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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 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.

• Java
• C++
• Python
• Go
• TypeScript

• 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;
      }
  };
  

• 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
  
  

• 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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