Add Digits
The task is to repeatedly add all the digits of a given number until the result has only one digit. This is a simple problem that can be solved by using a loop to add the digits repeatedly.
However, there’s an interesting mathematical property that can solve this problem without the need for a loop. It is based on the concept of “digital root.” The digital root of a number can be found using the formula:
[ \text{{Digital root of }} n = 1 + (n  1) % 9 ]
Here’s the code that implements both the straightforward loop approach and the mathematical approach:
Straightforward Loop Approach:


Mathematical Approach


Key Insights
 The loop approach is straightforward and iteratively sums the digits until a single digit is reached.
 The mathematical approach is more efficient, with a time complexity of (O(1)), as it leverages the property of digital root.
 Both approaches meet the given constraints and provide the correct result.
Identifying Problem Isomorphism
“Add Digits” can be approximately mapped to the problem “Happy Number”.
In “Add Digits”, you are given a nonnegative integer num, and you repeatedly add all its digits until the result has only one digit.
In the “Happy Number” problem, you are given a number n, and you have to determine if it’s “happy”. A happy number is defined as a number which eventually reaches 1 when replaced by the sum of the square of each digit. For example, 19 is a happy number because 1² + 9² = 82, 8² + 2² = 68, 6² + 8² = 100, and finally, 1² + 0² + 0² = 1.
The reasoning behind this mapping is that both problems involve the process of digit manipulation and the iterative process to reduce a number to a desired outcome. The main difference lies in the nature of the operation being used (simple addition vs squaring and then addition) and the desired outcome (single digit vs 1).
“Add Digits” is simpler because it involves a more straightforward operation, addition, and does not need to handle a case where an input number could lead to an infinite loop, unlike in the “Happy Number” problem.


Single Responsibility Function: The function is designed to do one thing only  reduce a number to a single digit by summing its digits repetitively.
Loop Invariant: The outer loop has an invariant condition
n > 9
, which ensures the process continues untiln
becomes a single digit. This semantic construct guarantees the function meets its objective.Digit Extraction and Summation: The inner loop along with the use of modulus and integer division operation provides a common technique to extract individual digits from a number and calculate their sum.
State Update: The function maintains and updates a state  the current sum of digits and the remaining part of the number (
n
). This state update allows the progress of the computation and eventual termination whenn
is a single digit.Result Production: Finally, the function is designed to produce a result  the reduced singledigit number, which is the semantic outcome of all its operations.
These elements give you a highlevel semantic understanding of what the code is designed to achieve and the logical steps it takes to get there.
The loop in the provided code continues until the given number n
is reduced to a singledigit number (i.e., n
<= 9). The progress through each iteration of the loop is made by gradually reducing n
to the sum of its digits.
Some of the key highlevel semantic concepts used in the code:
Reduction: The process of breaking down a larger problem (reducing a multidigit number to a single digit number) into a simpler problem (summing the digits of a number) until the solution is found. This is represented in the outer
while
loop which continues untiln
is reduced to a single digit number.Accumulation: This is a common pattern in many algorithms where a running total or accumulation of values is needed. In this case, it is used in the inner
while
loop to sum the digits ofn
. This sum is then used to updaten
in the outer loop, effectively accumulating the sum of digits untiln
is reduced to a single digit.
These concepts are quite fundamental in algorithm design and problemsolving and are very well observed in this piece of code.
Problem Classification
This problem can be classified under a few categories:
 Mathematical Problem: It involves performing arithmetic operations (addition) on the digits of an integer.
 Digit Manipulation: The problem revolves around operations on individual digits of a number.
 Iterative/Recursive Problem: The problem is solved by repeatedly performing an operation (adding digits) until a condition is met (the result is a single digit).
The problem involves taking an integer, breaking it down into its individual digits, performing arithmetic operations on those digits, and potentially repeating the process. It’s an iterative problem where you continue to repeat a process until you’ve reached a desired result.
Clarification Questions
What are the clarification questions we can ask about this problem?
Identifying Problem Isomorphism
Can you help me with finding the isomorphism for this problem?
Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?
Language Agnostic Coding Drills
Basic Arithmetic Operations: Understanding how basic arithmetic operations work, particularly addition and division, is crucial for this problem.
Modulo Operation: This operation returns the remainder of the division of the number by the divisor. In most languages, this operation is represented by the ‘%’ symbol.
Integer Division/Floor Division: This is the division operation, but the result is rounded down to the nearest integer value. It’s commonly represented by the ‘//’ symbol. This is essential for extracting individual digits from an integer.
Understanding Loops: A fundamental understanding of loops (while and for) is required as they are heavily used in this problem. A while loop is used to iterate until a certain condition is met. Here it is used twice, once to ensure we continue the process until we get a singledigit number and once to iterate through the digits of the number.
Control Flow: Ifelse and while loop constructs are used to direct the control flow of the program.
Understanding Problem Constraints: In this case, knowing when to stop  specifically, when we have a singledigit number.
With these units of learning, we can approach the problem in the following steps:
Start by checking if the given number is already a singledigit number. If it is, we can return it as the answer.
If the number is not a singledigit number, we initiate a loop that continues until we have a singledigit number. This is the outer while loop in the code, where the condition is ’num > 9’.
In the loop, we start by setting the sum to 0. This sum will hold the sum of the digits in our current number.
We then initiate another loop to iterate through the digits in our number. We get the last digit by using the modulo operation (num%10), add it to our sum, and then remove the last digit from our number using integer division (num = num//10).
We repeat the inner loop until all digits have been added to the sum. After that, we assign this sum to num.
We repeat the outer loop, meaning we start over with the new num, which is the sum of the digits in the original num. The process continues until num is a singledigit number.
Finally, we return the singledigit number.
Targeted Drills in Python
 Basic Arithmetic Operations: This is the fundamental concept of any programming language. Let’s consider an operation of addition and division:


In Python, both /
and //
are used for division operations, but they behave differently:
/
(Division operator): This operator returns the division of the operands. It returns a floating point number which is the exact division of the operands, regardless of the type of the operands.
Example:


//
(Floor Division or Integer Division operator): This operator returns the largest possible integer. This integer is either less than or equal to the exact division. In simple terms, it rounds down the result of division to the nearest whole number towards negative infinity.
Example:


So the difference between the two operators is that /
returns the exact quotient in floatingpoint value, while //
returns the nearest whole number to the quotient by rounding towards negative infinity.
 Modulo Operation: Here is a simple function demonstrating the modulo operation:


 Integer Division/Floor Division: Let’s demonstrate this concept:


 Understanding Loops: Here we will use a while loop to print numbers from 1 to 5:


 Control Flow: This code demonstrates the use of control flow in a program:


 Understanding Problem Constraints: Here is a code that keeps asking the user for a single digit number until they enter it correctly:


Now we can combine these skills to solve the final problem:


Coding Constructs
Reduce Number
 Modulo Operation: The modulo operation finds the remainder after division of one number by another. It is often used in programming to reduce a number by its remainder when divided by another number.
1 2 3
# Finding the remainder when 17 is divided by 5 remainder = 17 % 5 print(remainder) # Output: 2
 Integer Division: Integer division is the division in which the fractional part (remainder) is discarded. It is often used to reduce a number by dividing it by another number and keeping only the integer part.
1 2 3
# Dividing 17 by 5 and keeping only the integer part quotient = 17 // 5 print(quotient) # Output: 3
 Modulo Operation: The modulo operation finds the remainder after division of one number by another. It is often used in programming to reduce a number by its remainder when divided by another number.
Extract Digits
 Floor Division: This is another name for integer division. It can be used to extract the digits of a number by iteratively dividing the number by 10.
1 2 3 4 5 6 7
# Extracting the digits of a number using floor division number = 123 while number > 0: digit = number % 10 # get the last digit print(digit) number = number // 10 # remove the last digit # Output: 3, 2, 1
 Floor Division: This is another name for integer division. It can be used to extract the digits of a number by iteratively dividing the number by 10.
Track Sum: This involves keeping track of the sum of multiple numbers, usually in a loop. A variable is often initialized at zero and then incremented inside the loop.
1 2 3 4 5 6 7 8
# Summing the digits of a number number = 123 sum_of_digits = 0 while number > 0: digit = number % 10 # get the last digit sum_of_digits += digit number = number // 10 # remove the last digit print(sum_of_digits) # Output: 6
Accumulate Sum: This is similar to tracking sum, but it often involves keeping track of cumulative sums in a list. The
itertools.accumulate
function is useful for this, but it can also be done with a loop.1 2 3 4 5 6 7 8 9
# Accumulating the sums of the digits of a number number = 123 digits = [] while number > 0: digit = number % 10 # get the last digit digits.append(digit) number = number // 10 # remove the last digit cumulative_sums = list(itertools.accumulate(digits)) print(cumulative_sums) # Output: [3, 5, 6]
These basic constructs can be combined and modified to solve more complex problems, like the Add Digits problem.