush(h, nxt)
        return ans

// Accepted solution for LeetCode #264: Ugly Number II
struct Solution;

impl Solution {
    fn nth_ugly_number(n: i32) -> i32 {
        let n = n as usize;
        let mut ugly: Vec<i32> = vec![1];
        let mut i = 0;
        let mut j = 0;
        let mut k = 0;
        while ugly.len() <= n {
            let min = vec![ugly[i] * 2, ugly[j] * 3, ugly[k] * 5]
                .into_iter()
                .min()
                .unwrap();
            ugly.push(min);
            if ugly[i] * 2 == min {
                i += 1;
            }
            if ugly[j] * 3 == min {
                j += 1;
            }
            if ugly[k] * 5 == min {
                k += 1;
            }
        }
        ugly[n - 1]
    }
}

#[test]
fn test() {
    let n = 10;
    let res = 12;
    assert_eq!(Solution::nth_ugly_number(n), res);
}

// Accepted solution for LeetCode #264: Ugly Number II
// Auto-generated TypeScript example from java.
function exampleSolution(): void {
}
// Reference (java):
// // Accepted solution for LeetCode #264: Ugly Number II
// class Solution {
//     public int nthUglyNumber(int n) {
//         Set<Long> vis = new HashSet<>();
//         PriorityQueue<Long> q = new PriorityQueue<>();
//         int[] f = new int[] {2, 3, 5};
//         q.offer(1L);
//         vis.add(1L);
//         long ans = 0;
//         while (n-- > 0) {
//             ans = q.poll();
//             for (int v : f) {
//                 long next = ans * v;
//                 if (vis.add(next)) {
//                     q.offer(next);
//                 }
//             }
//         }
//         return (int) ans;
//     }
// }

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n × m)

Space

O(n × m)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.