# Dr. Nam Bo Phan approached QHR with a problem to solve. She has recently learnt of a process that takes a number and returns the sum of the squares of each of its digits. She asked her assistant Ken Nothad to build a list of results for the numbers 1 to 500. X digitSquareSum(x) 1 1 2 4 … … Ken got started on the list, but got tired of punching the digits in his calculator. Dr. Phan would like us to produce this list. 1,1 2,4 3,9 … 500,25 Dr. Phan has found that by performing multiple times, two numbers get back to their original number: She asked Ken to identify which of the original numbers, between 1 and 500, will arrive at 1 after multiple iterations of performing . Ken is threatening to quit. Ken has called QHR to say that Dr. Phan has been rocking in her chair and going on about world domination. Her latest request has Ken fearing for his sanity and his life.

Problem Statement

Dr. Nam Bo Phan has approached QHR with a problem related to a mathematical process she recently discovered. This process takes a number and returns the sum of the squares of each of its digits. To illustrate her problem, she asked her assistant, Ken Nothad, to build a list of results for the numbers 1 to 500 using this process. However, Ken grew tired of manually calculating the digit square sums and has requested our help in producing this list.

The digit square sum for each number from 1 to 500 should be calculated and recorded in the following format:

1, 1

2, 4

3, 9

…

500, 25

Dr. Phan has also discovered an interesting aspect of this process. Some numbers, after performing the digit square sum multiple times, return to their original value. She wants Ken to identify which of the original numbers, between 1 and 500, will eventually arrive at a value of 1 after multiple iterations of the digit square sum.

Ken is feeling overwhelmed and has expressed to QHR that Dr. Phan’s behavior is increasingly concerning. He described her sitting in her chair, rocking back and forth, and talking about world domination. Her latest request has left Ken fearing for his sanity and his life.

Analysis

The problem presented by Dr. Phan involves the calculation of the digit square sum for numbers ranging from 1 to 500. This process involves taking each digit of a given number, squaring it, and then summing up the squared digits. For example, the digit square sum of 159 would be calculated as follows:

1^2 + 5^2 + 9^2 = 1 + 25 + 81 = 107

It is important to note that this process must be performed repeatedly on the resulting sum until a single-digit number is obtained. The final single-digit number will be recorded as the digit square sum for the original number.

To solve this problem, we will need to develop an algorithm that can calculate the digit square sum for each number from 1 to 500, while also keeping track of the iterations required to reach a single-digit number. Additionally, the algorithm should identify which original numbers will eventually arrive at a sum of 1 after repeated iterations.

Once the algorithm is implemented, we can provide a solution to Dr. Phan that includes the required list of results and the identification of the relevant numbers. This will help to address Dr. Phan’s concerns and alleviate Ken’s growing apprehension.

In the next section, we will discuss the algorithm that can be used to calculate the digit square sum and identify the numbers that eventually arrive at a sum of 1. We will also provide a step-by-step process for implementing the algorithm.

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