2698. Find the Punishment Number of an Integer

Description

Given a positive integer n, return the punishment number of n.

The punishment number of n is defined as the sum of the squares of all integers i such that:

• 1 <= i <= n
• The decimal representation of i * i can be partitioned into contiguous substrings such that the sum of the integer values of these substrings equals i.

 

Example 1:

Input: n = 10
Output: 182
Explanation: There are exactly 3 integers i in the range [1, 10] that satisfy the conditions in the statement:
- 1 since 1 * 1 = 1
- 9 since 9 * 9 = 81 and 81 can be partitioned into 8 and 1 with a sum equal to 8 + 1 == 9.
- 10 since 10 * 10 = 100 and 100 can be partitioned into 10 and 0 with a sum equal to 10 + 0 == 10.
Hence, the punishment number of 10 is 1 + 81 + 100 = 182

Example 2:

Input: n = 37
Output: 1478
Explanation: There are exactly 4 integers i in the range [1, 37] that satisfy the conditions in the statement:
- 1 since 1 * 1 = 1. 
- 9 since 9 * 9 = 81 and 81 can be partitioned into 8 + 1. 
- 10 since 10 * 10 = 100 and 100 can be partitioned into 10 + 0. 
- 36 since 36 * 36 = 1296 and 1296 can be partitioned into 1 + 29 + 6.
Hence, the punishment number of 37 is 1 + 81 + 100 + 1296 = 1478

 

Constraints:

• 1 <= n <= 1000

Solutions

Solution 1: Enumeration + DFS

We enumerate \(i\), where \(1 \leq i \leq n\). For each \(i\), we split the decimal representation string of \(x = i^2\), and then check whether it meets the requirements of the problem. If it does, we add \(x\) to the answer.

After the enumeration ends, we return the answer.

The time complexity is \(O(n^{1 + 2 \log_{10}^2})\), and the space complexity is \(O(\log n)\), where \(n\) is the given positive integer.

Python3JavaC++GoTypeScript

class Solution:
    def punishmentNumber(self, n: int) -> int:
        def check(s: str, i: int, x: int) -> bool:
            m = len(s)
            if i >= m:
                return x == 0
            y = 0
            for j in range(i, m):
                y = y * 10 + int(s[j])
                if y > x:
                    break
                if check(s, j + 1, x - y):
                    return True
            return False

        ans = 0
        for i in range(1, n + 1):
            x = i * i
            if check(str(x), 0, i):
                ans += x
        return ans

class Solution {
    public int punishmentNumber(int n) {
        int ans = 0;
        for (int i = 1; i <= n; ++i) {
            int x = i * i;
            if (check(x + "", 0, i)) {
                ans += x;
            }
        }
        return ans;
    }

    private boolean check(String s, int i, int x) {
        int m = s.length();
        if (i >= m) {
            return x == 0;
        }
        int y = 0;
        for (int j = i; j < m; ++j) {
            y = y * 10 + (s.charAt(j) - '0');
            if (y > x) {
                break;
            }
            if (check(s, j + 1, x - y)) {
                return true;
            }
        }
        return false;
    }
}

class Solution {
public:
    int punishmentNumber(int n) {
        int ans = 0;
        for (int i = 1; i <= n; ++i) {
            int x = i * i;
            string s = to_string(x);
            if (check(s, 0, i)) {
                ans += x;
            }
        }
        return ans;
    }

    bool check(const string& s, int i, int x) {
        int m = s.size();
        if (i >= m) {
            return x == 0;
        }
        int y = 0;
        for (int j = i; j < m; ++j) {
            y = y * 10 + s[j] - '0';
            if (y > x) {
                break;
            }
            if (check(s, j + 1, x - y)) {
                return true;
            }
        }
        return false;
    }
};

func punishmentNumber(n int) (ans int) {
    var check func(string, int, int) bool
    check = func(s string, i, x int) bool {
        m := len(s)
        if i >= m {
            return x == 0
        }
        y := 0
        for j := i; j < m; j++ {
            y = y*10 + int(s[j]-'0')
            if y > x {
                break
            }
            if check(s, j+1, x-y) {
                return true
            }
        }
        return false
    }
    for i := 1; i <= n; i++ {
        x := i * i
        s := strconv.Itoa(x)
        if check(s, 0, i) {
            ans += x
        }
    }
    return
}

function punishmentNumber(n: number): number {
    const check = (s: string, i: number, x: number): boolean => {
        const m = s.length;
        if (i >= m) {
            return x === 0;
        }
        let y = 0;
        for (let j = i; j < m; ++j) {
            y = y * 10 + Number(s[j]);
            if (y > x) {
                break;
            }
            if (check(s, j + 1, x - y)) {
                return true;
            }
        }
        return false;
    };
    let ans = 0;
    for (let i = 1; i <= n; ++i) {
        const x = i * i;
        const s = x.toString();
        if (check(s, 0, i)) {
            ans += x;
        }
    }
    return ans;
}

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