#1025Easy~30 min
Divisor Game
Time O(1) · Space O(1) · Official statement on LeetCode
Solutions
// Time: O(1)
// Space: O(1)
// math
class Solution {
public:
bool divisorGame(int n) {
// 1. if we get an even, we can choose x = 1
// to make the opponent always get an odd
// 2. if the opponent gets an odd, he can only choose x = 1 or other odds
// and we can still get an even
// 3. at the end, the opponent can only choose x = 1 and we win
// 4. in summary, we win if only if we get an even and
// keeps even until the opponent loses
return n % 2 == 0;
}
};
// Time: O(nlogn)
// Space: O(nlogn)
// dp, number theory
class Solution2 {
public:
bool divisorGame(int n) {
const auto& factors = [](int n) {
vector<vector<int>> result(n + 1);
for (int i = 1; i <= n; ++i) {
for (int j = i; j <= n; j += i) {
result[j].emplace_back(i);
}
}
return result;
};
const auto& FACTORS = factors(n);
vector<bool> dp(n + 1);
for (int i = 2; i <= n; ++i) {
for (const auto& j : FACTORS[i]) {
if (j != i && !dp[i - j]) {
dp[i] = true;
break;
}
}
}
return dp[n];
}
};
// Time: O(nlogn)
// Space: O(nlogn)
// memoization, number theory
class Solution3 {
public:
bool divisorGame(int n) {
const auto& factors = [](int n) {
vector<vector<int>> result(n + 1);
for (int i = 1; i <= n; ++i) {
for (int j = i; j <= n; j += i) {
result[j].emplace_back(i);
}
}
return result;
};
const auto& FACTORS = factors(n);
vector<int> lookup(n + 1, -1);
const function<int (int)> memoization = [&](int n) {
if (lookup[n] == -1) {
int result = 0;
for (const auto& i : FACTORS[n]) {
if (i != n && !memoization(n - i)) {
result = 1;
break;
}
}
lookup[n] = result;
}
return lookup[n];
};
return memoization(n);
}
};
// Time: O(n^(3/2))
// Space: O(n)
// memoization
class Solution4 {
public:
bool divisorGame(int n) {
vector<int> lookup(n + 1, -1);
const function<int (int)> memoization = [&](int n) {
if (lookup[n] == -1) {
int result = 0;
for (auto i = 1; i * i <= n; ++i) {
if (n % i) {
continue;
}
if (i != n && !memoization(n - i)) {
result = 1;
break;
}
const int j = n / i;
if (j == i) {
continue;
}
if (j != n && !memoization(n - j)) {
result = 1;
break;
}
}
lookup[n] = result;
}
return lookup[n];
};
return memoization(n);
}
};
// Time: O(n^2)
// Space: O(n)
// memoization
class Solution5 {
public:
bool divisorGame(int n) {
vector<int> lookup(n + 1, -1);
const function<int (int)> memoization = [&](int n) {
if (lookup[n] == -1) {
int result = 0;
for (auto i = 1; i <= n; ++i) {
if (n % i) {
continue;
}
if (i != n && !memoization(n - i)) {
result = 1;
break;
}
}
lookup[n] = result;
}
return lookup[n];
};
return memoization(n);
}
};
Beginner Explanation
What is Divisor Game?
Divisor Game (LeetCode #1025) is a Easy problem that primarily trains math.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with dynamic programming.
- Only then translate the idea into code.
Why this problem matters
It builds core muscle memory you will reuse on harder variants. Official solution notes mention: DP.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Divisor Game
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to dynamic programming.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(1)) and space (O(1)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(1) time and O(1) space.
Pattern focus: dynamic programming
Use the pattern as a checklist:
- dynamic programming — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(1) |
| Space | O(1) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Divisor Game
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dynamic programming — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to dynamic programming:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: math.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Divisor Game in a second language (cpp, python).
- Drill 3–5 more problems tagged math.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Conceptual diagram for Divisor Game: show input structure (math), highlight the moving parts of the dynamic programming approach, and annotate each step with the maintained invariant and complexity.
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the dynamic programming approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Divisor Game (#1025) — Easy. Pattern: dynamic programming. Complexity: O(1) time / O(1) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Divisor Game?+
The reference solutions aim for O(1) time and O(1) space. Always re-derive complexity from the code you write in the interview.
What pattern does Divisor Game use?+
It primarily maps to dynamic programming, within the broader topic of math.
Is Divisor Game good for interviews?+
Yes — as a Easy problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/divisor-game/