LeetCode #753 — HARD

Cracking the Safe

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

Solve on LeetCode
The Problem

Problem Statement

There is a safe protected by a password. The password is a sequence of n digits where each digit can be in the range [0, k - 1].

The safe has a peculiar way of checking the password. When you enter in a sequence, it checks the most recent n digits that were entered each time you type a digit.

  • For example, the correct password is "345" and you enter in "012345":
    • After typing 0, the most recent 3 digits is "0", which is incorrect.
    • After typing 1, the most recent 3 digits is "01", which is incorrect.
    • After typing 2, the most recent 3 digits is "012", which is incorrect.
    • After typing 3, the most recent 3 digits is "123", which is incorrect.
    • After typing 4, the most recent 3 digits is "234", which is incorrect.
    • After typing 5, the most recent 3 digits is "345", which is correct and the safe unlocks.

Return any string of minimum length that will unlock the safe at some point of entering it.

Example 1:

Input: n = 1, k = 2
Output: "10"
Explanation: The password is a single digit, so enter each digit. "01" would also unlock the safe.

Example 2:

Input: n = 2, k = 2
Output: "01100"
Explanation: For each possible password:
- "00" is typed in starting from the 4th digit.
- "01" is typed in starting from the 1st digit.
- "10" is typed in starting from the 3rd digit.
- "11" is typed in starting from the 2nd digit.
Thus "01100" will unlock the safe. "10011", and "11001" would also unlock the safe.

Constraints:

  • 1 <= n <= 4
  • 1 <= k <= 10
  • 1 <= kn <= 4096

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: There is a safe protected by a password. The password is a sequence of n digits where each digit can be in the range [0, k - 1]. The safe has a peculiar way of checking the password. When you enter in a sequence, it checks the most recent n digits that were entered each time you type a digit. For example, the correct password is "345" and you enter in "012345": After typing 0, the most recent 3 digits is "0", which is incorrect. After typing 1, the most recent 3 digits is "01", which is incorrect. After typing 2, the most recent 3 digits is "012", which is incorrect. After typing 3, the most recent 3 digits is "123", which is incorrect. After typing 4, the most recent 3 digits is "234", which is incorrect. After typing 5, the most recent 3 digits is "345", which is correct and the safe unlocks. Return any string of minimum length that will unlock the safe at some point of entering it.

Baseline thinking

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

Pattern signal: General problem-solving

Example 1

1
2

Example 2

2
2
Step 02

Core Insight

What unlocks the optimal approach

  • We can think of this problem as the problem of finding an Euler path (a path visiting every edge exactly once) on the following graph: there are <code>k<sup>n-1</sup></code> nodes with each node having <code>k</code> edges. It turns out this graph always has an Eulerian circuit (path starting where it ends.) We should visit each node in "post-order" so as to not get stuck in the graph prematurely.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. 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
Largest constraint values
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 #753: Cracking the Safe
class Solution {
    private Set<Integer> vis = new HashSet<>();
    private StringBuilder ans = new StringBuilder();
    private int mod;

    public String crackSafe(int n, int k) {
        mod = (int) Math.pow(10, n - 1);
        dfs(0, k);
        ans.append("0".repeat(n - 1));
        return ans.toString();
    }

    private void dfs(int u, int k) {
        for (int x = 0; x < k; ++x) {
            int e = u * 10 + x;
            if (vis.add(e)) {
                int v = e % mod;
                dfs(v, k);
                ans.append(x);
            }
        }
    }
}
Step 06

Interactive Study Demo

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

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(k^n)
Space
O(k^n)

Approach Breakdown

BRUTE FORCE
O(n²) time
O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED
O(n) time
O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.
Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.