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.
Move from brute-force thinking to an efficient approach using math strategy.
Alice and Bob are playing a turn-based game on a field, with two lanes of flowers between them. There are x flowers in the first lane between Alice and Bob, and y flowers in the second lane between them.
The game proceeds as follows:
Given two integers, n and m, the task is to compute the number of possible pairs (x, y) that satisfy the conditions:
x in the first lane must be in the range [1,n].y in the second lane must be in the range [1,m].Return the number of possible pairs (x, y) that satisfy the conditions mentioned in the statement.
Example 1:
Input: n = 3, m = 2 Output: 3 Explanation: The following pairs satisfy conditions described in the statement: (1,2), (3,2), (2,1).
Example 2:
Input: n = 1, m = 1 Output: 0 Explanation: No pairs satisfy the conditions described in the statement.
Constraints:
1 <= n, m <= 105Problem summary: Alice and Bob are playing a turn-based game on a field, with two lanes of flowers between them. There are x flowers in the first lane between Alice and Bob, and y flowers in the second lane between them. The game proceeds as follows: Alice takes the first turn. In each turn, a player must choose either one of the lane and pick one flower from that side. At the end of the turn, if there are no flowers left at all in either lane, the current player captures their opponent and wins the game. Given two integers, n and m, the task is to compute the number of possible pairs (x, y) that satisfy the conditions: Alice must win the game according to the described rules. The number of flowers x in the first lane must be in the range [1,n]. The number of flowers y in the second lane must be in the range [1,m]. Return the number of possible pairs (x, y) that satisfy the conditions mentioned in the
Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.
Pattern signal: Math
3 2
1 1
Source-backed implementations are provided below for direct study and interview prep.
// Accepted solution for LeetCode #3021: Alice and Bob Playing Flower Game
class Solution {
public long flowerGame(int n, int m) {
long a1 = (n + 1) / 2;
long b1 = (m + 1) / 2;
long a2 = n / 2;
long b2 = m / 2;
return a1 * b2 + a2 * b1;
}
}
// Accepted solution for LeetCode #3021: Alice and Bob Playing Flower Game
func flowerGame(n int, m int) int64 {
a1, b1 := (n+1)/2, (m+1)/2
a2, b2 := n/2, m/2
return int64(a1*b2 + a2*b1)
}
# Accepted solution for LeetCode #3021: Alice and Bob Playing Flower Game
class Solution:
def flowerGame(self, n: int, m: int) -> int:
a1 = (n + 1) // 2
b1 = (m + 1) // 2
a2 = n // 2
b2 = m // 2
return a1 * b2 + a2 * b1
// Accepted solution for LeetCode #3021: Alice and Bob Playing Flower Game
impl Solution {
pub fn flower_game(n: i32, m: i32) -> i64 {
let a1 = ((n + 1) / 2) as i64;
let b1 = ((m + 1) / 2) as i64;
let a2 = (n / 2) as i64;
let b2 = (m / 2) as i64;
a1 * b2 + a2 * b1
}
}
// Accepted solution for LeetCode #3021: Alice and Bob Playing Flower Game
function flowerGame(n: number, m: number): number {
const [a1, b1] = [(n + 1) >> 1, (m + 1) >> 1];
const [a2, b2] = [n >> 1, m >> 1];
return a1 * b2 + a2 * b1;
}
Use this to step through a reusable interview workflow for this problem.
Simulate the process step by step — multiply n times, check each number up to n, or iterate through all possibilities. Each step is O(1), but doing it n times gives O(n). No extra space needed since we just track running state.
Math problems often have a closed-form or O(log n) solution hidden behind an O(n) simulation. Modular arithmetic, fast exponentiation (repeated squaring), GCD (Euclidean algorithm), and number theory properties can dramatically reduce complexity.
Review these before coding to avoid predictable interview regressions.
Wrong move: Temporary multiplications exceed integer bounds.
Usually fails on: Large inputs wrap around unexpectedly.
Fix: Use wider types, modular arithmetic, or rearranged operations.