Roman Numerals

For a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to be expressed in more than one way there is always a “best” way of writing a particular number.
For example, it would appear that there are at least six ways of writing the number sixteen:
IIIIIIIIIIIIIIII
VIIIIIIIIIII
VVIIIIII
XIIIIII
VVVI
XVI
However, according to the rules only XIIIIII and XVI are valid, and the last example is considered to be the most efficient, as it uses the least number of numerals.
The 11K text file, roman.txt (right click and ‘Save Link/Target As…’), contains one thousand numbers written in valid, but not necessarily minimal, Roman numerals; see About… Roman Numerals for the definitive rules for this problem.
Find the number of characters saved by writing each of these in their minimal form.
Note: You can assume that all the Roman numerals in the file contain no more than four consecutive identical units.

The solution found to this problem depends on the understanding of the rules of Roman numerals. Here are the rules used in the algorithm to solve this problem:

– “I” before “V” or “X” means one less than the next digit (i.e. IV is 4 and IX is 9).
– “X” before “L” or “C” means ten less than the next digit (i.e. XL is 40 and XC is 90).
– “C” before “D” or “M” means a hundred less than the next digit (i.e. CD is 400 and CM is 900).

We need an algorithm to convert Roman numerals to Arabic numerals, and then reconvert the Arabic numerals to minimal Roman numerals. The difference in length between the original – not necessarily minimal Roman numerals – and the newly – minimal – Roman numeral gives the number of character saved.

Here’s a Python implementation that does this.

“`python
def roman_to_int(s):
rom_val = {‘I’: 1, ‘V’: 5, ‘X’: 10, ‘L’: 50, ‘C’: 100, ‘D’: 500, ‘M’: 1000}
int_val = 0
for i in range(len(s)):
if i > 0 and rom_val[s[i]] > rom_val[s[i – 1]]:
int_val += rom_val[s[i]] – 2 * rom_val[s[i – 1]]
else:
int_val += rom_val[s[i]]
return int_val

def int_to_roman(num):
val = [1000, 900, 500, 400, 100, 90, 50, 40, 10, 9, 5, 4, 1]
syb = [“M”, “CM”, “D”, “CD”, “C”, “XC”, “L”, “XL”, “X”, “IX”, “V”, “IV”, “I”]
roman_num = ”
i = 0
while num > 0:
for _ in range(num // val[i]):
roman_num += syb[i]
num -= val[i]
i += 1
return roman_num

def characters_saved(s):
return len(s) – len(int_to_roman(roman_to_int(s)))

file = open(“roman.txt”, “r”)
lines = file.readlines()
total = 0
for line in lines:
total += characters_saved(line.strip())
print(total)
“`

Make sure to put “roman.txt” file in the same location where you run this script. The ROMAN.TXT file should contain the 1000 Roman numerals, one per line. The “strip()” method is used to remove the newline character at the end of each line read from the file.

The script opens the roman.txt file, reads it line by line, converts each Roman numeral to an integer, then converts it back to the minimal Roman numeral. By comparing the length of the original numerals to the minimal numerals, the script calculates the total number of characters saved.

Note that not all Roman numerals representations are valid according to the strict standards. The provided .txt file would need to only include Valid Roman numerals. When assuming that all the Roman numerals in the file contain no more than four consecutive identical units, this algorithm should work. If an invalid numeral is encountered, the conversion function may not behave as expected.

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