67 lines
2.4 KiB
Markdown
67 lines
2.4 KiB
Markdown
---
|
|
id: 5900f50b1000cf542c51001d
|
|
title: 'Problem 414: Kaprekar constant'
|
|
challengeType: 5
|
|
forumTopicId: 302083
|
|
dashedName: 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`.
|
|
|
|
```js
|
|
assert.strictEqual(kaprekarConstant(), 552506775824935500);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function kaprekarConstant() {
|
|
|
|
return true;
|
|
}
|
|
|
|
kaprekarConstant();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|