#### Table of Contents

Damn! Another 2 hour marathon to solve this problem! ðŸ˜…ðŸ’¦ The challenge today was to convert integers into a string of Roman Numerals. It was a really interesting problem that pushed my brain and my Elixir skills to the limit. Honestly, I found this one pretty hard and I’m not sure if I missed an easy way to do it! This one is rated as `easy`

on Exercism… I’m getting scared for any that are marked as `medium`

. That being said, I have found that within `easy`

there are problems I can do in 10 minutes and also problems like this, that take me 2 hours+.

## Live solution video

This is my daily live stream video solution to this problem.

- Video available here: https://youtu.be/97UtVhJvF00
- Also streaming on Twitch: https://www.twitch.tv/percygrunwald

## Explanation of the solution

### Function to split an integer into the integers that make it up

The idea of this function is to do something like this:

`split_integer(123) => [1, 2, 3]`

`split_integer(456) => [4, 5, 6]`

My solution to this problem was to convert the integer into a string, then split up that string into a list of individual characters before parsing each character back into an integer. Something like this:

```
123 => "123" => ["1", "2", "3"] => [1, 2, 3]
```

Here’s my solution:

```
defp split_integer(int) do
int
|> Kernel.to_string()
|> String.split("", trim: true)
|> Enum.map(fn int_string ->
Integer.parse(int_string)
|> case do
{int, _} ->
int
_ ->
0
end
end)
end
```

### Function to multiply each integer in the list by it’s correct order of magnitude

This combines with the previous function to do something like this:

`multiply_by_ten_power([1, 2, 3]) => [100, 20, 3]`

`multiply_by_ten_power([4, 5, 6, 7]) => [4000, 500, 60, 7]`

We’ll pass each one of those into the next function in order to get the correct Roman Numeral representation. Here’s my solution:

```
defp multiply_by_ten_power(int_list) do
int_list
|> Enum.with_index(1)
|> Enum.map(fn {int, index} ->
ten_factor = :math.pow(10, length(int_list) - index) |> :erlang.floor()
int * ten_factor
end)
end
```

### Recursive function to get the integers from a combination of a pool of integers adhering to the rules of Roman Numerals

This is a complex recursive function that is designed to get as input an integer of a particular order of magnitude – i.e. `100`

, `20`

or `3`

instead of `123`

– and return that number expressed as a list of integers following the rules of Roman Numerals:

```
100 => [100]
20 => [10, 10]
3 => [1, 1, 1]
```

The function tries to make the number up out of the pool of integers representing the roman numerals: `[1, 5, 10, 50, 100, 500, 1000]`

. It finds the closest number in the number pool and the difference, then puts the closest number into the accumulator `acc`

and calls itself again if it hasn’t reached the exact value. For example:

- Given
`100`

, the function will find`100`

in the pool of numbers, the difference is 0 so it’s done, returning`[100]`

. - Given
`20`

, the function first finds`10`

, adds it to`acc`

and calls itself again because the difference between`20`

and`10`

is`10`

. The next pass of the function is trying to find`10`

and finds it in the pool, returning`[10, 10]`

. - Given
`3`

, the function will find`1`

in the pool then call itself twice again finding`1`

each time, returning`[1, 1, 1]`

.

There are some cases when the first number passed to the function will be negative: e.g. `90`

. The function finds `100`

as the closest number from the pool, then calls itself again with `-10`

as the number to find. Because we’re dealing only with positive numbers, we find the closest positive number, then put it to the left of `100`

in the resulting list: `[10, 100]`

.

The bulk of the heavy lifting is done by the

```
defp get_combined_numbers(int, acc \\ {[], 0}) do
is_negative? = int < 0
abs_int = :erlang.abs(int)
number_pool = Map.keys(@int_to_roman_numeral_map)
{acc_list, last_index_inserted} = acc
{difference, closest_int} = get_difference_and_closest_int(abs_int, number_pool)
index_to_insert =
if is_negative? do
last_index_inserted - 1
else
last_index_inserted + 1
end
acc_list = List.insert_at(acc_list, index_to_insert, closest_int)
if difference == 0 do
acc_list
else
get_combined_numbers(difference, {acc_list, index_to_insert})
end
end
```

### (very ugly) Function to get the option from a pool with the lowest difference, but adhering to the rules of Roman Numerals

The purpose of this function is to find the closest item in a number pool to the one passed in, but with the crucial difference that it follows the rules of Roman Numerals. For example: if `80`

is passed in, the function should return `{30, 50}`

instead of `{-20, 100}`

, even though `-20`

is the smaller difference. This is because Roman Numerals expects `80`

to be represented as `LXXX`

, not `XXC`

. The general rule that this function follows is:

If the difference is

negative, the differencemustbe equal to asingleitem in the number pool, otherwise choose the item one before the lowest difference.

That is, choose `{30, 50}`

instead of `{-20, 100}`

because `-20`

**cannot** be expressed as a **single** item from the number pool (`10 + 10`

). But in the case of `90`

, we should choose `{-10, 100}`

instead of `{40, 50}`

because `-10`

can be made with a single item from the number pool (`10`

).

```
defp get_difference_and_closest_int(abs_int, number_pool) do
number_pool_with_differences =
number_pool
|> Enum.map(fn int_to_compare ->
{abs_int - int_to_compare, int_to_compare}
end)
{{difference, closest_int}, index} =
number_pool_with_differences
|> Enum.with_index()
|> Enum.min_by(fn {{difference, _int_to_compare}, _index} ->
:erlang.abs(difference)
end)
cond do
difference == 0 ->
{difference, closest_int}
difference > 0 ->
{difference, closest_int}
difference < 0 ->
# 85 => LXXXV
# 85 => ... ___{35, 50}___, {-15, 100} ...
#
# 90 => XVC
# 90 => ... {40, 50}, ___{-10, 100}___ ...
abs_difference = :erlang.abs(difference)
exact_match = Enum.find(number_pool, fn number -> number == abs_difference end)
if not is_nil(exact_match) do
{difference, closest_int}
else
Enum.at(number_pool_with_differences, index - 1)
end
end
end
```

### The main function tying it all together

The function below implements the following data flow:

- Split the integer into its parts
`2085 => [2, 0, 8, 5]`

- Multiply by the correct power of
`10`

depending on position`[2, 0, 8, 5] => [2000, 0, 80, 5]`

- Remove any
`0`

s`[2000, 0, 80, 5] => [2000, 80, 5]`

- Represent each number as the correct numbers from the number pool
`[2000, 80, 5] => [[1000, 1000], [50, 10, 10, 10], [5]]`

- Flatten the list
`[[1000, 1000], [50, 10, 10, 10], [5]] => [1000, 1000, 50, 10, 10, 10, 5]`

- Map each integer to its corresponding Roman Numeral
`[1000, 1000, 50, 10, 10, 10, 5] => ["M", "M", "L", "X", "X", "X", "V"]`

- Join the list of strings into a single string
`["M", "M", "L", "X", "X", "X", "V"] => "MMLXXXV"`

More succinctly:

```
2085 => [2, 0, 8, 5] => [2000, 0, 80, 5] => [2000, 80, 5]
=> [[1000, 1000], [50, 10, 10, 10], [5]]
=> [1000, 1000, 50, 10, 10, 10, 5]
=> ["M", "M", "L", "X", "X", "X", "V"]
=> "MMLXXXV"
```

And the solution in code:

```
@int_to_roman_numeral_map %{
1 => "I",
5 => "V",
10 => "X",
50 => "L",
100 => "C",
500 => "D",
1000 => "M"
}
@doc """
Convert the number to a roman number.
"""
@spec numerals(pos_integer) :: String.t()
def numerals(number) do
number
|> split_integer()
|> multiply_by_ten_power()
|> Enum.reject(fn int -> int == 0 end)
|> Enum.map(&get_combined_numbers/1)
|> List.flatten()
|> Enum.map(&Map.get(@int_to_roman_numeral_map, &1, ""))
|> Enum.join()
end
```

## Full solution in text form

Here’s the full solution in text form, in case you want to look over the final product:

```
defmodule Roman do
@int_to_roman_numeral_map %{
1 => "I",
5 => "V",
10 => "X",
50 => "L",
100 => "C",
500 => "D",
1000 => "M"
}
@doc """
Convert the number to a roman number.
"""
@spec numerals(pos_integer) :: String.t()
def numerals(number) do
number
|> split_integer()
|> multiply_by_ten_power()
|> Enum.reject(fn int -> int == 0 end)
|> Enum.map(&get_combined_numbers/1)
|> List.flatten()
|> Enum.map(&Map.get(@int_to_roman_numeral_map, &1, ""))
|> Enum.join()
end
defp split_integer(int) do
int
|> Kernel.to_string()
|> String.split("", trim: true)
|> Enum.map(fn int_string ->
Integer.parse(int_string)
|> case do
{int, _} ->
int
_ ->
0
end
end)
end
defp multiply_by_ten_power(int_list) do
int_list
|> Enum.with_index(1)
|> Enum.map(fn {int, index} ->
ten_factor = :math.pow(10, length(int_list) - index) |> :erlang.floor()
int * ten_factor
end)
end
defp get_combined_numbers(int, acc \\ {[], 0}) do
is_negative? = int < 0
abs_int = :erlang.abs(int)
number_pool = Map.keys(@int_to_roman_numeral_map)
{acc_list, last_index_inserted} = acc
{difference, closest_int} = get_difference_and_closest_int(abs_int, number_pool)
index_to_insert =
if is_negative? do
last_index_inserted - 1
else
last_index_inserted + 1
end
acc_list = List.insert_at(acc_list, index_to_insert, closest_int)
if difference == 0 do
acc_list
else
get_combined_numbers(difference, {acc_list, index_to_insert})
end
end
defp get_difference_and_closest_int(abs_int, number_pool) do
number_pool_with_differences =
number_pool
|> Enum.map(fn int_to_compare ->
{abs_int - int_to_compare, int_to_compare}
end)
{{difference, closest_int}, index} =
number_pool_with_differences
|> Enum.with_index()
|> Enum.min_by(fn {{difference, _int_to_compare}, _index} ->
:erlang.abs(difference)
end)
cond do
difference == 0 ->
{difference, closest_int}
difference > 0 ->
{difference, closest_int}
difference < 0 ->
# 85 => LXXXV
# 85 => ... ___{35, 50}___, {-15, 100} ...
#
# 90 => XVC
# 90 => ... {40, 50}, ___{-10, 100}___ ...
abs_difference = :erlang.abs(difference)
exact_match = Enum.find(number_pool, fn number -> number == abs_difference end)
if not is_nil(exact_match) do
{difference, closest_int}
else
Enum.at(number_pool_with_differences, index - 1)
end
end
end
end
```