This is my solution to the Kth Special Number Problem, which can be stated as follows:
A Special Number is defined as a positive integer whose digits add up to 10.
Write a function that takes an integer k and computes the kth special number.
The sequence is zero-indexed, so an input of k=0 should return 19, k=1 should return 28, etc.
Python 3.6.8 or greater.
You should regularly run $ flake8 from the project root. Certain warnings and errors are ignored according to the [flake8] section of setup.cfg.
The test compares the first 50,000 special numbers returned by the function src.kth_special_number.first_k_special_numbers with 50,000 numbers computed by a simple brute force method. Computing the numbers by brute force can be time-consuming, so they are provided in a json in the test module.
To run the test, simply call $ pytest from the project root.
The central problem is, given an integer whose digits we KNOW add up to 10, how can we easily find the next greatest integer?
Let's look at a subsequence of special numbers:
- 2331100
- 2332000
- 2340001
- 2340010
- 2340100
- 2341000
- 2350000
- 2400004
This sequence actually contains everything we need to know to develop an algorithm. Let's break it up even further:
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (i) | 2 | 3 | 3 | 1 | 1 | 0 | 0 |
| (ii) | 2 | 3 | 3 | 2 | 0 | 0 | 0 |
Here we've broken up each digit into a named column for easier reference. Row (i) contains the digits of the first number of the sequence, and row (ii) contains the second number. The transition from the first the second, in this case, is accomplished by changing column E from 1 to0, and column D from 1 to 2.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (ii) | 2 | 3 | 3 | 2 | 0 | 0 | 0 |
| (iii) | 2 | 3 | 4 | 0 | 0 | 0 | 1 |
The transition from (ii) to the (iii) is more complicated. C gains 1, D loses 2, and G gains 1.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (iii) | 2 | 3 | 4 | 0 | 0 | 0 | 1 |
| (iv) | 2 | 3 | 4 | 0 | 0 | 1 | 0 |
Back to the simple transition: F is incremented while G is decremented.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (iv) | 2 | 3 | 4 | 0 | 0 | 1 | 0 |
| (v) | 2 | 3 | 4 | 0 | 1 | 0 | 0 |
Simple again: Now E is incremented and F is decremented.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (v) | 2 | 3 | 4 | 0 | 1 | 0 | 0 |
| (vi) | 2 | 3 | 4 | 1 | 0 | 0 | 0 |
Another simple transition (you're probably noticing a pattern now). D is incremented while E is decremented.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (vi) | 2 | 3 | 4 | 1 | 0 | 0 | 0 |
| (vii) | 2 | 3 | 5 | 0 | 0 | 0 | 0 |
Simple again: C is incremented and D is decremented.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| (vii) | 2 | 3 | 5 | 0 | 0 | 0 | 0 |
| (viii) | 2 | 4 | 0 | 0 | 0 | 0 | 4 |
This one is complicated again: B is incremented, C is reset to 0, and G is increased by 4.
Every time we want to find the next special number, we need to start by finding the right-most (least significant) non-zero digit. Once we've found it, we do three things:
- Increment the digit immediately to the left of it (the next most significant digit)
- Set the value of the one's-digit to X - 1 (where X is the value of the digit)
- Set the value of the digit to 0