Call decimal number a monotone if:
Write a program which takes positive number n on input and returns number of decimal numbers of length n that are monotone. Numbers can’t start with 0.
Input : 1 Output : 9 Numbers are 1, 2, 3, ... 9 Input : 2 Output : 45 Numbers are 11, 12, 13, .... 22, 23 ...29, 33, 34, ... 39. Count is 9 + 8 + 7 ... + 1 = 45
Explanation: Let’s start by example of monotone numbers:
All those numbers are monotone as each digit on higher place is than the one before it.
What are the monotone numbers are of length 1 and digits 1 or 2? It is question to ask yourself at the very beginning. We can see that possible numbers are:
That was easy, now lets expand the question to digits 1, 2 and 3:
Now different question, what are the different monotone numbers consisting of only 1 and length 3 are there?
Lets try now draw this very simple observation in 2 dimensional array for number of length 3, where first column is the length of string and first row is possible digits:
Let’s try to fill 3rd row 3rd column(number of monotone numbers consisting from numbers 1 or 2 with length 2). This should be:
If we will look closer we already have subsets of this set i.e:
– Monotone numbers that has length 2 and consist of 1 or 2
– Monotone numbers of length 2 and consisting of number 2
We just need to add previous values to get the longer one.
Final matrix should look like this:
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