freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-414-kaprekar-constant.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
5900f50b1000cf542c51001d	Problem 414: Kaprekar constant	5	302083	problem-414-kaprekar-constant

--description--

6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get 7641 - 1467 = 6174.

Even more remarkable is that if we start from any 4 digit number and repeat this process of sorting and subtracting, we'll eventually end up with 6174 or immediately with 0 if all digits are equal.

This also works with numbers that have less than 4 digits if we pad the number with leading zeroes until we have 4 digits.

E.g. let's start with the number 0837:

$$\begin{align} & 8730 - 0378 = 8352 \\ & 8532 - 2358 = 6174 \end{align}$$

6174 is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the Kaprekar routine.

We can consider the Kaprekar routine for other bases and number of digits. Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers. However, it can be shown that for 5 digits and a base b = 6t + 3 ≠ 9, a Kaprekar constant exists.

E.g. base 15: {(10, 4, 14, 9, 5)}\_{15} base 21: (14, 6, 20, 13, 7)_{21}

Define C_b to be the Kaprekar constant in base b for 5 digits. Define the function sb(i) to be:

• 0 if i = C_b or if i written in base b consists of 5 identical digits
• the number of iterations it takes the Kaprekar routine in base b to arrive at C_b, otherwise

Note that we can define sb(i) for all integers i < b^5. If i written in base b takes less than 5 digits, the number is padded with leading zero digits until we have 5 digits before applying the Kaprekar routine.

Define S(b) as the sum of sb(i) for 0 < i < b^5. E.g. S(15) = 5\\,274\\,369 S(111) = 400\\,668\\,930\\,299

Find the sum of S(6k + 3) for 2 ≤ k ≤ 300. Give the last 18 digits as your answer.

--hints--

kaprekarConstant() should return 552506775824935500.

assert.strictEqual(kaprekarConstant(), 552506775824935500);

--seed--

--seed-contents--

function kaprekarConstant() {

  return true;
}

kaprekarConstant();

--solutions--

// solution required