Sum Game

Sum Game - Problem

Alice and Bob take turns playing a game, with Alice starting first.

You are given a string num of even length consisting of digits and '?' characters. On each turn, a player will do the following if there is still at least one '?' in num:

Choose an index i where num[i] == '?'.
Replace num[i] with any digit between '0' and '9'.

The game ends when there are no more '?' characters in num.

For Bob to win, the sum of the digits in the first half of num must be equal to the sum of the digits in the second half. For Alice to win, the sums must not be equal.

For example, if the game ended with num = "243801", then Bob wins because 2+4+3 = 8+0+1. If the game ended with num = "243803", then Alice wins because 2+4+3 != 8+0+3.

Assuming Alice and Bob play optimally, return true if Alice will win and false if Bob will win.

Input & Output

Example 1 — Alice Advantage

$ Input: num = "5?2?"

› Output: true

💡 Note: Left sum = 5, right sum = 2. Current diff = 3. Alice can maintain this difference since 2*3 + 9*0 = 6 ≠ 0.

Example 2 — Perfect Balance Possible

$ Input: num = "??1??2"

› Output: true

💡 Note: Left sum = 1, right sum = 2. Diff = -1, net_questions = 2-2 = 0. Since 2*(-1) + 9*0 = -2 ≠ 0, Alice wins.

Example 3 — Equal Question Marks

$ Input: num = "??22"

› Output: true

💡 Note: Left sum = 0, right sum = 4. Diff = -4, net_questions = 2-0 = 2. Since 2*(-4) + 9*2 = 10 ≠ 0, Alice wins.

Constraints

2 ≤ num.length ≤ 10⁵
num.length is even
num consists only of digits and '?'

Visualization

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Asked in

G Google 12 M Microsoft 8

The key insight is to use mathematical analysis instead of simulation. Calculate the current sum difference and question mark distribution, then apply the formula 2*diff + 9*net_questions ≠ 0 to determine if Alice wins. Best approach is Mathematical Analysis with Time: O(n), Space: O(1).

Common Approaches

✓ Mathematical Analysis

⏱️ Time: O(n) Space: O(1)

Calculate the current sum difference between halves and the number of question marks in each half. Use mathematical reasoning to determine if Alice can prevent Bob from balancing the sums.

Recursive Simulation

⏱️ Time: O(10^k) Space: O(k)

Recursively simulate the game by trying all possible digit placements for each player's turn. Use minimax approach where Alice maximizes her winning chances and Bob minimizes them.

Mathematical Analysis — Algorithm Steps

Calculate current sum difference between halves
Count question marks in each half
Determine if Alice can maintain imbalance

Visualization

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Step-by-Step Walkthrough

Calculate Sums

Sum known digits in each half

Count Questions

Count '?' marks in each half

Apply Formula

Use 2*diff + 9*net_questions != 0

Code -

solution.c — C

#include <stdio.h>
#include <string.h>
#include <stdbool.h>

bool solution(char* num) {
    int n = strlen(num);
    int half = n / 2;
    
    // Calculate current sums and count question marks
    int leftSum = 0;
    int rightSum = 0;
    int leftQuestions = 0;
    int rightQuestions = 0;
    
    for (int i = 0; i < half; i++) {
        if (num[i] == '?') {
            leftQuestions++;
        } else {
            leftSum += num[i] - '0';
        }
    }
    
    for (int i = half; i < n; i++) {
        if (num[i] == '?') {
            rightQuestions++;
        } else {
            rightSum += num[i] - '0';
        }
    }
    
    // Current difference (left - right)
    int diff = leftSum - rightSum;
    int netQuestions = leftQuestions - rightQuestions;
    
    // Alice wins if: 2*diff + 9*net_questions != 0
    return 2 * diff + 9 * netQuestions != 0;
}

int main() {
    char num[1001];
    fgets(num, sizeof(num), stdin);
    num[strcspn(num, "\n")] = 0;  // Remove newline
    bool result = solution(num);
    printf(result ? "true\n" : "false\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single pass through the string to calculate sums and count question marks

✓ Linear Growth

Space Complexity

O(1)

Only using constant extra variables for sums and counts

✓ Linear Space

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Mathematical Analysis — Algorithm Steps

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Code -

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