Find the Punishment Number of an Integer


10 Prerequisite LeetCode Problems
For this, the following are a good preparation:
“202. Happy Number”: This problem helps understand how to deal with the squares of individual digits of an integer which could be helpful for understanding the punishment number.
“9. Palindrome Number”: This problem requires converting integers to strings and vice versa which is a useful skill for problem 2698.
“168. Excel Sheet Column Title”: This problem helps with understanding how to handle conversions between different bases, which could be beneficial when dealing with decimal representations in problem 2698.
“412. Fizz Buzz”: This problem requires you to iterate over a range of numbers and apply a set of conditions, a similar approach can be used in problem 2698.
“728. Self Dividing Numbers”: This problem helps with understanding how to divide numbers into their individual digits which is an essential step in problem 2698.
“13. Roman to Integer”: This problem requires converting between two different representations of a number, a concept which could help solve problem 2698.
“415. Add Strings”: This problem helps to understand how to deal with string numbers, which is a crucial part of problem 2698.
“171. Excel Sheet Column Number”: Understanding the conversion of different number representations in this problem could be beneficial when solving problem 2698.
“204. Count Primes”: This problem requires iterating over a number range and applying a condition to each number, a similar approach is used in problem 2698.
“434. Number of Segments in a String”: This problem deals with partitioning a string into substrings, which can be directly applied to the “partitioned into contiguous substrings” part of problem 2698.
These cover concepts like conversion between different number representations, dealing with string numbers, partitioning a string into substrings, iterating over a range of numbers, and handling the squares of individual digits of an integer, all of which are useful for tackling problem 2698.


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
Problem Classification
This problem falls under the domain of number theory and string processing. It involves dealing with integer numbers, their square values, and analyzing the decimal representations of those square values.
‘What’ Components:
 Given: A positive integer
n
.  Required: Return the ‘punishment number’ of
n
. The punishment number is defined as the sum of squares of all integersi
such that1 <= i <= n
and the decimal representation ofi * i
can be partitioned into contiguous substrings such that the sum of the integer values of these substrings equalsi
.
Problem Classification: This problem can be classified as a Combinatorial Search problem, where you need to explore all possible partitions of the decimal representation of a square of a number to see if any of these combinations satisfy a certain condition. The problem requires understanding of integer manipulation (calculating squares and sums), number to string conversion, string partitioning, and combinatorial search strategy (to search through all possible partitions).
The main challenge of this problem comes from the need to examine all possible partitions of the decimal representation of a number’s square. This requires an efficient way to explore the combinations, especially when dealing with large numbers or ranges.
Language Agnostic Coding Drills
1. Dissect the code and identify each distinct concept it contains
This code consists of several distinct coding concepts:
Basic Arithmetic Operations: It involves basic arithmetic operations like multiplication, addition and subtraction.
Loops: The code uses for loops to iterate over a range of values and to create substrings of a string.
Conditional Statements: The if statements are used for checking certain conditions.
Type Conversions: The conversion of integer to string and back to integer is used in this code.
Recursion: Recursion is used to explore all possible partitions of the decimal representation of a number’s square.
Accumulator Variable: An accumulator variable (
ans
) is used to keep a running total of the sum of the squares of the numbers that meet the specified conditions.
2. List them out in order of increasing difficulty
 Basic Arithmetic Operations  Basic operations in any programming language, simple to understand.
 Type Conversions  This involves understanding that numbers can be represented as strings, and vice versa.
 Conditional Statements  These involve understanding the logic and flow control in the program.
 Loops  These are used to repeat certain operations, understanding them requires some level of control flow comprehension.
 Accumulator Variable  The concept of accumulating a result in a variable over iterations is slightly more complex as it involves understanding state that persists across iterations.
 Recursion  This is often considered one of the more complex topics for beginners as it requires thinking about problems in a way that allows for selfreferential solutions.
3. Problemsolving approach
The problemsolving approach of this code involves iterating over all numbers up to n
, squaring each number and converting the square to a string. Then, for each square, the code uses recursion to explore all possible contiguous substrings of the string representation of the square, converting each substring back to an integer and subtracting it from the original number. If it’s possible to find a set of substrings that sums to the original number, the square of the number is added to the total sum.
The concept drills associated with this problem can be seen as follows:
 Understanding basic arithmetic and loops allows for the iteration over the range of numbers and the calculation of each square.
 Understanding type conversion allows for the conversion of the square into a string and the substrings back into integers.
 Understanding conditionals allows for the decision of whether or not to add a square to the total sum.
 Understanding recursion is crucial for exploring all possible substrings.
 Understanding the concept of an accumulator variable allows for the accumulation of the total sum of the squares.
Targeted Drills in Python
Basic Arithmetic Operations: These are fundamental operations, such as addition, subtraction, multiplication, and division.
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a = 5 b = 3 print(a + b) # Addition print(a  b) # Subtraction print(a * b) # Multiplication print(a / b) # Division
Type Conversions: In Python, you can convert one data type to another using builtin functions.
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num = 123 str_num = str(num) # Convert integer to string print(str_num, type(str_num)) str_num = '456' num = int(str_num) # Convert string to integer print(num, type(num))
Conditional Statements: They are used to perform different computations or actions depending on whether a condition evaluates to true or false.
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x = 5 if x > 0: print("Positive") elif x < 0: print("Negative") else: print("Zero")
Loops: A loop statement allows us to execute a statement or group of statements multiple times.
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for i in range(5): # Loop over a range of numbers print(i)
Accumulator Variable: An accumulator is a variable that the program uses to calculate a sum or product iteratively in a loop.
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sum = 0 for i in range(1, 11): sum += i print(sum) # Prints the sum of numbers from 1 to 10
Recursion: Recursion is the process of defining something in terms of itself. A physical world example would be to place two parallel mirrors facing each other.
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def factorial(n): if n == 1: return 1 else: return n * factorial(n1) # Recursive call print(factorial(5))
Problemspecific Drills:
String slicing and loops: The problem specifically requires checking all contiguous substrings of a given string. This concept will be vital to create all possible substrings from the string representation of the square.
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s = 'hello' for i in range(len(s)): for j in range(i+1, len(s)+1): print(s[i:j]) # Prints all substrings of s
Recursive problemsolving: The problem involves checking whether the sum of some contiguous substrings equals the original number. This requires a recursive approach to explore all possible combinations of substrings.
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# A simplified recursive problem: Can a target number be obtained by the sum of any numbers in a list? def can_sum(target, numbers): if target == 0: return True if target < 0: return False for num in numbers: if can_sum(targetnum, numbers): return True return False print(can_sum(7, [2, 3])) # Prints True print(can_sum(7, [4, 2])) # Prints False
Integrating the Drills:
Starting with a basic understanding of arithmetic operations and loops, we can iterate over a range of numbers and calculate each number’s square. We then convert this square to a string, and for each
square, we use recursion to explore all possible contiguous substrings of the string representation of the square, converting each substring back to an integer and subtracting it from the original number. If it’s possible to find a set of substrings that sums to the original number, the square of the number is added to the total sum using an accumulator variable.
All these smaller drills together contribute to building up to the final solution. This approach is not only fundamental to this particular problem but also can be applied to a vast number of algorithmic problems involving recursion and dynamic programming.