Question

2698. Find the Punishment Number of an Integer

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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 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.
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 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.

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;
}