1927. Sum Game

Description

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.

Example 1:

Input: num = "5023"
Output: false
Explanation: There are no moves to be made.
The sum of the first half is equal to the sum of the second half: 5 + 0 = 2 + 3.

Example 2:

Input: num = "25??"
Output: true
Explanation: Alice can replace one of the '?'s with '9' and it will be impossible for Bob to make the sums equal.

Example 3:

Input: num = "?3295???"
Output: false
Explanation: It can be proven that Bob will always win. One possible outcome is:
- Alice replaces the first '?' with '9'. num = "93295???".
- Bob replaces one of the '?' in the right half with '9'. num = "932959??".
- Alice replaces one of the '?' in the right half with '2'. num = "9329592?".
- Bob replaces the last '?' in the right half with '7'. num = "93295927".
Bob wins because 9 + 3 + 2 + 9 = 5 + 9 + 2 + 7.

Constraints:

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

Solutions

Solution 1: Case Analysis

If the number of '?' is odd, Alice will definitely win because she can choose to replace the last '?' with any digit, making the sum of the first half different from the sum of the second half.

If the number of '?' is even, Alice will try to make the sums of the two halves different by placing \(9\) in the half with the larger current sum and \(0\) in the half with the smaller current sum. Bob, on the other hand, will try to make the sums equal by placing a digit in the other half that matches the digit Alice placed.

As a result, all remaining even-numbered '?' will be concentrated in one half. Suppose the current difference between the sums of the two halves is \(d\).

Let's consider the case where there are two remaining '?' and the difference is \(x\):

If \(x \lt 9\), Alice will definitely win because she can replace one of the '?' with \(9\), making the sums of the two halves different.
If \(x \gt 9\), Alice will definitely win because she can replace one of the '?' with \(0\), making the sums of the two halves different.
If \(x = 9\), Bob will definitely win. Suppose Alice replaces a digit with \(a\), then Bob can replace the other '?' with \(9 - a\), making the sums of the two halves equal.

Therefore, if the difference between the sums of the two halves is \(d = \frac{9 \times \textit{cnt}}{2}\), where \(\textit{cnt}\) is the number of remaining '?', Bob will definitely win; otherwise, Alice will definitely win.

The time complexity is \(O(n)\), where \(n\) is the length of the string. The space complexity is \(O(1)\).

Python3JavaC++GoTypeScript

class Solution:
    def sumGame(self, num: str) -> bool:
        n = len(num)
        cnt1 = num[: n // 2].count("?")
        cnt2 = num[n // 2 :].count("?")
        s1 = sum(int(x) for x in num[: n // 2] if x != "?")
        s2 = sum(int(x) for x in num[n // 2 :] if x != "?")
        return (cnt1 + cnt2) % 2 == 1 or s1 - s2 != 9 * (cnt2 - cnt1) // 2

class Solution {
    public boolean sumGame(String num) {
        int n = num.length();
        int cnt1 = 0, cnt2 = 0;
        int s1 = 0, s2 = 0;
        for (int i = 0; i < n / 2; ++i) {
            if (num.charAt(i) == '?') {
                cnt1++;
            } else {
                s1 += num.charAt(i) - '0';
            }
        }
        for (int i = n / 2; i < n; ++i) {
            if (num.charAt(i) == '?') {
                cnt2++;
            } else {
                s2 += num.charAt(i) - '0';
            }
        }
        return (cnt1 + cnt2) % 2 == 1 || s1 - s2 != 9 * (cnt2 - cnt1) / 2;
    }
}

class Solution {
public:
    bool sumGame(string num) {
        int n = num.size();
        int cnt1 = 0, cnt2 = 0;
        int s1 = 0, s2 = 0;
        for (int i = 0; i < n / 2; ++i) {
            if (num[i] == '?') {
                cnt1++;
            } else {
                s1 += num[i] - '0';
            }
        }
        for (int i = n / 2; i < n; ++i) {
            if (num[i] == '?') {
                cnt2++;
            } else {
                s2 += num[i] - '0';
            }
        }
        return (cnt1 + cnt2) % 2 == 1 || (s1 - s2) != 9 * (cnt2 - cnt1) / 2;
    }
};

func sumGame(num string) bool {
    n := len(num)
    var cnt1, cnt2, s1, s2 int
    for i := 0; i < n/2; i++ {
        if num[i] == '?' {
            cnt1++
        } else {
            s1 += int(num[i] - '0')
        }
    }
    for i := n / 2; i < n; i++ {
        if num[i] == '?' {
            cnt2++
        } else {
            s2 += int(num[i] - '0')
        }
    }
    return (cnt1+cnt2)%2 == 1 || s1-s2 != (cnt2-cnt1)*9/2
}

function sumGame(num: string): boolean {
    const n = num.length;
    let [cnt1, cnt2, s1, s2] = [0, 0, 0, 0];
    for (let i = 0; i < n >> 1; ++i) {
        if (num[i] === '?') {
            ++cnt1;
        } else {
            s1 += num[i].charCodeAt(0) - '0'.charCodeAt(0);
        }
    }
    for (let i = n >> 1; i < n; ++i) {
        if (num[i] === '?') {
            ++cnt2;
        } else {
            s2 += num[i].charCodeAt(0) - '0'.charCodeAt(0);
        }
    }
    return (cnt1 + cnt2) % 2 === 1 || 2 * (s1 - s2) !== 9 * (cnt2 - cnt1);
}

1927. Sum Game

Description

Solutions

Solution 1: Case Analysis

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